Orthogonal Frequency Division MultiplexingOrthogonal Frequency Division Multiplexing(OFDM)(OFDM)(OFDM)(OFDM)

Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

國立中山大學國立中山大學通訊所通訊所國立中山大學國立中山大學通訊所通訊所

李志鵬李志鵬

July 2010July 2010

Table of ContentsTable of Contentsloz Introduction to OFDM

loz Synchronization

loz Channel Estimation

Introduction to OFDM SystemsIntroduction to OFDM Systems

Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

OFDM OverviewOFDM Overview

loz OFDMloz Orthogonal Frequency Division Multiplexing

loz Frequency Division Multiplexing (FDM) or multi-tone systems have been employed in military applicationssystems have been employed in military applications since the 196Os

loz OFDM employs multiple carriers overlapping in the f d ifrequency domain

4

OFDM OverviewOFDM Overview

Single carrier (SC) vs multi-carrier (MC)

loz Single carrier data are Multi-carrier data are shared transmitted over only one carrier

among several carriers and simultaneously transmittedFl t f di b iloz Selective fading Flat fading per subcarrier

5

OFDM OverviewOFDM Overview

loz OFDM modulation

loz Featuresloz No intercarrier guard bandsloz Overlapping of bands

S t l ffi iloz Spectral efficiencyloz Easy implementation by IFFTsloz Very sensitive to synchronizationloz Very sensitive to synchronization

6

Applications of OFDM TechnologyApplications of OFDM Technologypp gypp gy

loz Broadband Wired Access AsymmetricDigital Subscriber Loop (ADSL) Digital Multi-tone(DMT)

loz Wireless LANs (IEEE 80211ag IEEE 80211n HIPERLAN-2)

loz Digital Broadcasting (DAB DVB-T DVB-H)loz WiMAX (IEEE 80216 Series) 3GPP Long Term ( ) g

Evolution (3GPP LTE) 4Gloz Wireless Personal Area Network (WPAN) IEEE ( )

80215aMBOAloz Power Line

7

OFDM System ModelOFDM System Modelyy

loz Multi-carrier Block Transmission

frequency

time

8T8

OFDM System ModelOFDM System Modelyy

loz OFDM A block modulation scheme that transmitsa block N source symbols in parallel by using subcarrierssubcarriersloz Sub-carriers are orthogonal in time but overlapped in

frequencyeque cyloz Frequency spacing 1

FFT

fT

Δ =

cos(2 )cos(2 ( ) ) 0FFTTf t f f t dtπ π + Δint 1 10

cos(2 )cos(2 ( ) ) 0f t f f t dtπ π + Δ =int

9

OFDM System ModelOFDM System Model

tNΔ

yy

( )tf02cos π

( )0a

( )0atT

tNΔ1+

( )tf2i

( )

( )0b

T

DataEncoder

tf s Δ=

1 ( ) ( ) njbna +

( )nd SP MUX ( )tDChannelInput

( )tf02sin π

( )tf N 12cos minusπbullbullbull

( )1minusNa

( )1minusNb( )0a1+

( )tf N 12sin minusπ

( )1Nb

tT

( )1a ( )1minusNa1minus

tΔ

Figure 1tNΔ

( )1a ( )1minusNa Figure 1

10

OFDM System ModelOFDM System Model

loz An OFDM system transmitter shown in Figure 1

yy

y gloz The transmitted waveform D(t) can be expressed as

)1( )2sin()()2cos()()(1

0summinus

=

+=N

nnn tfnbtfnatD ππ

where tNffnffn Δ=ΔΔ+=

1 and 0

loz Using a two-dimensional digital modulation format the data

tNΔ

g g symbols d(n) can be represented as a(n) + jb(n)loz a(n) in-phase componentloz b(n) quadrature component

11

OFDM System ModelOFDM System Model

loz The serial data elements spaced by are grouped and tΔ

yy

p y g pused to modulate N carriers Thus they are frequency division multiplexedp

loz The signaling interval is then increased to which tNΔmakes the system less susceptible to channel delay spread impairments

Small-scale fading(Based on multipath time delay spread)

Flat Fading1 BW of signal lt BW of channel

Frequency Selective Fading1 BW of signal gt BW of channel

2 Delay spread lt Symbol preiod 2 Delay spread gt Symbol preiod

12

OrthogonalityOrthogonalityg yg y

loz Consider a set of transmitted carriers as follows

(2) 1 1 0for )( 2 0

minus==⎟⎠⎞

⎜⎝⎛

Δ+

Nnett

tNnfj

n

πψ n

⎨⎧ =minus

intqpab

dtttb for )(

)()( ψψ⎩⎨ Δ=minusne

=int tNabqpdttt

a qp )( and for 0 )()( ψψ

13

OrthogonalityOrthogonality

b tqpjb

intint)(2π

g yg y

dtedtttajbj

b

atN

qpjb

a qp = Δminus

intint )()(

)(2)(2

)(2ψψπ

tNqpjee tN

qpjtN

qpj

Δminusminus

=Δ

minusΔ

minus

)(2

)(2)(2

π

ππ

ee

tNqpjba

tNqpj

tNbqpj

⎟⎟⎞

⎜⎜⎛minus

Δ

minusΔ

minusΔ

minus1

)(2)(1)(2)(2

π

ππ

tNqpj Δminus

⎟⎠

⎜⎝=

)(2πtNabqp Δ=minusne= )( and for 0

14

OrthogonalityOrthogonalityg yg y

tNΔ

15

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFTby using IDFTby using IDFT

loz According to the structure of Tx it must use Ng oscillators That increases the hardware complexity

loz The equivalent method is using IDFT (IFFT)

16

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

loz In general each carrier can be expressed as

by using IDFTby using IDFT

g p

( ) (6) )()( )(2 ttfjcc

ccetAtS φπ +=

loz We assume that there are N carriers in the OFDM signal Th th t t l l i l S (t) b t d bThen the total complex signal Ss(t) can be represented by

( ) (7))(1)(1

)(2etAtSN

ttfj nn φπ= summinus

+( )

where

(7) )()(

0

0

fnff

etAN

tSn

ns

Δ+=

= sum=

liihffrequency carrier phase amplitude are )( )( and

where 0

fttAfnff

nnn

n

φΔ+

lyrespective carrierth -of n17

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

loz Then we sample the signal at a sampling frequency tΔ1

by using IDFTby using IDFT

p g p g q y and An(t) and φn(t) becomes

(8))(t = φφ(9) )((8) )(

nn

nn

AtAt

== φφ

( ) (10) 1)(1

0

)(2 0summinus

=

+ΔΔ+=ΔN

n

tkfnfjns

neAN

tkS φπ

loz Then the sampled signal can be expressed as0=nN

( )( ) (11) 1)(1

0

22 0summinus

=

ΔΔ+Δ sdot=ΔN

n

tfnkjtkfjns eeA

NtkS n πφπ

0n

18

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

loz The inverse discrete Fourier transform (IDFT) is defined

by using IDFTby using IDFT

( )as the following

1 21

NnkjN

πsumminus

(12) )(1)( 2

0

Nnkj

nefnF

Ntkf πsum

=

Δ=Δ

loz Comparing eq(11) and eq(12) the condition must be satisfied in order to make eq(11) an inverse Fouriersatisfied in order to make eq(11) an inverse Fourier transform relationship

1 (13) 1tN

fΔ

=Δ

19

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

loz If eq(13) is satisfied

by using IDFTby using IDFT

q ( ) loz is the frequency domain signal loz is the time domain signal

( )ntkfjneA φπ +Δ02

)( tkSs Δ gloz is the sub-channel spacingloz is the symbol duration in each sub-channel

)(s

fΔtNΔloz s e sy bo du o e c sub c e

loz This outcome is the same as the result obtained in the

N

loz This outcome is the same as the result obtained in the system of Figure 1 Therefore IDFT can be used to generate an OFDM transmission signalgenerate an OFDM transmission signal

20

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

( )tf02cos π

( )0a

by using IDFTby using IDFT

( )nd ( )tDI( )tf02sin π

( )0a

( )0b

bull

tf s Δ=

1 ( ) ( ) njbna +

( )nd

tNΔffnffn Δ

=Δ+=1 0

( )tDChannelInput

( )tf N 12cos minusπ

( )1minusNa

bullbull

tNΔ

( )tf N 12sin minusπ

( )1minusNb

( )0d

( )nd ( )tDChannelInput

bull bull

( )( )1d

( )2d

tf s Δ=

1 ( ) ( ) njbna +

fNΔffnffn Δ

=Δ+=1 0

ChannelInput

bullbullbullbull

bullbullbullbull

( )1minusNd

21

Frequency Error Results in ICIFrequency Error Results in ICIq yq y

22

Synchronization Error Results in ICISynchronization Error Results in ICIyy

Not Orthogonal Any More

23

Cyclic PrefixCyclic Prefixyy

loz In multipath channel delayed replicas of previous OFDM signal lead to ISI between successive OFDM signals

loz Solution Insert a guard interval between successive OFDMloz Solution Insert a guard interval between successive OFDM signals

24

Cyclic PrefixCyclic Prefix

loz Guard interval leads to intercarrier interference (ICI) in OFDM d d l i

yy

demodulation

loz In DFT interval difference between two subcarriers does not maintain integer number of cycles loss of orthogonality

loz Delayed version of subcarrier 2 causes ICI in the process of d d l ti b i 1demodulating subcarrier 1

25

Cyclic PrefixCyclic Prefixloz Cyclic prefix (CP) A copy of the last part of OFDM signal is

attached to the front of itself

yy

attached to the front of itself

[ ]0d

[ ]1d

[ ]0D[ ]1D

[ ]nd[ ]kD~

symbols dataInput

bullbullbullbull

[ ]1d

[ ]2d

[ ][ ]2D

bullbullbullbullbull

y

bullbullbullbullbull

[ ]1minusNd [ ]1minusND

[ ]gNND minus

bullbullbull

26

Cyclic PrefixCyclic Prefix

loz All delayed replicas of subcarriers always have an

yy

y p yinteger number of cycles within DFT interval no ICI

27

Cyclic PrefixCyclic Prefix

loz Linear convolution vs circular convolution

yy

N

28

Cyclic PrefixCyclic Prefix

loz Channel effect with cyclic prefix

yy

Signal after removed CP

29

Cyclic PrefixCyclic Prefix

loz Time-Domain Explanation

yy

p

30

Cyclic PrefixCyclic Prefixyy

31

Cyclic PrefixCyclic Prefix

loz Spectrum of channel response with length (smaller [ ]nh hL

yy

than )h

gN[ ] nhFFTHk =

loz Received complete OFDM signal[ ] [ ] [ ] 20 ~~ minus++leleotimes= hg LNNnnhnDnr

loz Received useful part [ ]nr[ ] [ ] [ ]nhnDnr Notimes=

where is N-point circular convolution (due to CP)loz Received symbol at k-th subcarrier

[ ] [ ] [ ]N

Notimes

[ ] [ ] [ ] kkNk HXnhnDFFTnrFFTY =otimes== kYX =rArr ldquoUseful property for OFDM system to reduce

kk H

X =rArrcomplexity of channel equalizationrdquo

32

Cyclic Cyclic Prefix (OFDM Prefix (OFDM Receiver)Receiver)yy (( ))

bull bull( )( )nr~Symbols

dataOutput

bullbullbull

bullbullbull( )nr

bullbull

bullbull

33

Cyclic PrefixCyclic Prefix

loz One of the most important reasons to do OFDM is the

yy

pefficient way it deals with multipath delay spread

loz To eliminate inter-symbol interference (ISI) almostloz To eliminate inter symbol interference (ISI) almost completely a guard time is introduced for each OFDM symbolsymbol(The guard time is chosen larger than the delay spread)

34

Bandwidth EfficiencyBandwidth Efficiencyyy

loz In a classical parallel system the channel is divided into p y N non-overlapping sub-channels to avoid inter-carrier interference (ICI)( )

loz The diagram for bandwidth efficiency of OFDM system is shown below

35

SummarySummaryyy

loz The advantage of the FFT-based OFDM system g yloz The use of IFFTFFT can reduce the computation complexityloz The orthogonality between the adjacent sub-carriers will make g y j

the use of transmission bandwidth more efficientloz The guard interval is used to resist the inter-symbol interference

(ISI)loz The main advantage of the OFDM transmission technique is its

high performance even in frequency selective channels

loz The drawbacks of the OFDM system loz It is highly vulnerable to synchronization errorsloz Peak to Average Power Ratio (PAPR) problems

36

SynchronizationSynchronization

Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offset (TO)loz Impact of timing offset (TO)loz Impact of carrier frequency offset (CFO)loz TO estimation algorithmloz CFO estimation algorithmg

38 Synchronization2010722

Introduction (13)Introduction (13)( )( )

loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods

loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead

loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the

cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of

computational complexity therefore it may be not be available in short-burst wireless communication

39 Synchronization2010722

Introduction (23)Introduction (23)( )( )

loz Transcevier

40 Synchronization2010722

Introduction (33)Introduction (33)( )( )

loz Synchronization erroryloz Symbol timing offset (TO)

loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)

loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect

41 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg

42 Synchronization2010722

Signal model (14)Signal model (14)g ( )g ( )

loz Time domain sample p1

2

0

1 Nj nk N

n kk

x X eN

πminus

minus= sum

Xk the frequency domain data at the k-th subcarrier0kN =

loz The multipath fading channel

( ) ( ) ( )1L

h t h t δminus

sum

loz Received time domain signal

( ) ( ) ( )0

l ll

h t h tτ τ δ τ τ=

= minussum

loz Received time-domain signal

( ) ( )1

L

n n ly h n l x w nminus

minus= +sum43

( ) ( )0l=sum

Synchronization2010722

Signal model (24)Signal model (24)g ( )g ( )

loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting

12

0

Nj kn N

k nn

Y y e πminus

minus

=

= sum

( )1 1

2

0

N Lj kn N

n l kn l

h n l x e Wπminus minus

minusminus

=

= +sumsum

loz The equation above could be expressed in matrix form

0n l

loz The equation above could be expressed in matrix form

= +Y HX W

44 Synchronization2010722

Signal model (34)Signal model (34)g ( )g ( )

H= = = = +Y Fy FCx FCF X HX W

loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal

y

H=H FCFcirculant matrix leading to be a diagonal matrix

=H FCF

( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0

h h L L hh h

⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥

( )( ) ( )

( ) ( )

1 10 1 0 0i

h Lh L h N L

⎢ ⎥⎢ ⎥minus minus⎢ ⎥

equiv= minus minus⎢ ⎥⎢ ⎥

C( ) ( )

( ) ( )

0 1 1 100

0 0 1 1 0

h L h N L

h N L L h N

⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

45

( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦

Synchronization2010722

Signal model (44)Signal model (44)g ( )g ( )

loz In a time-invariant channel the channel matrix H would be a diagonal matrix

k k k kY H X W= +

( )1 1

2

0 0

N Lj kl N

kn l

H h n l e πminus minus

minus

= =

= sumsum

loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI

k k k k kY H X I Wα= + +k k k k k

( ) ( )1 1 1

2 2 N N L

j m k Nj ml Nk mI X h n l e e ππ

minus minus minusminusminus⎛ ⎞

= ⎜ ⎟⎝ ⎠

sum sum sumICI

46

0 0 0m n lm k= = =ne

⎝ ⎠

Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

47 Synchronization2010722

Impact of timing offset Impact of timing offset p gp g

loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively

( )I f s

Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ

( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr

2 fj k Nk k k kY X H e Wπδminus= +

loz For integer part of timing offset k k k kY X H e W+

I sTδ ( ) 0g IN L δminus minus le leFFT

2 Ij k NY X H e Wπδminus= +

( ) ( ) 2 I sFFT

j f TI sh t T H f e π δδ minusminus rarr

48

k k k kY X H e W= +

Synchronization2010722

Impact of timing offset derivationImpact of timing offset derivationp gp g

49 Synchronization2010722

Discussion of symbol boundaryDiscussion of symbol boundaryy yy y

50 Synchronization2010722

Impact of timing offsetImpact of timing offsetp gp g

loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI

2 Ij k NIk k k k

NY X H e ISI W

Nπδδ minusminus

= + +

loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk

e e

p kloz Phase Shift can be resolved by Differential Encoding

Decoding or carrier recovery with pilots

51 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

52 Synchronization2010722

Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )

53 Synchronization2010722

Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )

( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s

i n t i N N T N T nTy y t e

= + + +=

( ) ( )( ) ( ) ( ) 12

sin

sin

g gI f f

I I

i N N N Nj jf N Ni k i k k

f

Y X H e eN

π ε ε π ε

ε ε

πεπε

+ + minus+

minus minus=⎛ ⎞⎜ ⎟

Faded signal attenuated and rotated by CFO

( )( ) ( ) ( ) ( )11 2

sin

sin g gI f I f

i N N N NN j j l kI f

NN

l k π ε ε π ε επ ε ε + + minusminus + + + minus

⎜ ⎟⎝ ⎠

+ + minussum

( )( )( )

( ) ( )2

0

sin

I f I f

I

j j l kI f N Ni l l

l l k I f

X H e el k

NN

π ε ε π ε ε

ε π ε ε

+ + +

= ne minus

+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠

sum

i k

N

W

⎜ ⎟⎝ ⎠

+ Inter-Carrier Interference (ICI)

54 Synchronization2010722

Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )

loz An easier representation of received signal with only p g yfractional CFO is

1N

( ) ( )1

0

0N

k k k l l kl l k

Y H X C H X C l k Wminus

= ne

= + minus +⎡ ⎤⎣ ⎦sum

where the ICI coefficient is given by

( )( )( ) ( )

sin 1exp 1fl k

C l k j l kπ ε

π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )

( )exp 1sin

f

f

C l k j l kNN l k

N

π επ ε

minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠

55 Synchronization2010722

Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )

loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality

loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift

56 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

57 Synchronization2010722

TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )

loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq

58 Synchronization2010722

TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )

loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below

loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and

rN2 can be expressed by

0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+

( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +

59

( )

Synchronization2010722

TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )

loz Without CFO the first half is identical to the second half (in time order)

loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below

( ) 12 in discretesft T f NT

NTφ π π π ε πε

⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟

⎝ ⎠

helliphellip

sNT⎝ ⎠

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)

60

Length=N2

Synchronization2010722

TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )

loz Received signal without CFOg0

0 0

jr a e θ=jθ 2 0

2 2 0

Nj j

N Nr a e a e

θ θ= =

loz Received signal with CFO0 0

0 0

j jr a e eθ φ= Extra phase rotation due to CFO0 0

2 2 20

2 2 0

N N Nj j jj

N Nr a e e a e e

θ φ φθ= =

due to CFO

Phase difference 2 2 0N N

( )2 02 20 2 0 0

Nj j T fN

r r a e a eφ φ πminus Δ= =

which contains the information about CFO

61

0 2 0 0Nr r a e a e

Synchronization2010722

TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )

loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be

loz Note that d is a time index corresponding to the first( ) ( ) 2 1

20

Nd m d m Nm

P d r rminus

+ + +== sum

loz Note that d is a time index corresponding to the first sample in a window of N samples

loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by

( )22 1

20

Nd m NR d rminus

+ += sumloz A timing metric can be defined as

))(()(

)( 2

2

dRdP

dM =

( ) 20 d m Nm + +=sum

62

))(( dR

Synchronization2010722

TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )

63 Synchronization2010722

TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )

loz If drsquo is the correct symbol timing offset

( )2

2 2

0 1 12

j T f j T f j T fN

P d a e a e a eπ π πΔ Δ Δ

minus= + + +

22

2 2

0 1 12

j T fN

e a a aπ Δ

minus

⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭

loz If drsquo falls behind the correct symbol timing offset 1 sample

2⎪ ⎪⎩ ⎭

64 Synchronization2010722

TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )

65 Synchronization2010722

TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )

loz Drawback Plateau effect

Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)

((N2) ((N2) ((N2)

helliphellip

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)

Sliding window (length = N)

66 Synchronization2010722

TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )

loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula

( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus

扣掉相加

( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +

loz The method also called delay correlator

67 Synchronization2010722

IEEE 80211aIEEE 80211a

68 Synchronization2010722

Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )

loz Minnrsquos Method [2][ ]B B －B －B

( )( )

( )

22

2Minn

P dM d =

1 4 1N minus

( )( )( )2

2

MinnR d

( ) ( ) ( ) ( )

1 4 1

2 2 2 40 0

N

d N m k d N m k Nm k

P d r r+ + + + += =

= sum sum

( ) ( ) ( )2

21 4 1

2 40 0

N

d N m k Nk

R d rminus

+ + += sum sum69

0 0m k= =

Synchronization2010722

Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )

loz Parkrsquos Method [3][ ]

where D is the symmetric version of CC D C D

where D is the symmetric version of C

( )( ) 2

3k

P dM d =( )

( )( )23

ParkM dR d

2N

( ) 2

30

N

d k d kk

P d r rminus +=

= sdotsum

( )3

22

0

N

d kk

R d r += sum70

0k=

Synchronization2010722

Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )

loz Renrsquos Method [4][ ]

AS AS

where S is the PN sequence weighted of the original bl 2preamble

( )( )( )( )

24

Re 24

n

P dM d

R d=

( )( )4R d

( ) 2 1

4 2 2

N

k k N d k d k NP d s s r rminus

+ + + += sum0k=sum

( )21

41 N

d kR d rminus

= sum71

( )402 d k

kR d r +

=sum

Synchronization2010722

Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )

72 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

73 Synchronization2010722

Fractional CFO estimation Fractional CFO estimation

loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected

11 Rminus⎛ ⎞sum

h L i th di t b t t id ti l bl k R i

0

12 d m d m L

ms

f r rLTπ + + +

=

⎛ ⎞= ang⎜ ⎟⎝ ⎠sum

where L is the distance between two identical block R is the block size

loz For exampleNN cp

L = N R = NL = N R = Ncp74 Synchronization2010722

Estimation rangeEstimation rangegg

loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval

f ( ) ( )s s⎣ ⎦

loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)

loz DVB-T (L=N) 802 11 (L N4)

[ ]05 05s sf fminus

[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)

(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus

loz Note that is the subcarrier spacing1sf NT=

75

sNT

Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz

plusmng q y

loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn

76 Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz

plusmng q y

loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for

subcarrier spacing is required

plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn

p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs

77 Synchronization2010722

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

Table of ContentsTable of Contentsloz Introduction to OFDM

loz Synchronization

loz Channel Estimation

Introduction to OFDM SystemsIntroduction to OFDM Systems

Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

OFDM OverviewOFDM Overview

loz OFDMloz Orthogonal Frequency Division Multiplexing

loz Frequency Division Multiplexing (FDM) or multi-tone systems have been employed in military applicationssystems have been employed in military applications since the 196Os

loz OFDM employs multiple carriers overlapping in the f d ifrequency domain

4

OFDM OverviewOFDM Overview

Single carrier (SC) vs multi-carrier (MC)

loz Single carrier data are Multi-carrier data are shared transmitted over only one carrier

among several carriers and simultaneously transmittedFl t f di b iloz Selective fading Flat fading per subcarrier

5

OFDM OverviewOFDM Overview

loz OFDM modulation

loz Featuresloz No intercarrier guard bandsloz Overlapping of bands

S t l ffi iloz Spectral efficiencyloz Easy implementation by IFFTsloz Very sensitive to synchronizationloz Very sensitive to synchronization

6

Applications of OFDM TechnologyApplications of OFDM Technologypp gypp gy

loz Broadband Wired Access AsymmetricDigital Subscriber Loop (ADSL) Digital Multi-tone(DMT)

loz Wireless LANs (IEEE 80211ag IEEE 80211n HIPERLAN-2)

loz Digital Broadcasting (DAB DVB-T DVB-H)loz WiMAX (IEEE 80216 Series) 3GPP Long Term ( ) g

Evolution (3GPP LTE) 4Gloz Wireless Personal Area Network (WPAN) IEEE ( )

80215aMBOAloz Power Line

7

OFDM System ModelOFDM System Modelyy

loz Multi-carrier Block Transmission

frequency

time

8T8

OFDM System ModelOFDM System Modelyy

loz OFDM A block modulation scheme that transmitsa block N source symbols in parallel by using subcarrierssubcarriersloz Sub-carriers are orthogonal in time but overlapped in

frequencyeque cyloz Frequency spacing 1

FFT

fT

Δ =

cos(2 )cos(2 ( ) ) 0FFTTf t f f t dtπ π + Δint 1 10

cos(2 )cos(2 ( ) ) 0f t f f t dtπ π + Δ =int

9

OFDM System ModelOFDM System Model

tNΔ

yy

( )tf02cos π

( )0a

( )0atT

tNΔ1+

( )tf2i

( )

( )0b

T

DataEncoder

tf s Δ=

1 ( ) ( ) njbna +

( )nd SP MUX ( )tDChannelInput

( )tf02sin π

( )tf N 12cos minusπbullbullbull

( )1minusNa

( )1minusNb( )0a1+

( )tf N 12sin minusπ

( )1Nb

tT

( )1a ( )1minusNa1minus

tΔ

Figure 1tNΔ

( )1a ( )1minusNa Figure 1

10

OFDM System ModelOFDM System Model

loz An OFDM system transmitter shown in Figure 1

yy

y gloz The transmitted waveform D(t) can be expressed as

)1( )2sin()()2cos()()(1

0summinus

=

+=N

nnn tfnbtfnatD ππ

where tNffnffn Δ=ΔΔ+=

1 and 0

loz Using a two-dimensional digital modulation format the data

tNΔ

g g symbols d(n) can be represented as a(n) + jb(n)loz a(n) in-phase componentloz b(n) quadrature component

11

OFDM System ModelOFDM System Model

loz The serial data elements spaced by are grouped and tΔ

yy

p y g pused to modulate N carriers Thus they are frequency division multiplexedp

loz The signaling interval is then increased to which tNΔmakes the system less susceptible to channel delay spread impairments

Small-scale fading(Based on multipath time delay spread)

Flat Fading1 BW of signal lt BW of channel

Frequency Selective Fading1 BW of signal gt BW of channel

2 Delay spread lt Symbol preiod 2 Delay spread gt Symbol preiod

12

OrthogonalityOrthogonalityg yg y

loz Consider a set of transmitted carriers as follows

(2) 1 1 0for )( 2 0

minus==⎟⎠⎞

⎜⎝⎛

Δ+

Nnett

tNnfj

n

πψ n

⎨⎧ =minus

intqpab

dtttb for )(

)()( ψψ⎩⎨ Δ=minusne

=int tNabqpdttt

a qp )( and for 0 )()( ψψ

13

OrthogonalityOrthogonality

b tqpjb

intint)(2π

g yg y

dtedtttajbj

b

atN

qpjb

a qp = Δminus

intint )()(

)(2)(2

)(2ψψπ

tNqpjee tN

qpjtN

qpj

Δminusminus

=Δ

minusΔ

minus

)(2

)(2)(2

π

ππ

ee

tNqpjba

tNqpj

tNbqpj

⎟⎟⎞

⎜⎜⎛minus

Δ

minusΔ

minusΔ

minus1

)(2)(1)(2)(2

π

ππ

tNqpj Δminus

⎟⎠

⎜⎝=

)(2πtNabqp Δ=minusne= )( and for 0

14

OrthogonalityOrthogonalityg yg y

tNΔ

15

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFTby using IDFTby using IDFT

loz According to the structure of Tx it must use Ng oscillators That increases the hardware complexity

loz The equivalent method is using IDFT (IFFT)

16

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

loz In general each carrier can be expressed as

by using IDFTby using IDFT

g p

( ) (6) )()( )(2 ttfjcc

ccetAtS φπ +=

loz We assume that there are N carriers in the OFDM signal Th th t t l l i l S (t) b t d bThen the total complex signal Ss(t) can be represented by

( ) (7))(1)(1

)(2etAtSN

ttfj nn φπ= summinus

+( )

where

(7) )()(

0

0

fnff

etAN

tSn

ns

Δ+=

= sum=

liihffrequency carrier phase amplitude are )( )( and

where 0

fttAfnff

nnn

n

φΔ+

lyrespective carrierth -of n17

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

loz Then we sample the signal at a sampling frequency tΔ1

by using IDFTby using IDFT

p g p g q y and An(t) and φn(t) becomes

(8))(t = φφ(9) )((8) )(

nn

nn

AtAt

== φφ

( ) (10) 1)(1

0

)(2 0summinus

=

+ΔΔ+=ΔN

n

tkfnfjns

neAN

tkS φπ

loz Then the sampled signal can be expressed as0=nN

( )( ) (11) 1)(1

0

22 0summinus

=

ΔΔ+Δ sdot=ΔN

n

tfnkjtkfjns eeA

NtkS n πφπ

0n

18

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

loz The inverse discrete Fourier transform (IDFT) is defined

by using IDFTby using IDFT

( )as the following

1 21

NnkjN

πsumminus

(12) )(1)( 2

0

Nnkj

nefnF

Ntkf πsum

=

Δ=Δ

loz Comparing eq(11) and eq(12) the condition must be satisfied in order to make eq(11) an inverse Fouriersatisfied in order to make eq(11) an inverse Fourier transform relationship

1 (13) 1tN

fΔ

=Δ

19

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

loz If eq(13) is satisfied

by using IDFTby using IDFT

q ( ) loz is the frequency domain signal loz is the time domain signal

( )ntkfjneA φπ +Δ02

)( tkSs Δ gloz is the sub-channel spacingloz is the symbol duration in each sub-channel

)(s

fΔtNΔloz s e sy bo du o e c sub c e

loz This outcome is the same as the result obtained in the

N

loz This outcome is the same as the result obtained in the system of Figure 1 Therefore IDFT can be used to generate an OFDM transmission signalgenerate an OFDM transmission signal

20

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

( )tf02cos π

( )0a

by using IDFTby using IDFT

( )nd ( )tDI( )tf02sin π

( )0a

( )0b

bull

tf s Δ=

1 ( ) ( ) njbna +

( )nd

tNΔffnffn Δ

=Δ+=1 0

( )tDChannelInput

( )tf N 12cos minusπ

( )1minusNa

bullbull

tNΔ

( )tf N 12sin minusπ

( )1minusNb

( )0d

( )nd ( )tDChannelInput

bull bull

( )( )1d

( )2d

tf s Δ=

1 ( ) ( ) njbna +

fNΔffnffn Δ

=Δ+=1 0

ChannelInput

bullbullbullbull

bullbullbullbull

( )1minusNd

21

Frequency Error Results in ICIFrequency Error Results in ICIq yq y

22

Synchronization Error Results in ICISynchronization Error Results in ICIyy

Not Orthogonal Any More

23

Cyclic PrefixCyclic Prefixyy

loz In multipath channel delayed replicas of previous OFDM signal lead to ISI between successive OFDM signals

loz Solution Insert a guard interval between successive OFDMloz Solution Insert a guard interval between successive OFDM signals

24

Cyclic PrefixCyclic Prefix

loz Guard interval leads to intercarrier interference (ICI) in OFDM d d l i

yy

demodulation

loz In DFT interval difference between two subcarriers does not maintain integer number of cycles loss of orthogonality

loz Delayed version of subcarrier 2 causes ICI in the process of d d l ti b i 1demodulating subcarrier 1

25

Cyclic PrefixCyclic Prefixloz Cyclic prefix (CP) A copy of the last part of OFDM signal is

attached to the front of itself

yy

attached to the front of itself

[ ]0d

[ ]1d

[ ]0D[ ]1D

[ ]nd[ ]kD~

symbols dataInput

bullbullbullbull

[ ]1d

[ ]2d

[ ][ ]2D

bullbullbullbullbull

y

bullbullbullbullbull

[ ]1minusNd [ ]1minusND

[ ]gNND minus

bullbullbull

26

Cyclic PrefixCyclic Prefix

loz All delayed replicas of subcarriers always have an

yy

y p yinteger number of cycles within DFT interval no ICI

27

Cyclic PrefixCyclic Prefix

loz Linear convolution vs circular convolution

yy

N

28

Cyclic PrefixCyclic Prefix

loz Channel effect with cyclic prefix

yy

Signal after removed CP

29

Cyclic PrefixCyclic Prefix

loz Time-Domain Explanation

yy

p

30

Cyclic PrefixCyclic Prefixyy

31

Cyclic PrefixCyclic Prefix

loz Spectrum of channel response with length (smaller [ ]nh hL

yy

than )h

gN[ ] nhFFTHk =

loz Received complete OFDM signal[ ] [ ] [ ] 20 ~~ minus++leleotimes= hg LNNnnhnDnr

loz Received useful part [ ]nr[ ] [ ] [ ]nhnDnr Notimes=

where is N-point circular convolution (due to CP)loz Received symbol at k-th subcarrier

[ ] [ ] [ ]N

Notimes

[ ] [ ] [ ] kkNk HXnhnDFFTnrFFTY =otimes== kYX =rArr ldquoUseful property for OFDM system to reduce

kk H

X =rArrcomplexity of channel equalizationrdquo

32

Cyclic Cyclic Prefix (OFDM Prefix (OFDM Receiver)Receiver)yy (( ))

bull bull( )( )nr~Symbols

dataOutput

bullbullbull

bullbullbull( )nr

bullbull

bullbull

33

Cyclic PrefixCyclic Prefix

loz One of the most important reasons to do OFDM is the

yy

pefficient way it deals with multipath delay spread

loz To eliminate inter-symbol interference (ISI) almostloz To eliminate inter symbol interference (ISI) almost completely a guard time is introduced for each OFDM symbolsymbol(The guard time is chosen larger than the delay spread)

34

Bandwidth EfficiencyBandwidth Efficiencyyy

loz In a classical parallel system the channel is divided into p y N non-overlapping sub-channels to avoid inter-carrier interference (ICI)( )

loz The diagram for bandwidth efficiency of OFDM system is shown below

35

SummarySummaryyy

loz The advantage of the FFT-based OFDM system g yloz The use of IFFTFFT can reduce the computation complexityloz The orthogonality between the adjacent sub-carriers will make g y j

the use of transmission bandwidth more efficientloz The guard interval is used to resist the inter-symbol interference

(ISI)loz The main advantage of the OFDM transmission technique is its

high performance even in frequency selective channels

loz The drawbacks of the OFDM system loz It is highly vulnerable to synchronization errorsloz Peak to Average Power Ratio (PAPR) problems

36

SynchronizationSynchronization

Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offset (TO)loz Impact of timing offset (TO)loz Impact of carrier frequency offset (CFO)loz TO estimation algorithmloz CFO estimation algorithmg

38 Synchronization2010722

Introduction (13)Introduction (13)( )( )

loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods

loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead

loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the

cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of

computational complexity therefore it may be not be available in short-burst wireless communication

39 Synchronization2010722

Introduction (23)Introduction (23)( )( )

loz Transcevier

40 Synchronization2010722

Introduction (33)Introduction (33)( )( )

loz Synchronization erroryloz Symbol timing offset (TO)

loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)

loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect

41 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg

42 Synchronization2010722

Signal model (14)Signal model (14)g ( )g ( )

loz Time domain sample p1

2

0

1 Nj nk N

n kk

x X eN

πminus

minus= sum

Xk the frequency domain data at the k-th subcarrier0kN =

loz The multipath fading channel

( ) ( ) ( )1L

h t h t δminus

sum

loz Received time domain signal

( ) ( ) ( )0

l ll

h t h tτ τ δ τ τ=

= minussum

loz Received time-domain signal

( ) ( )1

L

n n ly h n l x w nminus

minus= +sum43

( ) ( )0l=sum

Synchronization2010722

Signal model (24)Signal model (24)g ( )g ( )

loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting

12

0

Nj kn N

k nn

Y y e πminus

minus

=

= sum

( )1 1

2

0

N Lj kn N

n l kn l

h n l x e Wπminus minus

minusminus

=

= +sumsum

loz The equation above could be expressed in matrix form

0n l

loz The equation above could be expressed in matrix form

= +Y HX W

44 Synchronization2010722

Signal model (34)Signal model (34)g ( )g ( )

H= = = = +Y Fy FCx FCF X HX W

loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal

y

H=H FCFcirculant matrix leading to be a diagonal matrix

=H FCF

( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0

h h L L hh h

⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥

( )( ) ( )

( ) ( )

1 10 1 0 0i

h Lh L h N L

⎢ ⎥⎢ ⎥minus minus⎢ ⎥

equiv= minus minus⎢ ⎥⎢ ⎥

C( ) ( )

( ) ( )

0 1 1 100

0 0 1 1 0

h L h N L

h N L L h N

⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

45

( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦

Synchronization2010722

Signal model (44)Signal model (44)g ( )g ( )

loz In a time-invariant channel the channel matrix H would be a diagonal matrix

k k k kY H X W= +

( )1 1

2

0 0

N Lj kl N

kn l

H h n l e πminus minus

minus

= =

= sumsum

loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI

k k k k kY H X I Wα= + +k k k k k

( ) ( )1 1 1

2 2 N N L

j m k Nj ml Nk mI X h n l e e ππ

minus minus minusminusminus⎛ ⎞

= ⎜ ⎟⎝ ⎠

sum sum sumICI

46

0 0 0m n lm k= = =ne

⎝ ⎠

Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

47 Synchronization2010722

Impact of timing offset Impact of timing offset p gp g

loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively

( )I f s

Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ

( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr

2 fj k Nk k k kY X H e Wπδminus= +

loz For integer part of timing offset k k k kY X H e W+

I sTδ ( ) 0g IN L δminus minus le leFFT

2 Ij k NY X H e Wπδminus= +

( ) ( ) 2 I sFFT

j f TI sh t T H f e π δδ minusminus rarr

48

k k k kY X H e W= +

Synchronization2010722

Impact of timing offset derivationImpact of timing offset derivationp gp g

49 Synchronization2010722

Discussion of symbol boundaryDiscussion of symbol boundaryy yy y

50 Synchronization2010722

Impact of timing offsetImpact of timing offsetp gp g

loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI

2 Ij k NIk k k k

NY X H e ISI W

Nπδδ minusminus

= + +

loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk

e e

p kloz Phase Shift can be resolved by Differential Encoding

Decoding or carrier recovery with pilots

51 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

52 Synchronization2010722

Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )

53 Synchronization2010722

Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )

( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s

i n t i N N T N T nTy y t e

= + + +=

( ) ( )( ) ( ) ( ) 12

sin

sin

g gI f f

I I

i N N N Nj jf N Ni k i k k

f

Y X H e eN

π ε ε π ε

ε ε

πεπε

+ + minus+

minus minus=⎛ ⎞⎜ ⎟

Faded signal attenuated and rotated by CFO

( )( ) ( ) ( ) ( )11 2

sin

sin g gI f I f

i N N N NN j j l kI f

NN

l k π ε ε π ε επ ε ε + + minusminus + + + minus

⎜ ⎟⎝ ⎠

+ + minussum

( )( )( )

( ) ( )2

0

sin

I f I f

I

j j l kI f N Ni l l

l l k I f

X H e el k

NN

π ε ε π ε ε

ε π ε ε

+ + +

= ne minus

+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠

sum

i k

N

W

⎜ ⎟⎝ ⎠

+ Inter-Carrier Interference (ICI)

54 Synchronization2010722

Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )

loz An easier representation of received signal with only p g yfractional CFO is

1N

( ) ( )1

0

0N

k k k l l kl l k

Y H X C H X C l k Wminus

= ne

= + minus +⎡ ⎤⎣ ⎦sum

where the ICI coefficient is given by

( )( )( ) ( )

sin 1exp 1fl k

C l k j l kπ ε

π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )

( )exp 1sin

f

f

C l k j l kNN l k

N

π επ ε

minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠

55 Synchronization2010722

Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )

loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality

loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift

56 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

57 Synchronization2010722

TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )

loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq

58 Synchronization2010722

TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )

loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below

loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and

rN2 can be expressed by

0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+

( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +

59

( )

Synchronization2010722

TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )

loz Without CFO the first half is identical to the second half (in time order)

loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below

( ) 12 in discretesft T f NT

NTφ π π π ε πε

⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟

⎝ ⎠

helliphellip

sNT⎝ ⎠

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)

60

Length=N2

Synchronization2010722

TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )

loz Received signal without CFOg0

0 0

jr a e θ=jθ 2 0

2 2 0

Nj j

N Nr a e a e

θ θ= =

loz Received signal with CFO0 0

0 0

j jr a e eθ φ= Extra phase rotation due to CFO0 0

2 2 20

2 2 0

N N Nj j jj

N Nr a e e a e e

θ φ φθ= =

due to CFO

Phase difference 2 2 0N N

( )2 02 20 2 0 0

Nj j T fN

r r a e a eφ φ πminus Δ= =

which contains the information about CFO

61

0 2 0 0Nr r a e a e

Synchronization2010722

TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )

loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be

loz Note that d is a time index corresponding to the first( ) ( ) 2 1

20

Nd m d m Nm

P d r rminus

+ + +== sum

loz Note that d is a time index corresponding to the first sample in a window of N samples

loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by

( )22 1

20

Nd m NR d rminus

+ += sumloz A timing metric can be defined as

))(()(

)( 2

2

dRdP

dM =

( ) 20 d m Nm + +=sum

62

))(( dR

Synchronization2010722

TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )

63 Synchronization2010722

TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )

loz If drsquo is the correct symbol timing offset

( )2

2 2

0 1 12

j T f j T f j T fN

P d a e a e a eπ π πΔ Δ Δ

minus= + + +

22

2 2

0 1 12

j T fN

e a a aπ Δ

minus

⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭

loz If drsquo falls behind the correct symbol timing offset 1 sample

2⎪ ⎪⎩ ⎭

64 Synchronization2010722

TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )

65 Synchronization2010722

TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )

loz Drawback Plateau effect

Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)

((N2) ((N2) ((N2)

helliphellip

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)

Sliding window (length = N)

66 Synchronization2010722

TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )

loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula

( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus

扣掉相加

( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +

loz The method also called delay correlator

67 Synchronization2010722

IEEE 80211aIEEE 80211a

68 Synchronization2010722

Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )

loz Minnrsquos Method [2][ ]B B －B －B

( )( )

( )

22

2Minn

P dM d =

1 4 1N minus

( )( )( )2

2

MinnR d

( ) ( ) ( ) ( )

1 4 1

2 2 2 40 0

N

d N m k d N m k Nm k

P d r r+ + + + += =

= sum sum

( ) ( ) ( )2

21 4 1

2 40 0

N

d N m k Nk

R d rminus

+ + += sum sum69

0 0m k= =

Synchronization2010722

Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )

loz Parkrsquos Method [3][ ]

where D is the symmetric version of CC D C D

where D is the symmetric version of C

( )( ) 2

3k

P dM d =( )

( )( )23

ParkM dR d

2N

( ) 2

30

N

d k d kk

P d r rminus +=

= sdotsum

( )3

22

0

N

d kk

R d r += sum70

0k=

Synchronization2010722

Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )

loz Renrsquos Method [4][ ]

AS AS

where S is the PN sequence weighted of the original bl 2preamble

( )( )( )( )

24

Re 24

n

P dM d

R d=

( )( )4R d

( ) 2 1

4 2 2

N

k k N d k d k NP d s s r rminus

+ + + += sum0k=sum

( )21

41 N

d kR d rminus

= sum71

( )402 d k

kR d r +

=sum

Synchronization2010722

Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )

72 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

73 Synchronization2010722

Fractional CFO estimation Fractional CFO estimation

loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected

11 Rminus⎛ ⎞sum

h L i th di t b t t id ti l bl k R i

0

12 d m d m L

ms

f r rLTπ + + +

=

⎛ ⎞= ang⎜ ⎟⎝ ⎠sum

where L is the distance between two identical block R is the block size

loz For exampleNN cp

L = N R = NL = N R = Ncp74 Synchronization2010722

Estimation rangeEstimation rangegg

loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval

f ( ) ( )s s⎣ ⎦

loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)

loz DVB-T (L=N) 802 11 (L N4)

[ ]05 05s sf fminus

[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)

(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus

loz Note that is the subcarrier spacing1sf NT=

75

sNT

Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz

plusmng q y

loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn

76 Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz

plusmng q y

loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for

subcarrier spacing is required

plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn

p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs

77 Synchronization2010722

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

Introduction to OFDM SystemsIntroduction to OFDM Systems

Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

OFDM OverviewOFDM Overview

loz OFDMloz Orthogonal Frequency Division Multiplexing

loz Frequency Division Multiplexing (FDM) or multi-tone systems have been employed in military applicationssystems have been employed in military applications since the 196Os

loz OFDM employs multiple carriers overlapping in the f d ifrequency domain

4

OFDM OverviewOFDM Overview

Single carrier (SC) vs multi-carrier (MC)

loz Single carrier data are Multi-carrier data are shared transmitted over only one carrier

among several carriers and simultaneously transmittedFl t f di b iloz Selective fading Flat fading per subcarrier

5

OFDM OverviewOFDM Overview

loz OFDM modulation

loz Featuresloz No intercarrier guard bandsloz Overlapping of bands

S t l ffi iloz Spectral efficiencyloz Easy implementation by IFFTsloz Very sensitive to synchronizationloz Very sensitive to synchronization

6

Applications of OFDM TechnologyApplications of OFDM Technologypp gypp gy

loz Broadband Wired Access AsymmetricDigital Subscriber Loop (ADSL) Digital Multi-tone(DMT)

loz Wireless LANs (IEEE 80211ag IEEE 80211n HIPERLAN-2)

loz Digital Broadcasting (DAB DVB-T DVB-H)loz WiMAX (IEEE 80216 Series) 3GPP Long Term ( ) g

Evolution (3GPP LTE) 4Gloz Wireless Personal Area Network (WPAN) IEEE ( )

80215aMBOAloz Power Line

7

OFDM System ModelOFDM System Modelyy

loz Multi-carrier Block Transmission

frequency

time

8T8

OFDM System ModelOFDM System Modelyy

loz OFDM A block modulation scheme that transmitsa block N source symbols in parallel by using subcarrierssubcarriersloz Sub-carriers are orthogonal in time but overlapped in

frequencyeque cyloz Frequency spacing 1

FFT

fT

Δ =

cos(2 )cos(2 ( ) ) 0FFTTf t f f t dtπ π + Δint 1 10

cos(2 )cos(2 ( ) ) 0f t f f t dtπ π + Δ =int

9

OFDM System ModelOFDM System Model

tNΔ

yy

( )tf02cos π

( )0a

( )0atT

tNΔ1+

( )tf2i

( )

( )0b

T

DataEncoder

tf s Δ=

1 ( ) ( ) njbna +

( )nd SP MUX ( )tDChannelInput

( )tf02sin π

( )tf N 12cos minusπbullbullbull

( )1minusNa

( )1minusNb( )0a1+

( )tf N 12sin minusπ

( )1Nb

tT

( )1a ( )1minusNa1minus

tΔ

Figure 1tNΔ

( )1a ( )1minusNa Figure 1

10

OFDM System ModelOFDM System Model

loz An OFDM system transmitter shown in Figure 1

yy

y gloz The transmitted waveform D(t) can be expressed as

)1( )2sin()()2cos()()(1

0summinus

=

+=N

nnn tfnbtfnatD ππ

where tNffnffn Δ=ΔΔ+=

1 and 0

loz Using a two-dimensional digital modulation format the data

tNΔ

g g symbols d(n) can be represented as a(n) + jb(n)loz a(n) in-phase componentloz b(n) quadrature component

11

OFDM System ModelOFDM System Model

loz The serial data elements spaced by are grouped and tΔ

yy

p y g pused to modulate N carriers Thus they are frequency division multiplexedp

loz The signaling interval is then increased to which tNΔmakes the system less susceptible to channel delay spread impairments

Small-scale fading(Based on multipath time delay spread)

Flat Fading1 BW of signal lt BW of channel

Frequency Selective Fading1 BW of signal gt BW of channel

2 Delay spread lt Symbol preiod 2 Delay spread gt Symbol preiod

12

OrthogonalityOrthogonalityg yg y

loz Consider a set of transmitted carriers as follows

(2) 1 1 0for )( 2 0

minus==⎟⎠⎞

⎜⎝⎛

Δ+

Nnett

tNnfj

n

πψ n

⎨⎧ =minus

intqpab

dtttb for )(

)()( ψψ⎩⎨ Δ=minusne

=int tNabqpdttt

a qp )( and for 0 )()( ψψ

13

OrthogonalityOrthogonality

b tqpjb

intint)(2π

g yg y

dtedtttajbj

b

atN

qpjb

a qp = Δminus

intint )()(

)(2)(2

)(2ψψπ

tNqpjee tN

qpjtN

qpj

Δminusminus

=Δ

minusΔ

minus

)(2

)(2)(2

π

ππ

ee

tNqpjba

tNqpj

tNbqpj

⎟⎟⎞

⎜⎜⎛minus

Δ

minusΔ

minusΔ

minus1

)(2)(1)(2)(2

π

ππ

tNqpj Δminus

⎟⎠

⎜⎝=

)(2πtNabqp Δ=minusne= )( and for 0

14

OrthogonalityOrthogonalityg yg y

tNΔ

15

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFTby using IDFTby using IDFT

loz According to the structure of Tx it must use Ng oscillators That increases the hardware complexity

loz The equivalent method is using IDFT (IFFT)

16

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

loz In general each carrier can be expressed as

by using IDFTby using IDFT

g p

( ) (6) )()( )(2 ttfjcc

ccetAtS φπ +=

loz We assume that there are N carriers in the OFDM signal Th th t t l l i l S (t) b t d bThen the total complex signal Ss(t) can be represented by

( ) (7))(1)(1

)(2etAtSN

ttfj nn φπ= summinus

+( )

where

(7) )()(

0

0

fnff

etAN

tSn

ns

Δ+=

= sum=

liihffrequency carrier phase amplitude are )( )( and

where 0

fttAfnff

nnn

n

φΔ+

lyrespective carrierth -of n17

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

loz Then we sample the signal at a sampling frequency tΔ1

by using IDFTby using IDFT

p g p g q y and An(t) and φn(t) becomes

(8))(t = φφ(9) )((8) )(

nn

nn

AtAt

== φφ

( ) (10) 1)(1

0

)(2 0summinus

=

+ΔΔ+=ΔN

n

tkfnfjns

neAN

tkS φπ

loz Then the sampled signal can be expressed as0=nN

( )( ) (11) 1)(1

0

22 0summinus

=

ΔΔ+Δ sdot=ΔN

n

tfnkjtkfjns eeA

NtkS n πφπ

0n

18

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

loz The inverse discrete Fourier transform (IDFT) is defined

by using IDFTby using IDFT

( )as the following

1 21

NnkjN

πsumminus

(12) )(1)( 2

0

Nnkj

nefnF

Ntkf πsum

=

Δ=Δ

loz Comparing eq(11) and eq(12) the condition must be satisfied in order to make eq(11) an inverse Fouriersatisfied in order to make eq(11) an inverse Fourier transform relationship

1 (13) 1tN

fΔ

=Δ

19

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

loz If eq(13) is satisfied

by using IDFTby using IDFT

q ( ) loz is the frequency domain signal loz is the time domain signal

( )ntkfjneA φπ +Δ02

)( tkSs Δ gloz is the sub-channel spacingloz is the symbol duration in each sub-channel

)(s

fΔtNΔloz s e sy bo du o e c sub c e

loz This outcome is the same as the result obtained in the

N

loz This outcome is the same as the result obtained in the system of Figure 1 Therefore IDFT can be used to generate an OFDM transmission signalgenerate an OFDM transmission signal

20

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

( )tf02cos π

( )0a

by using IDFTby using IDFT

( )nd ( )tDI( )tf02sin π

( )0a

( )0b

bull

tf s Δ=

1 ( ) ( ) njbna +

( )nd

tNΔffnffn Δ

=Δ+=1 0

( )tDChannelInput

( )tf N 12cos minusπ

( )1minusNa

bullbull

tNΔ

( )tf N 12sin minusπ

( )1minusNb

( )0d

( )nd ( )tDChannelInput

bull bull

( )( )1d

( )2d

tf s Δ=

1 ( ) ( ) njbna +

fNΔffnffn Δ

=Δ+=1 0

ChannelInput

bullbullbullbull

bullbullbullbull

( )1minusNd

21

Frequency Error Results in ICIFrequency Error Results in ICIq yq y

22

Synchronization Error Results in ICISynchronization Error Results in ICIyy

Not Orthogonal Any More

23

Cyclic PrefixCyclic Prefixyy

loz In multipath channel delayed replicas of previous OFDM signal lead to ISI between successive OFDM signals

loz Solution Insert a guard interval between successive OFDMloz Solution Insert a guard interval between successive OFDM signals

24

Cyclic PrefixCyclic Prefix

loz Guard interval leads to intercarrier interference (ICI) in OFDM d d l i

yy

demodulation

loz In DFT interval difference between two subcarriers does not maintain integer number of cycles loss of orthogonality

loz Delayed version of subcarrier 2 causes ICI in the process of d d l ti b i 1demodulating subcarrier 1

25

Cyclic PrefixCyclic Prefixloz Cyclic prefix (CP) A copy of the last part of OFDM signal is

attached to the front of itself

yy

attached to the front of itself

[ ]0d

[ ]1d

[ ]0D[ ]1D

[ ]nd[ ]kD~

symbols dataInput

bullbullbullbull

[ ]1d

[ ]2d

[ ][ ]2D

bullbullbullbullbull

y

bullbullbullbullbull

[ ]1minusNd [ ]1minusND

[ ]gNND minus

bullbullbull

26

Cyclic PrefixCyclic Prefix

loz All delayed replicas of subcarriers always have an

yy

y p yinteger number of cycles within DFT interval no ICI

27

Cyclic PrefixCyclic Prefix

loz Linear convolution vs circular convolution

yy

N

28

Cyclic PrefixCyclic Prefix

loz Channel effect with cyclic prefix

yy

Signal after removed CP

29

Cyclic PrefixCyclic Prefix

loz Time-Domain Explanation

yy

p

30

Cyclic PrefixCyclic Prefixyy

31

Cyclic PrefixCyclic Prefix

loz Spectrum of channel response with length (smaller [ ]nh hL

yy

than )h

gN[ ] nhFFTHk =

loz Received complete OFDM signal[ ] [ ] [ ] 20 ~~ minus++leleotimes= hg LNNnnhnDnr

loz Received useful part [ ]nr[ ] [ ] [ ]nhnDnr Notimes=

where is N-point circular convolution (due to CP)loz Received symbol at k-th subcarrier

[ ] [ ] [ ]N

Notimes

[ ] [ ] [ ] kkNk HXnhnDFFTnrFFTY =otimes== kYX =rArr ldquoUseful property for OFDM system to reduce

kk H

X =rArrcomplexity of channel equalizationrdquo

32

Cyclic Cyclic Prefix (OFDM Prefix (OFDM Receiver)Receiver)yy (( ))

bull bull( )( )nr~Symbols

dataOutput

bullbullbull

bullbullbull( )nr

bullbull

bullbull

33

Cyclic PrefixCyclic Prefix

loz One of the most important reasons to do OFDM is the

yy

pefficient way it deals with multipath delay spread

loz To eliminate inter-symbol interference (ISI) almostloz To eliminate inter symbol interference (ISI) almost completely a guard time is introduced for each OFDM symbolsymbol(The guard time is chosen larger than the delay spread)

34

Bandwidth EfficiencyBandwidth Efficiencyyy

loz In a classical parallel system the channel is divided into p y N non-overlapping sub-channels to avoid inter-carrier interference (ICI)( )

loz The diagram for bandwidth efficiency of OFDM system is shown below

35

SummarySummaryyy

loz The advantage of the FFT-based OFDM system g yloz The use of IFFTFFT can reduce the computation complexityloz The orthogonality between the adjacent sub-carriers will make g y j

the use of transmission bandwidth more efficientloz The guard interval is used to resist the inter-symbol interference

(ISI)loz The main advantage of the OFDM transmission technique is its

high performance even in frequency selective channels

loz The drawbacks of the OFDM system loz It is highly vulnerable to synchronization errorsloz Peak to Average Power Ratio (PAPR) problems

36

SynchronizationSynchronization

Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offset (TO)loz Impact of timing offset (TO)loz Impact of carrier frequency offset (CFO)loz TO estimation algorithmloz CFO estimation algorithmg

38 Synchronization2010722

Introduction (13)Introduction (13)( )( )

loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods

loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead

loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the

cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of

computational complexity therefore it may be not be available in short-burst wireless communication

39 Synchronization2010722

Introduction (23)Introduction (23)( )( )

loz Transcevier

40 Synchronization2010722

Introduction (33)Introduction (33)( )( )

loz Synchronization erroryloz Symbol timing offset (TO)

loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)

loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect

41 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg

42 Synchronization2010722

Signal model (14)Signal model (14)g ( )g ( )

loz Time domain sample p1

2

0

1 Nj nk N

n kk

x X eN

πminus

minus= sum

Xk the frequency domain data at the k-th subcarrier0kN =

loz The multipath fading channel

( ) ( ) ( )1L

h t h t δminus

sum

loz Received time domain signal

( ) ( ) ( )0

l ll

h t h tτ τ δ τ τ=

= minussum

loz Received time-domain signal

( ) ( )1

L

n n ly h n l x w nminus

minus= +sum43

( ) ( )0l=sum

Synchronization2010722

Signal model (24)Signal model (24)g ( )g ( )

loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting

12

0

Nj kn N

k nn

Y y e πminus

minus

=

= sum

( )1 1

2

0

N Lj kn N

n l kn l

h n l x e Wπminus minus

minusminus

=

= +sumsum

loz The equation above could be expressed in matrix form

0n l

loz The equation above could be expressed in matrix form

= +Y HX W

44 Synchronization2010722

Signal model (34)Signal model (34)g ( )g ( )

H= = = = +Y Fy FCx FCF X HX W

loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal

y

H=H FCFcirculant matrix leading to be a diagonal matrix

=H FCF

( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0

h h L L hh h

⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥

( )( ) ( )

( ) ( )

1 10 1 0 0i

h Lh L h N L

⎢ ⎥⎢ ⎥minus minus⎢ ⎥

equiv= minus minus⎢ ⎥⎢ ⎥

C( ) ( )

( ) ( )

0 1 1 100

0 0 1 1 0

h L h N L

h N L L h N

⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

45

( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦

Synchronization2010722

Signal model (44)Signal model (44)g ( )g ( )

loz In a time-invariant channel the channel matrix H would be a diagonal matrix

k k k kY H X W= +

( )1 1

2

0 0

N Lj kl N

kn l

H h n l e πminus minus

minus

= =

= sumsum

loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI

k k k k kY H X I Wα= + +k k k k k

( ) ( )1 1 1

2 2 N N L

j m k Nj ml Nk mI X h n l e e ππ

minus minus minusminusminus⎛ ⎞

= ⎜ ⎟⎝ ⎠

sum sum sumICI

46

0 0 0m n lm k= = =ne

⎝ ⎠

Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

47 Synchronization2010722

Impact of timing offset Impact of timing offset p gp g

loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively

( )I f s

Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ

( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr

2 fj k Nk k k kY X H e Wπδminus= +

loz For integer part of timing offset k k k kY X H e W+

I sTδ ( ) 0g IN L δminus minus le leFFT

2 Ij k NY X H e Wπδminus= +

( ) ( ) 2 I sFFT

j f TI sh t T H f e π δδ minusminus rarr

48

k k k kY X H e W= +

Synchronization2010722

Impact of timing offset derivationImpact of timing offset derivationp gp g

49 Synchronization2010722

Discussion of symbol boundaryDiscussion of symbol boundaryy yy y

50 Synchronization2010722

Impact of timing offsetImpact of timing offsetp gp g

loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI

2 Ij k NIk k k k

NY X H e ISI W

Nπδδ minusminus

= + +

loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk

e e

p kloz Phase Shift can be resolved by Differential Encoding

Decoding or carrier recovery with pilots

51 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

52 Synchronization2010722

Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )

53 Synchronization2010722

Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )

( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s

i n t i N N T N T nTy y t e

= + + +=

( ) ( )( ) ( ) ( ) 12

sin

sin

g gI f f

I I

i N N N Nj jf N Ni k i k k

f

Y X H e eN

π ε ε π ε

ε ε

πεπε

+ + minus+

minus minus=⎛ ⎞⎜ ⎟

Faded signal attenuated and rotated by CFO

( )( ) ( ) ( ) ( )11 2

sin

sin g gI f I f

i N N N NN j j l kI f

NN

l k π ε ε π ε επ ε ε + + minusminus + + + minus

⎜ ⎟⎝ ⎠

+ + minussum

( )( )( )

( ) ( )2

0

sin

I f I f

I

j j l kI f N Ni l l

l l k I f

X H e el k

NN

π ε ε π ε ε

ε π ε ε

+ + +

= ne minus

+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠

sum

i k

N

W

⎜ ⎟⎝ ⎠

+ Inter-Carrier Interference (ICI)

54 Synchronization2010722

Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )

loz An easier representation of received signal with only p g yfractional CFO is

1N

( ) ( )1

0

0N

k k k l l kl l k

Y H X C H X C l k Wminus

= ne

= + minus +⎡ ⎤⎣ ⎦sum

where the ICI coefficient is given by

( )( )( ) ( )

sin 1exp 1fl k

C l k j l kπ ε

π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )

( )exp 1sin

f

f

C l k j l kNN l k

N

π επ ε

minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠

55 Synchronization2010722

Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )

loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality

loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift

56 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

57 Synchronization2010722

TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )

loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq

58 Synchronization2010722

TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )

loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below

loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and

rN2 can be expressed by

0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+

( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +

59

( )

Synchronization2010722

TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )

loz Without CFO the first half is identical to the second half (in time order)

loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below

( ) 12 in discretesft T f NT

NTφ π π π ε πε

⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟

⎝ ⎠

helliphellip

sNT⎝ ⎠

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)

60

Length=N2

Synchronization2010722

TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )

loz Received signal without CFOg0

0 0

jr a e θ=jθ 2 0

2 2 0

Nj j

N Nr a e a e

θ θ= =

loz Received signal with CFO0 0

0 0

j jr a e eθ φ= Extra phase rotation due to CFO0 0

2 2 20

2 2 0

N N Nj j jj

N Nr a e e a e e

θ φ φθ= =

due to CFO

Phase difference 2 2 0N N

( )2 02 20 2 0 0

Nj j T fN

r r a e a eφ φ πminus Δ= =

which contains the information about CFO

61

0 2 0 0Nr r a e a e

Synchronization2010722

TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )

loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be

loz Note that d is a time index corresponding to the first( ) ( ) 2 1

20

Nd m d m Nm

P d r rminus

+ + +== sum

loz Note that d is a time index corresponding to the first sample in a window of N samples

loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by

( )22 1

20

Nd m NR d rminus

+ += sumloz A timing metric can be defined as

))(()(

)( 2

2

dRdP

dM =

( ) 20 d m Nm + +=sum

62

))(( dR

Synchronization2010722

TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )

63 Synchronization2010722

TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )

loz If drsquo is the correct symbol timing offset

( )2

2 2

0 1 12

j T f j T f j T fN

P d a e a e a eπ π πΔ Δ Δ

minus= + + +

22

2 2

0 1 12

j T fN

e a a aπ Δ

minus

⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭

loz If drsquo falls behind the correct symbol timing offset 1 sample

2⎪ ⎪⎩ ⎭

64 Synchronization2010722

TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )

65 Synchronization2010722

TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )

loz Drawback Plateau effect

Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)

((N2) ((N2) ((N2)

helliphellip

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)

Sliding window (length = N)

66 Synchronization2010722

TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )

loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula

( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus

扣掉相加

( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +

loz The method also called delay correlator

67 Synchronization2010722

IEEE 80211aIEEE 80211a

68 Synchronization2010722

Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )

loz Minnrsquos Method [2][ ]B B －B －B

( )( )

( )

22

2Minn

P dM d =

1 4 1N minus

( )( )( )2

2

MinnR d

( ) ( ) ( ) ( )

1 4 1

2 2 2 40 0

N

d N m k d N m k Nm k

P d r r+ + + + += =

= sum sum

( ) ( ) ( )2

21 4 1

2 40 0

N

d N m k Nk

R d rminus

+ + += sum sum69

0 0m k= =

Synchronization2010722

Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )

loz Parkrsquos Method [3][ ]

where D is the symmetric version of CC D C D

where D is the symmetric version of C

( )( ) 2

3k

P dM d =( )

( )( )23

ParkM dR d

2N

( ) 2

30

N

d k d kk

P d r rminus +=

= sdotsum

( )3

22

0

N

d kk

R d r += sum70

0k=

Synchronization2010722

Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )

loz Renrsquos Method [4][ ]

AS AS

where S is the PN sequence weighted of the original bl 2preamble

( )( )( )( )

24

Re 24

n

P dM d

R d=

( )( )4R d

( ) 2 1

4 2 2

N

k k N d k d k NP d s s r rminus

+ + + += sum0k=sum

( )21

41 N

d kR d rminus

= sum71

( )402 d k

kR d r +

=sum

Synchronization2010722

Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )

72 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

73 Synchronization2010722

Fractional CFO estimation Fractional CFO estimation

loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected

11 Rminus⎛ ⎞sum

h L i th di t b t t id ti l bl k R i

0

12 d m d m L

ms

f r rLTπ + + +

=

⎛ ⎞= ang⎜ ⎟⎝ ⎠sum

where L is the distance between two identical block R is the block size

loz For exampleNN cp

L = N R = NL = N R = Ncp74 Synchronization2010722

Estimation rangeEstimation rangegg

loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval

f ( ) ( )s s⎣ ⎦

loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)

loz DVB-T (L=N) 802 11 (L N4)

[ ]05 05s sf fminus

[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)

(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus

loz Note that is the subcarrier spacing1sf NT=

75

sNT

Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz

plusmng q y

loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn

76 Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz

plusmng q y

loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for

subcarrier spacing is required

plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn

p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs

77 Synchronization2010722

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

OFDM OverviewOFDM Overview

loz OFDMloz Orthogonal Frequency Division Multiplexing

loz Frequency Division Multiplexing (FDM) or multi-tone systems have been employed in military applicationssystems have been employed in military applications since the 196Os

loz OFDM employs multiple carriers overlapping in the f d ifrequency domain

4

OFDM OverviewOFDM Overview

Single carrier (SC) vs multi-carrier (MC)

loz Single carrier data are Multi-carrier data are shared transmitted over only one carrier

among several carriers and simultaneously transmittedFl t f di b iloz Selective fading Flat fading per subcarrier

5

OFDM OverviewOFDM Overview

loz OFDM modulation

loz Featuresloz No intercarrier guard bandsloz Overlapping of bands

S t l ffi iloz Spectral efficiencyloz Easy implementation by IFFTsloz Very sensitive to synchronizationloz Very sensitive to synchronization

6

Applications of OFDM TechnologyApplications of OFDM Technologypp gypp gy

loz Broadband Wired Access AsymmetricDigital Subscriber Loop (ADSL) Digital Multi-tone(DMT)

loz Wireless LANs (IEEE 80211ag IEEE 80211n HIPERLAN-2)

loz Digital Broadcasting (DAB DVB-T DVB-H)loz WiMAX (IEEE 80216 Series) 3GPP Long Term ( ) g

Evolution (3GPP LTE) 4Gloz Wireless Personal Area Network (WPAN) IEEE ( )

80215aMBOAloz Power Line

7

OFDM System ModelOFDM System Modelyy

loz Multi-carrier Block Transmission

frequency

time

8T8

OFDM System ModelOFDM System Modelyy

loz OFDM A block modulation scheme that transmitsa block N source symbols in parallel by using subcarrierssubcarriersloz Sub-carriers are orthogonal in time but overlapped in

frequencyeque cyloz Frequency spacing 1

FFT

fT

Δ =

cos(2 )cos(2 ( ) ) 0FFTTf t f f t dtπ π + Δint 1 10

cos(2 )cos(2 ( ) ) 0f t f f t dtπ π + Δ =int

9

OFDM System ModelOFDM System Model

tNΔ

yy

( )tf02cos π

( )0a

( )0atT

tNΔ1+

( )tf2i

( )

( )0b

T

DataEncoder

tf s Δ=

1 ( ) ( ) njbna +

( )nd SP MUX ( )tDChannelInput

( )tf02sin π

( )tf N 12cos minusπbullbullbull

( )1minusNa

( )1minusNb( )0a1+

( )tf N 12sin minusπ

( )1Nb

tT

( )1a ( )1minusNa1minus

tΔ

Figure 1tNΔ

( )1a ( )1minusNa Figure 1

10

OFDM System ModelOFDM System Model

loz An OFDM system transmitter shown in Figure 1

yy

y gloz The transmitted waveform D(t) can be expressed as

)1( )2sin()()2cos()()(1

0summinus

=

+=N

nnn tfnbtfnatD ππ

where tNffnffn Δ=ΔΔ+=

1 and 0

loz Using a two-dimensional digital modulation format the data

tNΔ

g g symbols d(n) can be represented as a(n) + jb(n)loz a(n) in-phase componentloz b(n) quadrature component

11

OFDM System ModelOFDM System Model

loz The serial data elements spaced by are grouped and tΔ

yy

p y g pused to modulate N carriers Thus they are frequency division multiplexedp

loz The signaling interval is then increased to which tNΔmakes the system less susceptible to channel delay spread impairments

Small-scale fading(Based on multipath time delay spread)

Flat Fading1 BW of signal lt BW of channel

Frequency Selective Fading1 BW of signal gt BW of channel

2 Delay spread lt Symbol preiod 2 Delay spread gt Symbol preiod

12

OrthogonalityOrthogonalityg yg y

loz Consider a set of transmitted carriers as follows

(2) 1 1 0for )( 2 0

minus==⎟⎠⎞

⎜⎝⎛

Δ+

Nnett

tNnfj

n

πψ n

⎨⎧ =minus

intqpab

dtttb for )(

)()( ψψ⎩⎨ Δ=minusne

=int tNabqpdttt

a qp )( and for 0 )()( ψψ

13

OrthogonalityOrthogonality

b tqpjb

intint)(2π

g yg y

dtedtttajbj

b

atN

qpjb

a qp = Δminus

intint )()(

)(2)(2

)(2ψψπ

tNqpjee tN

qpjtN

qpj

Δminusminus

=Δ

minusΔ

minus

)(2

)(2)(2

π

ππ

ee

tNqpjba

tNqpj

tNbqpj

⎟⎟⎞

⎜⎜⎛minus

Δ

minusΔ

minusΔ

minus1

)(2)(1)(2)(2

π

ππ

tNqpj Δminus

⎟⎠

⎜⎝=

)(2πtNabqp Δ=minusne= )( and for 0

14

OrthogonalityOrthogonalityg yg y

tNΔ

15

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFTby using IDFTby using IDFT

loz According to the structure of Tx it must use Ng oscillators That increases the hardware complexity

loz The equivalent method is using IDFT (IFFT)

16

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

loz In general each carrier can be expressed as

by using IDFTby using IDFT

g p

( ) (6) )()( )(2 ttfjcc

ccetAtS φπ +=

loz We assume that there are N carriers in the OFDM signal Th th t t l l i l S (t) b t d bThen the total complex signal Ss(t) can be represented by

( ) (7))(1)(1

)(2etAtSN

ttfj nn φπ= summinus

+( )

where

(7) )()(

0

0

fnff

etAN

tSn

ns

Δ+=

= sum=

liihffrequency carrier phase amplitude are )( )( and

where 0

fttAfnff

nnn

n

φΔ+

lyrespective carrierth -of n17

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

loz Then we sample the signal at a sampling frequency tΔ1

by using IDFTby using IDFT

p g p g q y and An(t) and φn(t) becomes

(8))(t = φφ(9) )((8) )(

nn

nn

AtAt

== φφ

( ) (10) 1)(1

0

)(2 0summinus

=

+ΔΔ+=ΔN

n

tkfnfjns

neAN

tkS φπ

loz Then the sampled signal can be expressed as0=nN

( )( ) (11) 1)(1

0

22 0summinus

=

ΔΔ+Δ sdot=ΔN

n

tfnkjtkfjns eeA

NtkS n πφπ

0n

18

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

loz The inverse discrete Fourier transform (IDFT) is defined

by using IDFTby using IDFT

( )as the following

1 21

NnkjN

πsumminus

(12) )(1)( 2

0

Nnkj

nefnF

Ntkf πsum

=

Δ=Δ

loz Comparing eq(11) and eq(12) the condition must be satisfied in order to make eq(11) an inverse Fouriersatisfied in order to make eq(11) an inverse Fourier transform relationship

1 (13) 1tN

fΔ

=Δ

19

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

loz If eq(13) is satisfied

by using IDFTby using IDFT

q ( ) loz is the frequency domain signal loz is the time domain signal

( )ntkfjneA φπ +Δ02

)( tkSs Δ gloz is the sub-channel spacingloz is the symbol duration in each sub-channel

)(s

fΔtNΔloz s e sy bo du o e c sub c e

loz This outcome is the same as the result obtained in the

N

loz This outcome is the same as the result obtained in the system of Figure 1 Therefore IDFT can be used to generate an OFDM transmission signalgenerate an OFDM transmission signal

20

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

( )tf02cos π

( )0a

by using IDFTby using IDFT

( )nd ( )tDI( )tf02sin π

( )0a

( )0b

bull

tf s Δ=

1 ( ) ( ) njbna +

( )nd

tNΔffnffn Δ

=Δ+=1 0

( )tDChannelInput

( )tf N 12cos minusπ

( )1minusNa

bullbull

tNΔ

( )tf N 12sin minusπ

( )1minusNb

( )0d

( )nd ( )tDChannelInput

bull bull

( )( )1d

( )2d

tf s Δ=

1 ( ) ( ) njbna +

fNΔffnffn Δ

=Δ+=1 0

ChannelInput

bullbullbullbull

bullbullbullbull

( )1minusNd

21

Frequency Error Results in ICIFrequency Error Results in ICIq yq y

22

Synchronization Error Results in ICISynchronization Error Results in ICIyy

Not Orthogonal Any More

23

Cyclic PrefixCyclic Prefixyy

loz In multipath channel delayed replicas of previous OFDM signal lead to ISI between successive OFDM signals

loz Solution Insert a guard interval between successive OFDMloz Solution Insert a guard interval between successive OFDM signals

24

Cyclic PrefixCyclic Prefix

loz Guard interval leads to intercarrier interference (ICI) in OFDM d d l i

yy

demodulation

loz In DFT interval difference between two subcarriers does not maintain integer number of cycles loss of orthogonality

loz Delayed version of subcarrier 2 causes ICI in the process of d d l ti b i 1demodulating subcarrier 1

25

Cyclic PrefixCyclic Prefixloz Cyclic prefix (CP) A copy of the last part of OFDM signal is

attached to the front of itself

yy

attached to the front of itself

[ ]0d

[ ]1d

[ ]0D[ ]1D

[ ]nd[ ]kD~

symbols dataInput

bullbullbullbull

[ ]1d

[ ]2d

[ ][ ]2D

bullbullbullbullbull

y

bullbullbullbullbull

[ ]1minusNd [ ]1minusND

[ ]gNND minus

bullbullbull

26

Cyclic PrefixCyclic Prefix

loz All delayed replicas of subcarriers always have an

yy

y p yinteger number of cycles within DFT interval no ICI

27

Cyclic PrefixCyclic Prefix

loz Linear convolution vs circular convolution

yy

N

28

Cyclic PrefixCyclic Prefix

loz Channel effect with cyclic prefix

yy

Signal after removed CP

29

Cyclic PrefixCyclic Prefix

loz Time-Domain Explanation

yy

p

30

Cyclic PrefixCyclic Prefixyy

31

Cyclic PrefixCyclic Prefix

loz Spectrum of channel response with length (smaller [ ]nh hL

yy

than )h

gN[ ] nhFFTHk =

loz Received complete OFDM signal[ ] [ ] [ ] 20 ~~ minus++leleotimes= hg LNNnnhnDnr

loz Received useful part [ ]nr[ ] [ ] [ ]nhnDnr Notimes=

where is N-point circular convolution (due to CP)loz Received symbol at k-th subcarrier

[ ] [ ] [ ]N

Notimes

[ ] [ ] [ ] kkNk HXnhnDFFTnrFFTY =otimes== kYX =rArr ldquoUseful property for OFDM system to reduce

kk H

X =rArrcomplexity of channel equalizationrdquo

32

Cyclic Cyclic Prefix (OFDM Prefix (OFDM Receiver)Receiver)yy (( ))

bull bull( )( )nr~Symbols

dataOutput

bullbullbull

bullbullbull( )nr

bullbull

bullbull

33

Cyclic PrefixCyclic Prefix

loz One of the most important reasons to do OFDM is the

yy

pefficient way it deals with multipath delay spread

loz To eliminate inter-symbol interference (ISI) almostloz To eliminate inter symbol interference (ISI) almost completely a guard time is introduced for each OFDM symbolsymbol(The guard time is chosen larger than the delay spread)

34

Bandwidth EfficiencyBandwidth Efficiencyyy

loz In a classical parallel system the channel is divided into p y N non-overlapping sub-channels to avoid inter-carrier interference (ICI)( )

loz The diagram for bandwidth efficiency of OFDM system is shown below

35

SummarySummaryyy

loz The advantage of the FFT-based OFDM system g yloz The use of IFFTFFT can reduce the computation complexityloz The orthogonality between the adjacent sub-carriers will make g y j

the use of transmission bandwidth more efficientloz The guard interval is used to resist the inter-symbol interference

(ISI)loz The main advantage of the OFDM transmission technique is its

high performance even in frequency selective channels

loz The drawbacks of the OFDM system loz It is highly vulnerable to synchronization errorsloz Peak to Average Power Ratio (PAPR) problems

36

SynchronizationSynchronization

Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offset (TO)loz Impact of timing offset (TO)loz Impact of carrier frequency offset (CFO)loz TO estimation algorithmloz CFO estimation algorithmg

38 Synchronization2010722

Introduction (13)Introduction (13)( )( )

loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods

loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead

loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the

cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of

computational complexity therefore it may be not be available in short-burst wireless communication

39 Synchronization2010722

Introduction (23)Introduction (23)( )( )

loz Transcevier

40 Synchronization2010722

Introduction (33)Introduction (33)( )( )

loz Synchronization erroryloz Symbol timing offset (TO)

loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)

loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect

41 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg

42 Synchronization2010722

Signal model (14)Signal model (14)g ( )g ( )

loz Time domain sample p1

2

0

1 Nj nk N

n kk

x X eN

πminus

minus= sum

Xk the frequency domain data at the k-th subcarrier0kN =

loz The multipath fading channel

( ) ( ) ( )1L

h t h t δminus

sum

loz Received time domain signal

( ) ( ) ( )0

l ll

h t h tτ τ δ τ τ=

= minussum

loz Received time-domain signal

( ) ( )1

L

n n ly h n l x w nminus

minus= +sum43

( ) ( )0l=sum

Synchronization2010722

Signal model (24)Signal model (24)g ( )g ( )

loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting

12

0

Nj kn N

k nn

Y y e πminus

minus

=

= sum

( )1 1

2

0

N Lj kn N

n l kn l

h n l x e Wπminus minus

minusminus

=

= +sumsum

loz The equation above could be expressed in matrix form

0n l

loz The equation above could be expressed in matrix form

= +Y HX W

44 Synchronization2010722

Signal model (34)Signal model (34)g ( )g ( )

H= = = = +Y Fy FCx FCF X HX W

loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal

y

H=H FCFcirculant matrix leading to be a diagonal matrix

=H FCF

( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0

h h L L hh h

⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥

( )( ) ( )

( ) ( )

1 10 1 0 0i

h Lh L h N L

⎢ ⎥⎢ ⎥minus minus⎢ ⎥

equiv= minus minus⎢ ⎥⎢ ⎥

C( ) ( )

( ) ( )

0 1 1 100

0 0 1 1 0

h L h N L

h N L L h N

⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

45

( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦

Synchronization2010722

Signal model (44)Signal model (44)g ( )g ( )

loz In a time-invariant channel the channel matrix H would be a diagonal matrix

k k k kY H X W= +

( )1 1

2

0 0

N Lj kl N

kn l

H h n l e πminus minus

minus

= =

= sumsum

loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI

k k k k kY H X I Wα= + +k k k k k

( ) ( )1 1 1

2 2 N N L

j m k Nj ml Nk mI X h n l e e ππ

minus minus minusminusminus⎛ ⎞

= ⎜ ⎟⎝ ⎠

sum sum sumICI

46

0 0 0m n lm k= = =ne

⎝ ⎠

Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

47 Synchronization2010722

Impact of timing offset Impact of timing offset p gp g

loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively

( )I f s

Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ

( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr

2 fj k Nk k k kY X H e Wπδminus= +

loz For integer part of timing offset k k k kY X H e W+

I sTδ ( ) 0g IN L δminus minus le leFFT

2 Ij k NY X H e Wπδminus= +

( ) ( ) 2 I sFFT

j f TI sh t T H f e π δδ minusminus rarr

48

k k k kY X H e W= +

Synchronization2010722

Impact of timing offset derivationImpact of timing offset derivationp gp g

49 Synchronization2010722

Discussion of symbol boundaryDiscussion of symbol boundaryy yy y

50 Synchronization2010722

Impact of timing offsetImpact of timing offsetp gp g

loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI

2 Ij k NIk k k k

NY X H e ISI W

Nπδδ minusminus

= + +

loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk

e e

p kloz Phase Shift can be resolved by Differential Encoding

Decoding or carrier recovery with pilots

51 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

52 Synchronization2010722

Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )

53 Synchronization2010722

Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )

( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s

i n t i N N T N T nTy y t e

= + + +=

( ) ( )( ) ( ) ( ) 12

sin

sin

g gI f f

I I

i N N N Nj jf N Ni k i k k

f

Y X H e eN

π ε ε π ε

ε ε

πεπε

+ + minus+

minus minus=⎛ ⎞⎜ ⎟

Faded signal attenuated and rotated by CFO

( )( ) ( ) ( ) ( )11 2

sin

sin g gI f I f

i N N N NN j j l kI f

NN

l k π ε ε π ε επ ε ε + + minusminus + + + minus

⎜ ⎟⎝ ⎠

+ + minussum

( )( )( )

( ) ( )2

0

sin

I f I f

I

j j l kI f N Ni l l

l l k I f

X H e el k

NN

π ε ε π ε ε

ε π ε ε

+ + +

= ne minus

+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠

sum

i k

N

W

⎜ ⎟⎝ ⎠

+ Inter-Carrier Interference (ICI)

54 Synchronization2010722

Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )

loz An easier representation of received signal with only p g yfractional CFO is

1N

( ) ( )1

0

0N

k k k l l kl l k

Y H X C H X C l k Wminus

= ne

= + minus +⎡ ⎤⎣ ⎦sum

where the ICI coefficient is given by

( )( )( ) ( )

sin 1exp 1fl k

C l k j l kπ ε

π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )

( )exp 1sin

f

f

C l k j l kNN l k

N

π επ ε

minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠

55 Synchronization2010722

Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )

loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality

loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift

56 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

57 Synchronization2010722

TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )

loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq

58 Synchronization2010722

TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )

loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below

loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and

rN2 can be expressed by

0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+

( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +

59

( )

Synchronization2010722

TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )

loz Without CFO the first half is identical to the second half (in time order)

loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below

( ) 12 in discretesft T f NT

NTφ π π π ε πε

⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟

⎝ ⎠

helliphellip

sNT⎝ ⎠

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)

60

Length=N2

Synchronization2010722

TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )

loz Received signal without CFOg0

0 0

jr a e θ=jθ 2 0

2 2 0

Nj j

N Nr a e a e

θ θ= =

loz Received signal with CFO0 0

0 0

j jr a e eθ φ= Extra phase rotation due to CFO0 0

2 2 20

2 2 0

N N Nj j jj

N Nr a e e a e e

θ φ φθ= =

due to CFO

Phase difference 2 2 0N N

( )2 02 20 2 0 0

Nj j T fN

r r a e a eφ φ πminus Δ= =

which contains the information about CFO

61

0 2 0 0Nr r a e a e

Synchronization2010722

TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )

loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be

loz Note that d is a time index corresponding to the first( ) ( ) 2 1

20

Nd m d m Nm

P d r rminus

+ + +== sum

loz Note that d is a time index corresponding to the first sample in a window of N samples

loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by

( )22 1

20

Nd m NR d rminus

+ += sumloz A timing metric can be defined as

))(()(

)( 2

2

dRdP

dM =

( ) 20 d m Nm + +=sum

62

))(( dR

Synchronization2010722

TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )

63 Synchronization2010722

TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )

loz If drsquo is the correct symbol timing offset

( )2

2 2

0 1 12

j T f j T f j T fN

P d a e a e a eπ π πΔ Δ Δ

minus= + + +

22

2 2

0 1 12

j T fN

e a a aπ Δ

minus

⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭

loz If drsquo falls behind the correct symbol timing offset 1 sample

2⎪ ⎪⎩ ⎭

64 Synchronization2010722

TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )

65 Synchronization2010722

TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )

loz Drawback Plateau effect

Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)

((N2) ((N2) ((N2)

helliphellip

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)

Sliding window (length = N)

66 Synchronization2010722

TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )

loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula

( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus

扣掉相加

( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +

loz The method also called delay correlator

67 Synchronization2010722

IEEE 80211aIEEE 80211a

68 Synchronization2010722

Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )

loz Minnrsquos Method [2][ ]B B －B －B

( )( )

( )

22

2Minn

P dM d =

1 4 1N minus

( )( )( )2

2

MinnR d

( ) ( ) ( ) ( )

1 4 1

2 2 2 40 0

N

d N m k d N m k Nm k

P d r r+ + + + += =

= sum sum

( ) ( ) ( )2

21 4 1

2 40 0

N

d N m k Nk

R d rminus

+ + += sum sum69

0 0m k= =

Synchronization2010722

Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )

loz Parkrsquos Method [3][ ]

where D is the symmetric version of CC D C D

where D is the symmetric version of C

( )( ) 2

3k

P dM d =( )

( )( )23

ParkM dR d

2N

( ) 2

30

N

d k d kk

P d r rminus +=

= sdotsum

( )3

22

0

N

d kk

R d r += sum70

0k=

Synchronization2010722

Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )

loz Renrsquos Method [4][ ]

AS AS

where S is the PN sequence weighted of the original bl 2preamble

( )( )( )( )

24

Re 24

n

P dM d

R d=

( )( )4R d

( ) 2 1

4 2 2

N

k k N d k d k NP d s s r rminus

+ + + += sum0k=sum

( )21

41 N

d kR d rminus

= sum71

( )402 d k

kR d r +

=sum

Synchronization2010722

Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )

72 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

73 Synchronization2010722

Fractional CFO estimation Fractional CFO estimation

loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected

11 Rminus⎛ ⎞sum

h L i th di t b t t id ti l bl k R i

0

12 d m d m L

ms

f r rLTπ + + +

=

⎛ ⎞= ang⎜ ⎟⎝ ⎠sum

where L is the distance between two identical block R is the block size

loz For exampleNN cp

L = N R = NL = N R = Ncp74 Synchronization2010722

Estimation rangeEstimation rangegg

loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval

f ( ) ( )s s⎣ ⎦

loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)

loz DVB-T (L=N) 802 11 (L N4)

[ ]05 05s sf fminus

[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)

(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus

loz Note that is the subcarrier spacing1sf NT=

75

sNT

Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz

plusmng q y

loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn

76 Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz

plusmng q y

loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for

subcarrier spacing is required

plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn

p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs

77 Synchronization2010722

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

OFDM OverviewOFDM Overview

Single carrier (SC) vs multi-carrier (MC)

loz Single carrier data are Multi-carrier data are shared transmitted over only one carrier

among several carriers and simultaneously transmittedFl t f di b iloz Selective fading Flat fading per subcarrier

5

OFDM OverviewOFDM Overview

loz OFDM modulation

loz Featuresloz No intercarrier guard bandsloz Overlapping of bands

S t l ffi iloz Spectral efficiencyloz Easy implementation by IFFTsloz Very sensitive to synchronizationloz Very sensitive to synchronization

6

Applications of OFDM TechnologyApplications of OFDM Technologypp gypp gy

loz Broadband Wired Access AsymmetricDigital Subscriber Loop (ADSL) Digital Multi-tone(DMT)

loz Wireless LANs (IEEE 80211ag IEEE 80211n HIPERLAN-2)

loz Digital Broadcasting (DAB DVB-T DVB-H)loz WiMAX (IEEE 80216 Series) 3GPP Long Term ( ) g

Evolution (3GPP LTE) 4Gloz Wireless Personal Area Network (WPAN) IEEE ( )

80215aMBOAloz Power Line

7

OFDM System ModelOFDM System Modelyy

loz Multi-carrier Block Transmission

frequency

time

8T8

OFDM System ModelOFDM System Modelyy

loz OFDM A block modulation scheme that transmitsa block N source symbols in parallel by using subcarrierssubcarriersloz Sub-carriers are orthogonal in time but overlapped in

frequencyeque cyloz Frequency spacing 1

FFT

fT

Δ =

cos(2 )cos(2 ( ) ) 0FFTTf t f f t dtπ π + Δint 1 10

cos(2 )cos(2 ( ) ) 0f t f f t dtπ π + Δ =int

9

OFDM System ModelOFDM System Model

tNΔ

yy

( )tf02cos π

( )0a

( )0atT

tNΔ1+

( )tf2i

( )

( )0b

T

DataEncoder

tf s Δ=

1 ( ) ( ) njbna +

( )nd SP MUX ( )tDChannelInput

( )tf02sin π

( )tf N 12cos minusπbullbullbull

( )1minusNa

( )1minusNb( )0a1+

( )tf N 12sin minusπ

( )1Nb

tT

( )1a ( )1minusNa1minus

tΔ

Figure 1tNΔ

( )1a ( )1minusNa Figure 1

10

OFDM System ModelOFDM System Model

loz An OFDM system transmitter shown in Figure 1

yy

y gloz The transmitted waveform D(t) can be expressed as

)1( )2sin()()2cos()()(1

0summinus

=

+=N

nnn tfnbtfnatD ππ

where tNffnffn Δ=ΔΔ+=

1 and 0

loz Using a two-dimensional digital modulation format the data

tNΔ

g g symbols d(n) can be represented as a(n) + jb(n)loz a(n) in-phase componentloz b(n) quadrature component

11

OFDM System ModelOFDM System Model

loz The serial data elements spaced by are grouped and tΔ

yy

p y g pused to modulate N carriers Thus they are frequency division multiplexedp

loz The signaling interval is then increased to which tNΔmakes the system less susceptible to channel delay spread impairments

Small-scale fading(Based on multipath time delay spread)

Flat Fading1 BW of signal lt BW of channel

Frequency Selective Fading1 BW of signal gt BW of channel

2 Delay spread lt Symbol preiod 2 Delay spread gt Symbol preiod

12

OrthogonalityOrthogonalityg yg y

loz Consider a set of transmitted carriers as follows

(2) 1 1 0for )( 2 0

minus==⎟⎠⎞

⎜⎝⎛

Δ+

Nnett

tNnfj

n

πψ n

⎨⎧ =minus

intqpab

dtttb for )(

)()( ψψ⎩⎨ Δ=minusne

=int tNabqpdttt

a qp )( and for 0 )()( ψψ

13

OrthogonalityOrthogonality

b tqpjb

intint)(2π

g yg y

dtedtttajbj

b

atN

qpjb

a qp = Δminus

intint )()(

)(2)(2

)(2ψψπ

tNqpjee tN

qpjtN

qpj

Δminusminus

=Δ

minusΔ

minus

)(2

)(2)(2

π

ππ

ee

tNqpjba

tNqpj

tNbqpj

⎟⎟⎞

⎜⎜⎛minus

Δ

minusΔ

minusΔ

minus1

)(2)(1)(2)(2

π

ππ

tNqpj Δminus

⎟⎠

⎜⎝=

)(2πtNabqp Δ=minusne= )( and for 0

14

OrthogonalityOrthogonalityg yg y

tNΔ

15

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFTby using IDFTby using IDFT

loz According to the structure of Tx it must use Ng oscillators That increases the hardware complexity

loz The equivalent method is using IDFT (IFFT)

16

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

loz In general each carrier can be expressed as

by using IDFTby using IDFT

g p

( ) (6) )()( )(2 ttfjcc

ccetAtS φπ +=

loz We assume that there are N carriers in the OFDM signal Th th t t l l i l S (t) b t d bThen the total complex signal Ss(t) can be represented by

( ) (7))(1)(1

)(2etAtSN

ttfj nn φπ= summinus

+( )

where

(7) )()(

0

0

fnff

etAN

tSn

ns

Δ+=

= sum=

liihffrequency carrier phase amplitude are )( )( and

where 0

fttAfnff

nnn

n

φΔ+

lyrespective carrierth -of n17

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

loz Then we sample the signal at a sampling frequency tΔ1

by using IDFTby using IDFT

p g p g q y and An(t) and φn(t) becomes

(8))(t = φφ(9) )((8) )(

nn

nn

AtAt

== φφ

( ) (10) 1)(1

0

)(2 0summinus

=

+ΔΔ+=ΔN

n

tkfnfjns

neAN

tkS φπ

loz Then the sampled signal can be expressed as0=nN

( )( ) (11) 1)(1

0

22 0summinus

=

ΔΔ+Δ sdot=ΔN

n

tfnkjtkfjns eeA

NtkS n πφπ

0n

18

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

loz The inverse discrete Fourier transform (IDFT) is defined

by using IDFTby using IDFT

( )as the following

1 21

NnkjN

πsumminus

(12) )(1)( 2

0

Nnkj

nefnF

Ntkf πsum

=

Δ=Δ

loz Comparing eq(11) and eq(12) the condition must be satisfied in order to make eq(11) an inverse Fouriersatisfied in order to make eq(11) an inverse Fourier transform relationship

1 (13) 1tN

fΔ

=Δ

19

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

loz If eq(13) is satisfied

by using IDFTby using IDFT

q ( ) loz is the frequency domain signal loz is the time domain signal

( )ntkfjneA φπ +Δ02

)( tkSs Δ gloz is the sub-channel spacingloz is the symbol duration in each sub-channel

)(s

fΔtNΔloz s e sy bo du o e c sub c e

loz This outcome is the same as the result obtained in the

N

loz This outcome is the same as the result obtained in the system of Figure 1 Therefore IDFT can be used to generate an OFDM transmission signalgenerate an OFDM transmission signal

20

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

( )tf02cos π

( )0a

by using IDFTby using IDFT

( )nd ( )tDI( )tf02sin π

( )0a

( )0b

bull

tf s Δ=

1 ( ) ( ) njbna +

( )nd

tNΔffnffn Δ

=Δ+=1 0

( )tDChannelInput

( )tf N 12cos minusπ

( )1minusNa

bullbull

tNΔ

( )tf N 12sin minusπ

( )1minusNb

( )0d

( )nd ( )tDChannelInput

bull bull

( )( )1d

( )2d

tf s Δ=

1 ( ) ( ) njbna +

fNΔffnffn Δ

=Δ+=1 0

ChannelInput

bullbullbullbull

bullbullbullbull

( )1minusNd

21

Frequency Error Results in ICIFrequency Error Results in ICIq yq y

22

Synchronization Error Results in ICISynchronization Error Results in ICIyy

Not Orthogonal Any More

23

Cyclic PrefixCyclic Prefixyy

loz In multipath channel delayed replicas of previous OFDM signal lead to ISI between successive OFDM signals

loz Solution Insert a guard interval between successive OFDMloz Solution Insert a guard interval between successive OFDM signals

24

Cyclic PrefixCyclic Prefix

loz Guard interval leads to intercarrier interference (ICI) in OFDM d d l i

yy

demodulation

loz In DFT interval difference between two subcarriers does not maintain integer number of cycles loss of orthogonality

loz Delayed version of subcarrier 2 causes ICI in the process of d d l ti b i 1demodulating subcarrier 1

25

Cyclic PrefixCyclic Prefixloz Cyclic prefix (CP) A copy of the last part of OFDM signal is

attached to the front of itself

yy

attached to the front of itself

[ ]0d

[ ]1d

[ ]0D[ ]1D

[ ]nd[ ]kD~

symbols dataInput

bullbullbullbull

[ ]1d

[ ]2d

[ ][ ]2D

bullbullbullbullbull

y

bullbullbullbullbull

[ ]1minusNd [ ]1minusND

[ ]gNND minus

bullbullbull

26

Cyclic PrefixCyclic Prefix

loz All delayed replicas of subcarriers always have an

yy

y p yinteger number of cycles within DFT interval no ICI

27

Cyclic PrefixCyclic Prefix

loz Linear convolution vs circular convolution

yy

N

28

Cyclic PrefixCyclic Prefix

loz Channel effect with cyclic prefix

yy

Signal after removed CP

29

Cyclic PrefixCyclic Prefix

loz Time-Domain Explanation

yy

p

30

Cyclic PrefixCyclic Prefixyy

31

Cyclic PrefixCyclic Prefix

loz Spectrum of channel response with length (smaller [ ]nh hL

yy

than )h

gN[ ] nhFFTHk =

loz Received complete OFDM signal[ ] [ ] [ ] 20 ~~ minus++leleotimes= hg LNNnnhnDnr

loz Received useful part [ ]nr[ ] [ ] [ ]nhnDnr Notimes=

where is N-point circular convolution (due to CP)loz Received symbol at k-th subcarrier

[ ] [ ] [ ]N

Notimes

[ ] [ ] [ ] kkNk HXnhnDFFTnrFFTY =otimes== kYX =rArr ldquoUseful property for OFDM system to reduce

kk H

X =rArrcomplexity of channel equalizationrdquo

32

Cyclic Cyclic Prefix (OFDM Prefix (OFDM Receiver)Receiver)yy (( ))

bull bull( )( )nr~Symbols

dataOutput

bullbullbull

bullbullbull( )nr

bullbull

bullbull

33

Cyclic PrefixCyclic Prefix

loz One of the most important reasons to do OFDM is the

yy

pefficient way it deals with multipath delay spread

loz To eliminate inter-symbol interference (ISI) almostloz To eliminate inter symbol interference (ISI) almost completely a guard time is introduced for each OFDM symbolsymbol(The guard time is chosen larger than the delay spread)

34

Bandwidth EfficiencyBandwidth Efficiencyyy

loz In a classical parallel system the channel is divided into p y N non-overlapping sub-channels to avoid inter-carrier interference (ICI)( )

loz The diagram for bandwidth efficiency of OFDM system is shown below

35

SummarySummaryyy

loz The advantage of the FFT-based OFDM system g yloz The use of IFFTFFT can reduce the computation complexityloz The orthogonality between the adjacent sub-carriers will make g y j

the use of transmission bandwidth more efficientloz The guard interval is used to resist the inter-symbol interference

(ISI)loz The main advantage of the OFDM transmission technique is its

high performance even in frequency selective channels

loz The drawbacks of the OFDM system loz It is highly vulnerable to synchronization errorsloz Peak to Average Power Ratio (PAPR) problems

36

SynchronizationSynchronization

Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offset (TO)loz Impact of timing offset (TO)loz Impact of carrier frequency offset (CFO)loz TO estimation algorithmloz CFO estimation algorithmg

38 Synchronization2010722

Introduction (13)Introduction (13)( )( )

loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods

loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead

loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the

cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of

computational complexity therefore it may be not be available in short-burst wireless communication

39 Synchronization2010722

Introduction (23)Introduction (23)( )( )

loz Transcevier

40 Synchronization2010722

Introduction (33)Introduction (33)( )( )

loz Synchronization erroryloz Symbol timing offset (TO)

loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)

loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect

41 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg

42 Synchronization2010722

Signal model (14)Signal model (14)g ( )g ( )

loz Time domain sample p1

2

0

1 Nj nk N

n kk

x X eN

πminus

minus= sum

Xk the frequency domain data at the k-th subcarrier0kN =

loz The multipath fading channel

( ) ( ) ( )1L

h t h t δminus

sum

loz Received time domain signal

( ) ( ) ( )0

l ll

h t h tτ τ δ τ τ=

= minussum

loz Received time-domain signal

( ) ( )1

L

n n ly h n l x w nminus

minus= +sum43

( ) ( )0l=sum

Synchronization2010722

Signal model (24)Signal model (24)g ( )g ( )

loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting

12

0

Nj kn N

k nn

Y y e πminus

minus

=

= sum

( )1 1

2

0

N Lj kn N

n l kn l

h n l x e Wπminus minus

minusminus

=

= +sumsum

loz The equation above could be expressed in matrix form

0n l

loz The equation above could be expressed in matrix form

= +Y HX W

44 Synchronization2010722

Signal model (34)Signal model (34)g ( )g ( )

H= = = = +Y Fy FCx FCF X HX W

loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal

y

H=H FCFcirculant matrix leading to be a diagonal matrix

=H FCF

( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0

h h L L hh h

⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥

( )( ) ( )

( ) ( )

1 10 1 0 0i

h Lh L h N L

⎢ ⎥⎢ ⎥minus minus⎢ ⎥

equiv= minus minus⎢ ⎥⎢ ⎥

C( ) ( )

( ) ( )

0 1 1 100

0 0 1 1 0

h L h N L

h N L L h N

⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

45

( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦

Synchronization2010722

Signal model (44)Signal model (44)g ( )g ( )

loz In a time-invariant channel the channel matrix H would be a diagonal matrix

k k k kY H X W= +

( )1 1

2

0 0

N Lj kl N

kn l

H h n l e πminus minus

minus

= =

= sumsum

loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI

k k k k kY H X I Wα= + +k k k k k

( ) ( )1 1 1

2 2 N N L

j m k Nj ml Nk mI X h n l e e ππ

minus minus minusminusminus⎛ ⎞

= ⎜ ⎟⎝ ⎠

sum sum sumICI

46

0 0 0m n lm k= = =ne

⎝ ⎠

Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

47 Synchronization2010722

Impact of timing offset Impact of timing offset p gp g

loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively

( )I f s

Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ

( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr

2 fj k Nk k k kY X H e Wπδminus= +

loz For integer part of timing offset k k k kY X H e W+

I sTδ ( ) 0g IN L δminus minus le leFFT

2 Ij k NY X H e Wπδminus= +

( ) ( ) 2 I sFFT

j f TI sh t T H f e π δδ minusminus rarr

48

k k k kY X H e W= +

Synchronization2010722

Impact of timing offset derivationImpact of timing offset derivationp gp g

49 Synchronization2010722

Discussion of symbol boundaryDiscussion of symbol boundaryy yy y

50 Synchronization2010722

Impact of timing offsetImpact of timing offsetp gp g

loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI

2 Ij k NIk k k k

NY X H e ISI W

Nπδδ minusminus

= + +

loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk

e e

p kloz Phase Shift can be resolved by Differential Encoding

Decoding or carrier recovery with pilots

51 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

52 Synchronization2010722

Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )

53 Synchronization2010722

Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )

( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s

i n t i N N T N T nTy y t e

= + + +=

( ) ( )( ) ( ) ( ) 12

sin

sin

g gI f f

I I

i N N N Nj jf N Ni k i k k

f

Y X H e eN

π ε ε π ε

ε ε

πεπε

+ + minus+

minus minus=⎛ ⎞⎜ ⎟

Faded signal attenuated and rotated by CFO

( )( ) ( ) ( ) ( )11 2

sin

sin g gI f I f

i N N N NN j j l kI f

NN

l k π ε ε π ε επ ε ε + + minusminus + + + minus

⎜ ⎟⎝ ⎠

+ + minussum

( )( )( )

( ) ( )2

0

sin

I f I f

I

j j l kI f N Ni l l

l l k I f

X H e el k

NN

π ε ε π ε ε

ε π ε ε

+ + +

= ne minus

+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠

sum

i k

N

W

⎜ ⎟⎝ ⎠

+ Inter-Carrier Interference (ICI)

54 Synchronization2010722

Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )

loz An easier representation of received signal with only p g yfractional CFO is

1N

( ) ( )1

0

0N

k k k l l kl l k

Y H X C H X C l k Wminus

= ne

= + minus +⎡ ⎤⎣ ⎦sum

where the ICI coefficient is given by

( )( )( ) ( )

sin 1exp 1fl k

C l k j l kπ ε

π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )

( )exp 1sin

f

f

C l k j l kNN l k

N

π επ ε

minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠

55 Synchronization2010722

Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )

loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality

loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift

56 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

57 Synchronization2010722

TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )

loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq

58 Synchronization2010722

TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )

loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below

loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and

rN2 can be expressed by

0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+

( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +

59

( )

Synchronization2010722

TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )

loz Without CFO the first half is identical to the second half (in time order)

loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below

( ) 12 in discretesft T f NT

NTφ π π π ε πε

⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟

⎝ ⎠

helliphellip

sNT⎝ ⎠

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)

60

Length=N2

Synchronization2010722

TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )

loz Received signal without CFOg0

0 0

jr a e θ=jθ 2 0

2 2 0

Nj j

N Nr a e a e

θ θ= =

loz Received signal with CFO0 0

0 0

j jr a e eθ φ= Extra phase rotation due to CFO0 0

2 2 20

2 2 0

N N Nj j jj

N Nr a e e a e e

θ φ φθ= =

due to CFO

Phase difference 2 2 0N N

( )2 02 20 2 0 0

Nj j T fN

r r a e a eφ φ πminus Δ= =

which contains the information about CFO

61

0 2 0 0Nr r a e a e

Synchronization2010722

TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )

loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be

loz Note that d is a time index corresponding to the first( ) ( ) 2 1

20

Nd m d m Nm

P d r rminus

+ + +== sum

loz Note that d is a time index corresponding to the first sample in a window of N samples

loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by

( )22 1

20

Nd m NR d rminus

+ += sumloz A timing metric can be defined as

))(()(

)( 2

2

dRdP

dM =

( ) 20 d m Nm + +=sum

62

))(( dR

Synchronization2010722

TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )

63 Synchronization2010722

TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )

loz If drsquo is the correct symbol timing offset

( )2

2 2

0 1 12

j T f j T f j T fN

P d a e a e a eπ π πΔ Δ Δ

minus= + + +

22

2 2

0 1 12

j T fN

e a a aπ Δ

minus

⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭

loz If drsquo falls behind the correct symbol timing offset 1 sample

2⎪ ⎪⎩ ⎭

64 Synchronization2010722

TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )

65 Synchronization2010722

TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )

loz Drawback Plateau effect

Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)

((N2) ((N2) ((N2)

helliphellip

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)

Sliding window (length = N)

66 Synchronization2010722

TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )

loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula

( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus

扣掉相加

( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +

loz The method also called delay correlator

67 Synchronization2010722

IEEE 80211aIEEE 80211a

68 Synchronization2010722

Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )

loz Minnrsquos Method [2][ ]B B －B －B

( )( )

( )

22

2Minn

P dM d =

1 4 1N minus

( )( )( )2

2

MinnR d

( ) ( ) ( ) ( )

1 4 1

2 2 2 40 0

N

d N m k d N m k Nm k

P d r r+ + + + += =

= sum sum

( ) ( ) ( )2

21 4 1

2 40 0

N

d N m k Nk

R d rminus

+ + += sum sum69

0 0m k= =

Synchronization2010722

Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )

loz Parkrsquos Method [3][ ]

where D is the symmetric version of CC D C D

where D is the symmetric version of C

( )( ) 2

3k

P dM d =( )

( )( )23

ParkM dR d

2N

( ) 2

30

N

d k d kk

P d r rminus +=

= sdotsum

( )3

22

0

N

d kk

R d r += sum70

0k=

Synchronization2010722

Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )

loz Renrsquos Method [4][ ]

AS AS

where S is the PN sequence weighted of the original bl 2preamble

( )( )( )( )

24

Re 24

n

P dM d

R d=

( )( )4R d

( ) 2 1

4 2 2

N

k k N d k d k NP d s s r rminus

+ + + += sum0k=sum

( )21

41 N

d kR d rminus

= sum71

( )402 d k

kR d r +

=sum

Synchronization2010722

Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )

72 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

73 Synchronization2010722

Fractional CFO estimation Fractional CFO estimation

loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected

11 Rminus⎛ ⎞sum

h L i th di t b t t id ti l bl k R i

0

12 d m d m L

ms

f r rLTπ + + +

=

⎛ ⎞= ang⎜ ⎟⎝ ⎠sum

where L is the distance between two identical block R is the block size

loz For exampleNN cp

L = N R = NL = N R = Ncp74 Synchronization2010722

Estimation rangeEstimation rangegg

loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval

f ( ) ( )s s⎣ ⎦

loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)

loz DVB-T (L=N) 802 11 (L N4)

[ ]05 05s sf fminus

[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)

(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus

loz Note that is the subcarrier spacing1sf NT=

75

sNT

Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz

plusmng q y

loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn

76 Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz

plusmng q y

loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for

subcarrier spacing is required

plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn

p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs

77 Synchronization2010722

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

OFDM OverviewOFDM Overview

loz OFDM modulation

loz Featuresloz No intercarrier guard bandsloz Overlapping of bands

S t l ffi iloz Spectral efficiencyloz Easy implementation by IFFTsloz Very sensitive to synchronizationloz Very sensitive to synchronization

6

Applications of OFDM TechnologyApplications of OFDM Technologypp gypp gy

loz Broadband Wired Access AsymmetricDigital Subscriber Loop (ADSL) Digital Multi-tone(DMT)

loz Wireless LANs (IEEE 80211ag IEEE 80211n HIPERLAN-2)

loz Digital Broadcasting (DAB DVB-T DVB-H)loz WiMAX (IEEE 80216 Series) 3GPP Long Term ( ) g

Evolution (3GPP LTE) 4Gloz Wireless Personal Area Network (WPAN) IEEE ( )

80215aMBOAloz Power Line

7

OFDM System ModelOFDM System Modelyy

loz Multi-carrier Block Transmission

frequency

time

8T8

OFDM System ModelOFDM System Modelyy

loz OFDM A block modulation scheme that transmitsa block N source symbols in parallel by using subcarrierssubcarriersloz Sub-carriers are orthogonal in time but overlapped in

frequencyeque cyloz Frequency spacing 1

FFT

fT

Δ =

cos(2 )cos(2 ( ) ) 0FFTTf t f f t dtπ π + Δint 1 10

cos(2 )cos(2 ( ) ) 0f t f f t dtπ π + Δ =int

9

OFDM System ModelOFDM System Model

tNΔ

yy

( )tf02cos π

( )0a

( )0atT

tNΔ1+

( )tf2i

( )

( )0b

T

DataEncoder

tf s Δ=

1 ( ) ( ) njbna +

( )nd SP MUX ( )tDChannelInput

( )tf02sin π

( )tf N 12cos minusπbullbullbull

( )1minusNa

( )1minusNb( )0a1+

( )tf N 12sin minusπ

( )1Nb

tT

( )1a ( )1minusNa1minus

tΔ

Figure 1tNΔ

( )1a ( )1minusNa Figure 1

10

OFDM System ModelOFDM System Model

loz An OFDM system transmitter shown in Figure 1

yy

y gloz The transmitted waveform D(t) can be expressed as

)1( )2sin()()2cos()()(1

0summinus

=

+=N

nnn tfnbtfnatD ππ

where tNffnffn Δ=ΔΔ+=

1 and 0

loz Using a two-dimensional digital modulation format the data

tNΔ

g g symbols d(n) can be represented as a(n) + jb(n)loz a(n) in-phase componentloz b(n) quadrature component

11

OFDM System ModelOFDM System Model

loz The serial data elements spaced by are grouped and tΔ

yy

p y g pused to modulate N carriers Thus they are frequency division multiplexedp

loz The signaling interval is then increased to which tNΔmakes the system less susceptible to channel delay spread impairments

Small-scale fading(Based on multipath time delay spread)

Flat Fading1 BW of signal lt BW of channel

Frequency Selective Fading1 BW of signal gt BW of channel

2 Delay spread lt Symbol preiod 2 Delay spread gt Symbol preiod

12

OrthogonalityOrthogonalityg yg y

loz Consider a set of transmitted carriers as follows

(2) 1 1 0for )( 2 0

minus==⎟⎠⎞

⎜⎝⎛

Δ+

Nnett

tNnfj

n

πψ n

⎨⎧ =minus

intqpab

dtttb for )(

)()( ψψ⎩⎨ Δ=minusne

=int tNabqpdttt

a qp )( and for 0 )()( ψψ

13

OrthogonalityOrthogonality

b tqpjb

intint)(2π

g yg y

dtedtttajbj

b

atN

qpjb

a qp = Δminus

intint )()(

)(2)(2

)(2ψψπ

tNqpjee tN

qpjtN

qpj

Δminusminus

=Δ

minusΔ

minus

)(2

)(2)(2

π

ππ

ee

tNqpjba

tNqpj

tNbqpj

⎟⎟⎞

⎜⎜⎛minus

Δ

minusΔ

minusΔ

minus1

)(2)(1)(2)(2

π

ππ

tNqpj Δminus

⎟⎠

⎜⎝=

)(2πtNabqp Δ=minusne= )( and for 0

14

OrthogonalityOrthogonalityg yg y

tNΔ

15

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFTby using IDFTby using IDFT

loz According to the structure of Tx it must use Ng oscillators That increases the hardware complexity

loz The equivalent method is using IDFT (IFFT)

16

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

loz In general each carrier can be expressed as

by using IDFTby using IDFT

g p

( ) (6) )()( )(2 ttfjcc

ccetAtS φπ +=

loz We assume that there are N carriers in the OFDM signal Th th t t l l i l S (t) b t d bThen the total complex signal Ss(t) can be represented by

( ) (7))(1)(1

)(2etAtSN

ttfj nn φπ= summinus

+( )

where

(7) )()(

0

0

fnff

etAN

tSn

ns

Δ+=

= sum=

liihffrequency carrier phase amplitude are )( )( and

where 0

fttAfnff

nnn

n

φΔ+

lyrespective carrierth -of n17

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

loz Then we sample the signal at a sampling frequency tΔ1

by using IDFTby using IDFT

p g p g q y and An(t) and φn(t) becomes

(8))(t = φφ(9) )((8) )(

nn

nn

AtAt

== φφ

( ) (10) 1)(1

0

)(2 0summinus

=

+ΔΔ+=ΔN

n

tkfnfjns

neAN

tkS φπ

loz Then the sampled signal can be expressed as0=nN

( )( ) (11) 1)(1

0

22 0summinus

=

ΔΔ+Δ sdot=ΔN

n

tfnkjtkfjns eeA

NtkS n πφπ

0n

18

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

loz The inverse discrete Fourier transform (IDFT) is defined

by using IDFTby using IDFT

( )as the following

1 21

NnkjN

πsumminus

(12) )(1)( 2

0

Nnkj

nefnF

Ntkf πsum

=

Δ=Δ

loz Comparing eq(11) and eq(12) the condition must be satisfied in order to make eq(11) an inverse Fouriersatisfied in order to make eq(11) an inverse Fourier transform relationship

1 (13) 1tN

fΔ

=Δ

19

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

loz If eq(13) is satisfied

by using IDFTby using IDFT

q ( ) loz is the frequency domain signal loz is the time domain signal

( )ntkfjneA φπ +Δ02

)( tkSs Δ gloz is the sub-channel spacingloz is the symbol duration in each sub-channel

)(s

fΔtNΔloz s e sy bo du o e c sub c e

loz This outcome is the same as the result obtained in the

N

loz This outcome is the same as the result obtained in the system of Figure 1 Therefore IDFT can be used to generate an OFDM transmission signalgenerate an OFDM transmission signal

20

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

( )tf02cos π

( )0a

by using IDFTby using IDFT

( )nd ( )tDI( )tf02sin π

( )0a

( )0b

bull

tf s Δ=

1 ( ) ( ) njbna +

( )nd

tNΔffnffn Δ

=Δ+=1 0

( )tDChannelInput

( )tf N 12cos minusπ

( )1minusNa

bullbull

tNΔ

( )tf N 12sin minusπ

( )1minusNb

( )0d

( )nd ( )tDChannelInput

bull bull

( )( )1d

( )2d

tf s Δ=

1 ( ) ( ) njbna +

fNΔffnffn Δ

=Δ+=1 0

ChannelInput

bullbullbullbull

bullbullbullbull

( )1minusNd

21

Frequency Error Results in ICIFrequency Error Results in ICIq yq y

22

Synchronization Error Results in ICISynchronization Error Results in ICIyy

Not Orthogonal Any More

23

Cyclic PrefixCyclic Prefixyy

loz In multipath channel delayed replicas of previous OFDM signal lead to ISI between successive OFDM signals

loz Solution Insert a guard interval between successive OFDMloz Solution Insert a guard interval between successive OFDM signals

24

Cyclic PrefixCyclic Prefix

loz Guard interval leads to intercarrier interference (ICI) in OFDM d d l i

yy

demodulation

loz In DFT interval difference between two subcarriers does not maintain integer number of cycles loss of orthogonality

loz Delayed version of subcarrier 2 causes ICI in the process of d d l ti b i 1demodulating subcarrier 1

25

Cyclic PrefixCyclic Prefixloz Cyclic prefix (CP) A copy of the last part of OFDM signal is

attached to the front of itself

yy

attached to the front of itself

[ ]0d

[ ]1d

[ ]0D[ ]1D

[ ]nd[ ]kD~

symbols dataInput

bullbullbullbull

[ ]1d

[ ]2d

[ ][ ]2D

bullbullbullbullbull

y

bullbullbullbullbull

[ ]1minusNd [ ]1minusND

[ ]gNND minus

bullbullbull

26

Cyclic PrefixCyclic Prefix

loz All delayed replicas of subcarriers always have an

yy

y p yinteger number of cycles within DFT interval no ICI

27

Cyclic PrefixCyclic Prefix

loz Linear convolution vs circular convolution

yy

N

28

Cyclic PrefixCyclic Prefix

loz Channel effect with cyclic prefix

yy

Signal after removed CP

29

Cyclic PrefixCyclic Prefix

loz Time-Domain Explanation

yy

p

30

Cyclic PrefixCyclic Prefixyy

31

Cyclic PrefixCyclic Prefix

loz Spectrum of channel response with length (smaller [ ]nh hL

yy

than )h

gN[ ] nhFFTHk =

loz Received complete OFDM signal[ ] [ ] [ ] 20 ~~ minus++leleotimes= hg LNNnnhnDnr

loz Received useful part [ ]nr[ ] [ ] [ ]nhnDnr Notimes=

where is N-point circular convolution (due to CP)loz Received symbol at k-th subcarrier

[ ] [ ] [ ]N

Notimes

[ ] [ ] [ ] kkNk HXnhnDFFTnrFFTY =otimes== kYX =rArr ldquoUseful property for OFDM system to reduce

kk H

X =rArrcomplexity of channel equalizationrdquo

32

Cyclic Cyclic Prefix (OFDM Prefix (OFDM Receiver)Receiver)yy (( ))

bull bull( )( )nr~Symbols

dataOutput

bullbullbull

bullbullbull( )nr

bullbull

bullbull

33

Cyclic PrefixCyclic Prefix

loz One of the most important reasons to do OFDM is the

yy

pefficient way it deals with multipath delay spread

loz To eliminate inter-symbol interference (ISI) almostloz To eliminate inter symbol interference (ISI) almost completely a guard time is introduced for each OFDM symbolsymbol(The guard time is chosen larger than the delay spread)

34

Bandwidth EfficiencyBandwidth Efficiencyyy

loz In a classical parallel system the channel is divided into p y N non-overlapping sub-channels to avoid inter-carrier interference (ICI)( )

loz The diagram for bandwidth efficiency of OFDM system is shown below

35

SummarySummaryyy

loz The advantage of the FFT-based OFDM system g yloz The use of IFFTFFT can reduce the computation complexityloz The orthogonality between the adjacent sub-carriers will make g y j

the use of transmission bandwidth more efficientloz The guard interval is used to resist the inter-symbol interference

(ISI)loz The main advantage of the OFDM transmission technique is its

high performance even in frequency selective channels

loz The drawbacks of the OFDM system loz It is highly vulnerable to synchronization errorsloz Peak to Average Power Ratio (PAPR) problems

36

SynchronizationSynchronization

Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offset (TO)loz Impact of timing offset (TO)loz Impact of carrier frequency offset (CFO)loz TO estimation algorithmloz CFO estimation algorithmg

38 Synchronization2010722

Introduction (13)Introduction (13)( )( )

loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods

loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead

loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the

cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of

computational complexity therefore it may be not be available in short-burst wireless communication

39 Synchronization2010722

Introduction (23)Introduction (23)( )( )

loz Transcevier

40 Synchronization2010722

Introduction (33)Introduction (33)( )( )

loz Synchronization erroryloz Symbol timing offset (TO)

loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)

loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect

41 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg

42 Synchronization2010722

Signal model (14)Signal model (14)g ( )g ( )

loz Time domain sample p1

2

0

1 Nj nk N

n kk

x X eN

πminus

minus= sum

Xk the frequency domain data at the k-th subcarrier0kN =

loz The multipath fading channel

( ) ( ) ( )1L

h t h t δminus

sum

loz Received time domain signal

( ) ( ) ( )0

l ll

h t h tτ τ δ τ τ=

= minussum

loz Received time-domain signal

( ) ( )1

L

n n ly h n l x w nminus

minus= +sum43

( ) ( )0l=sum

Synchronization2010722

Signal model (24)Signal model (24)g ( )g ( )

loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting

12

0

Nj kn N

k nn

Y y e πminus

minus

=

= sum

( )1 1

2

0

N Lj kn N

n l kn l

h n l x e Wπminus minus

minusminus

=

= +sumsum

loz The equation above could be expressed in matrix form

0n l

loz The equation above could be expressed in matrix form

= +Y HX W

44 Synchronization2010722

Signal model (34)Signal model (34)g ( )g ( )

H= = = = +Y Fy FCx FCF X HX W

loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal

y

H=H FCFcirculant matrix leading to be a diagonal matrix

=H FCF

( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0

h h L L hh h

⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥

( )( ) ( )

( ) ( )

1 10 1 0 0i

h Lh L h N L

⎢ ⎥⎢ ⎥minus minus⎢ ⎥

equiv= minus minus⎢ ⎥⎢ ⎥

C( ) ( )

( ) ( )

0 1 1 100

0 0 1 1 0

h L h N L

h N L L h N

⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

45

( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦

Synchronization2010722

Signal model (44)Signal model (44)g ( )g ( )

loz In a time-invariant channel the channel matrix H would be a diagonal matrix

k k k kY H X W= +

( )1 1

2

0 0

N Lj kl N

kn l

H h n l e πminus minus

minus

= =

= sumsum

loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI

k k k k kY H X I Wα= + +k k k k k

( ) ( )1 1 1

2 2 N N L

j m k Nj ml Nk mI X h n l e e ππ

minus minus minusminusminus⎛ ⎞

= ⎜ ⎟⎝ ⎠

sum sum sumICI

46

0 0 0m n lm k= = =ne

⎝ ⎠

Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

47 Synchronization2010722

Impact of timing offset Impact of timing offset p gp g

loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively

( )I f s

Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ

( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr

2 fj k Nk k k kY X H e Wπδminus= +

loz For integer part of timing offset k k k kY X H e W+

I sTδ ( ) 0g IN L δminus minus le leFFT

2 Ij k NY X H e Wπδminus= +

( ) ( ) 2 I sFFT

j f TI sh t T H f e π δδ minusminus rarr

48

k k k kY X H e W= +

Synchronization2010722

Impact of timing offset derivationImpact of timing offset derivationp gp g

49 Synchronization2010722

Discussion of symbol boundaryDiscussion of symbol boundaryy yy y

50 Synchronization2010722

Impact of timing offsetImpact of timing offsetp gp g

loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI

2 Ij k NIk k k k

NY X H e ISI W

Nπδδ minusminus

= + +

loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk

e e

p kloz Phase Shift can be resolved by Differential Encoding

Decoding or carrier recovery with pilots

51 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

52 Synchronization2010722

Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )

53 Synchronization2010722

Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )

( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s

i n t i N N T N T nTy y t e

= + + +=

( ) ( )( ) ( ) ( ) 12

sin

sin

g gI f f

I I

i N N N Nj jf N Ni k i k k

f

Y X H e eN

π ε ε π ε

ε ε

πεπε

+ + minus+

minus minus=⎛ ⎞⎜ ⎟

Faded signal attenuated and rotated by CFO

( )( ) ( ) ( ) ( )11 2

sin

sin g gI f I f

i N N N NN j j l kI f

NN

l k π ε ε π ε επ ε ε + + minusminus + + + minus

⎜ ⎟⎝ ⎠

+ + minussum

( )( )( )

( ) ( )2

0

sin

I f I f

I

j j l kI f N Ni l l

l l k I f

X H e el k

NN

π ε ε π ε ε

ε π ε ε

+ + +

= ne minus

+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠

sum

i k

N

W

⎜ ⎟⎝ ⎠

+ Inter-Carrier Interference (ICI)

54 Synchronization2010722

Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )

loz An easier representation of received signal with only p g yfractional CFO is

1N

( ) ( )1

0

0N

k k k l l kl l k

Y H X C H X C l k Wminus

= ne

= + minus +⎡ ⎤⎣ ⎦sum

where the ICI coefficient is given by

( )( )( ) ( )

sin 1exp 1fl k

C l k j l kπ ε

π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )

( )exp 1sin

f

f

C l k j l kNN l k

N

π επ ε

minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠

55 Synchronization2010722

Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )

loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality

loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift

56 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

57 Synchronization2010722

TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )

loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq

58 Synchronization2010722

TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )

loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below

loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and

rN2 can be expressed by

0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+

( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +

59

( )

Synchronization2010722

TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )

loz Without CFO the first half is identical to the second half (in time order)

loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below

( ) 12 in discretesft T f NT

NTφ π π π ε πε

⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟

⎝ ⎠

helliphellip

sNT⎝ ⎠

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)

60

Length=N2

Synchronization2010722

TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )

loz Received signal without CFOg0

0 0

jr a e θ=jθ 2 0

2 2 0

Nj j

N Nr a e a e

θ θ= =

loz Received signal with CFO0 0

0 0

j jr a e eθ φ= Extra phase rotation due to CFO0 0

2 2 20

2 2 0

N N Nj j jj

N Nr a e e a e e

θ φ φθ= =

due to CFO

Phase difference 2 2 0N N

( )2 02 20 2 0 0

Nj j T fN

r r a e a eφ φ πminus Δ= =

which contains the information about CFO

61

0 2 0 0Nr r a e a e

Synchronization2010722

TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )

loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be

loz Note that d is a time index corresponding to the first( ) ( ) 2 1

20

Nd m d m Nm

P d r rminus

+ + +== sum

loz Note that d is a time index corresponding to the first sample in a window of N samples

loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by

( )22 1

20

Nd m NR d rminus

+ += sumloz A timing metric can be defined as

))(()(

)( 2

2

dRdP

dM =

( ) 20 d m Nm + +=sum

62

))(( dR

Synchronization2010722

TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )

63 Synchronization2010722

TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )

loz If drsquo is the correct symbol timing offset

( )2

2 2

0 1 12

j T f j T f j T fN

P d a e a e a eπ π πΔ Δ Δ

minus= + + +

22

2 2

0 1 12

j T fN

e a a aπ Δ

minus

⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭

loz If drsquo falls behind the correct symbol timing offset 1 sample

2⎪ ⎪⎩ ⎭

64 Synchronization2010722

TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )

65 Synchronization2010722

TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )

loz Drawback Plateau effect

Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)

((N2) ((N2) ((N2)

helliphellip

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)

Sliding window (length = N)

66 Synchronization2010722

TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )

loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula

( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus

扣掉相加

( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +

loz The method also called delay correlator

67 Synchronization2010722

IEEE 80211aIEEE 80211a

68 Synchronization2010722

Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )

loz Minnrsquos Method [2][ ]B B －B －B

( )( )

( )

22

2Minn

P dM d =

1 4 1N minus

( )( )( )2

2

MinnR d

( ) ( ) ( ) ( )

1 4 1

2 2 2 40 0

N

d N m k d N m k Nm k

P d r r+ + + + += =

= sum sum

( ) ( ) ( )2

21 4 1

2 40 0

N

d N m k Nk

R d rminus

+ + += sum sum69

0 0m k= =

Synchronization2010722

Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )

loz Parkrsquos Method [3][ ]

where D is the symmetric version of CC D C D

where D is the symmetric version of C

( )( ) 2

3k

P dM d =( )

( )( )23

ParkM dR d

2N

( ) 2

30

N

d k d kk

P d r rminus +=

= sdotsum

( )3

22

0

N

d kk

R d r += sum70

0k=

Synchronization2010722

Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )

loz Renrsquos Method [4][ ]

AS AS

where S is the PN sequence weighted of the original bl 2preamble

( )( )( )( )

24

Re 24

n

P dM d

R d=

( )( )4R d

( ) 2 1

4 2 2

N

k k N d k d k NP d s s r rminus

+ + + += sum0k=sum

( )21

41 N

d kR d rminus

= sum71

( )402 d k

kR d r +

=sum

Synchronization2010722

Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )

72 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

73 Synchronization2010722

Fractional CFO estimation Fractional CFO estimation

loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected

11 Rminus⎛ ⎞sum

h L i th di t b t t id ti l bl k R i

0

12 d m d m L

ms

f r rLTπ + + +

=

⎛ ⎞= ang⎜ ⎟⎝ ⎠sum

where L is the distance between two identical block R is the block size

loz For exampleNN cp

L = N R = NL = N R = Ncp74 Synchronization2010722

Estimation rangeEstimation rangegg

loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval

f ( ) ( )s s⎣ ⎦

loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)

loz DVB-T (L=N) 802 11 (L N4)

[ ]05 05s sf fminus

[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)

(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus

loz Note that is the subcarrier spacing1sf NT=

75

sNT

Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz

plusmng q y

loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn

76 Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz

plusmng q y

loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for

subcarrier spacing is required

plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn

p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs

77 Synchronization2010722

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

Applications of OFDM TechnologyApplications of OFDM Technologypp gypp gy

loz Broadband Wired Access AsymmetricDigital Subscriber Loop (ADSL) Digital Multi-tone(DMT)

loz Wireless LANs (IEEE 80211ag IEEE 80211n HIPERLAN-2)

loz Digital Broadcasting (DAB DVB-T DVB-H)loz WiMAX (IEEE 80216 Series) 3GPP Long Term ( ) g

Evolution (3GPP LTE) 4Gloz Wireless Personal Area Network (WPAN) IEEE ( )

80215aMBOAloz Power Line

7

OFDM System ModelOFDM System Modelyy

loz Multi-carrier Block Transmission

frequency

time

8T8

OFDM System ModelOFDM System Modelyy

loz OFDM A block modulation scheme that transmitsa block N source symbols in parallel by using subcarrierssubcarriersloz Sub-carriers are orthogonal in time but overlapped in

frequencyeque cyloz Frequency spacing 1

FFT

fT

Δ =

cos(2 )cos(2 ( ) ) 0FFTTf t f f t dtπ π + Δint 1 10

cos(2 )cos(2 ( ) ) 0f t f f t dtπ π + Δ =int

9

OFDM System ModelOFDM System Model

tNΔ

yy

( )tf02cos π

( )0a

( )0atT

tNΔ1+

( )tf2i

( )

( )0b

T

DataEncoder

tf s Δ=

1 ( ) ( ) njbna +

( )nd SP MUX ( )tDChannelInput

( )tf02sin π

( )tf N 12cos minusπbullbullbull

( )1minusNa

( )1minusNb( )0a1+

( )tf N 12sin minusπ

( )1Nb

tT

( )1a ( )1minusNa1minus

tΔ

Figure 1tNΔ

( )1a ( )1minusNa Figure 1

10

OFDM System ModelOFDM System Model

loz An OFDM system transmitter shown in Figure 1

yy

y gloz The transmitted waveform D(t) can be expressed as

)1( )2sin()()2cos()()(1

0summinus

=

+=N

nnn tfnbtfnatD ππ

where tNffnffn Δ=ΔΔ+=

1 and 0

loz Using a two-dimensional digital modulation format the data

tNΔ

g g symbols d(n) can be represented as a(n) + jb(n)loz a(n) in-phase componentloz b(n) quadrature component

11

OFDM System ModelOFDM System Model

loz The serial data elements spaced by are grouped and tΔ

yy

p y g pused to modulate N carriers Thus they are frequency division multiplexedp

loz The signaling interval is then increased to which tNΔmakes the system less susceptible to channel delay spread impairments

Small-scale fading(Based on multipath time delay spread)

Flat Fading1 BW of signal lt BW of channel

Frequency Selective Fading1 BW of signal gt BW of channel

2 Delay spread lt Symbol preiod 2 Delay spread gt Symbol preiod

12

OrthogonalityOrthogonalityg yg y

loz Consider a set of transmitted carriers as follows

(2) 1 1 0for )( 2 0

minus==⎟⎠⎞

⎜⎝⎛

Δ+

Nnett

tNnfj

n

πψ n

⎨⎧ =minus

intqpab

dtttb for )(

)()( ψψ⎩⎨ Δ=minusne

=int tNabqpdttt

a qp )( and for 0 )()( ψψ

13

OrthogonalityOrthogonality

b tqpjb

intint)(2π

g yg y

dtedtttajbj

b

atN

qpjb

a qp = Δminus

intint )()(

)(2)(2

)(2ψψπ

tNqpjee tN

qpjtN

qpj

Δminusminus

=Δ

minusΔ

minus

)(2

)(2)(2

π

ππ

ee

tNqpjba

tNqpj

tNbqpj

⎟⎟⎞

⎜⎜⎛minus

Δ

minusΔ

minusΔ

minus1

)(2)(1)(2)(2

π

ππ

tNqpj Δminus

⎟⎠

⎜⎝=

)(2πtNabqp Δ=minusne= )( and for 0

14

OrthogonalityOrthogonalityg yg y

tNΔ

15

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFTby using IDFTby using IDFT

loz According to the structure of Tx it must use Ng oscillators That increases the hardware complexity

loz The equivalent method is using IDFT (IFFT)

16

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

loz In general each carrier can be expressed as

by using IDFTby using IDFT

g p

( ) (6) )()( )(2 ttfjcc

ccetAtS φπ +=

loz We assume that there are N carriers in the OFDM signal Th th t t l l i l S (t) b t d bThen the total complex signal Ss(t) can be represented by

( ) (7))(1)(1

)(2etAtSN

ttfj nn φπ= summinus

+( )

where

(7) )()(

0

0

fnff

etAN

tSn

ns

Δ+=

= sum=

liihffrequency carrier phase amplitude are )( )( and

where 0

fttAfnff

nnn

n

φΔ+

lyrespective carrierth -of n17

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

loz Then we sample the signal at a sampling frequency tΔ1

by using IDFTby using IDFT

p g p g q y and An(t) and φn(t) becomes

(8))(t = φφ(9) )((8) )(

nn

nn

AtAt

== φφ

( ) (10) 1)(1

0

)(2 0summinus

=

+ΔΔ+=ΔN

n

tkfnfjns

neAN

tkS φπ

loz Then the sampled signal can be expressed as0=nN

( )( ) (11) 1)(1

0

22 0summinus

=

ΔΔ+Δ sdot=ΔN

n

tfnkjtkfjns eeA

NtkS n πφπ

0n

18

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

loz The inverse discrete Fourier transform (IDFT) is defined

by using IDFTby using IDFT

( )as the following

1 21

NnkjN

πsumminus

(12) )(1)( 2

0

Nnkj

nefnF

Ntkf πsum

=

Δ=Δ

loz Comparing eq(11) and eq(12) the condition must be satisfied in order to make eq(11) an inverse Fouriersatisfied in order to make eq(11) an inverse Fourier transform relationship

1 (13) 1tN

fΔ

=Δ

19

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

loz If eq(13) is satisfied

by using IDFTby using IDFT

q ( ) loz is the frequency domain signal loz is the time domain signal

( )ntkfjneA φπ +Δ02

)( tkSs Δ gloz is the sub-channel spacingloz is the symbol duration in each sub-channel

)(s

fΔtNΔloz s e sy bo du o e c sub c e

loz This outcome is the same as the result obtained in the

N

loz This outcome is the same as the result obtained in the system of Figure 1 Therefore IDFT can be used to generate an OFDM transmission signalgenerate an OFDM transmission signal

20

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

( )tf02cos π

( )0a

by using IDFTby using IDFT

( )nd ( )tDI( )tf02sin π

( )0a

( )0b

bull

tf s Δ=

1 ( ) ( ) njbna +

( )nd

tNΔffnffn Δ

=Δ+=1 0

( )tDChannelInput

( )tf N 12cos minusπ

( )1minusNa

bullbull

tNΔ

( )tf N 12sin minusπ

( )1minusNb

( )0d

( )nd ( )tDChannelInput

bull bull

( )( )1d

( )2d

tf s Δ=

1 ( ) ( ) njbna +

fNΔffnffn Δ

=Δ+=1 0

ChannelInput

bullbullbullbull

bullbullbullbull

( )1minusNd

21

Frequency Error Results in ICIFrequency Error Results in ICIq yq y

22

Synchronization Error Results in ICISynchronization Error Results in ICIyy

Not Orthogonal Any More

23

Cyclic PrefixCyclic Prefixyy

loz In multipath channel delayed replicas of previous OFDM signal lead to ISI between successive OFDM signals

loz Solution Insert a guard interval between successive OFDMloz Solution Insert a guard interval between successive OFDM signals

24

Cyclic PrefixCyclic Prefix

loz Guard interval leads to intercarrier interference (ICI) in OFDM d d l i

yy

demodulation

loz In DFT interval difference between two subcarriers does not maintain integer number of cycles loss of orthogonality

loz Delayed version of subcarrier 2 causes ICI in the process of d d l ti b i 1demodulating subcarrier 1

25

Cyclic PrefixCyclic Prefixloz Cyclic prefix (CP) A copy of the last part of OFDM signal is

attached to the front of itself

yy

attached to the front of itself

[ ]0d

[ ]1d

[ ]0D[ ]1D

[ ]nd[ ]kD~

symbols dataInput

bullbullbullbull

[ ]1d

[ ]2d

[ ][ ]2D

bullbullbullbullbull

y

bullbullbullbullbull

[ ]1minusNd [ ]1minusND

[ ]gNND minus

bullbullbull

26

Cyclic PrefixCyclic Prefix

loz All delayed replicas of subcarriers always have an

yy

y p yinteger number of cycles within DFT interval no ICI

27

Cyclic PrefixCyclic Prefix

loz Linear convolution vs circular convolution

yy

N

28

Cyclic PrefixCyclic Prefix

loz Channel effect with cyclic prefix

yy

Signal after removed CP

29

Cyclic PrefixCyclic Prefix

loz Time-Domain Explanation

yy

p

30

Cyclic PrefixCyclic Prefixyy

31

Cyclic PrefixCyclic Prefix

loz Spectrum of channel response with length (smaller [ ]nh hL

yy

than )h

gN[ ] nhFFTHk =

loz Received complete OFDM signal[ ] [ ] [ ] 20 ~~ minus++leleotimes= hg LNNnnhnDnr

loz Received useful part [ ]nr[ ] [ ] [ ]nhnDnr Notimes=

where is N-point circular convolution (due to CP)loz Received symbol at k-th subcarrier

[ ] [ ] [ ]N

Notimes

[ ] [ ] [ ] kkNk HXnhnDFFTnrFFTY =otimes== kYX =rArr ldquoUseful property for OFDM system to reduce

kk H

X =rArrcomplexity of channel equalizationrdquo

32

Cyclic Cyclic Prefix (OFDM Prefix (OFDM Receiver)Receiver)yy (( ))

bull bull( )( )nr~Symbols

dataOutput

bullbullbull

bullbullbull( )nr

bullbull

bullbull

33

Cyclic PrefixCyclic Prefix

loz One of the most important reasons to do OFDM is the

yy

pefficient way it deals with multipath delay spread

loz To eliminate inter-symbol interference (ISI) almostloz To eliminate inter symbol interference (ISI) almost completely a guard time is introduced for each OFDM symbolsymbol(The guard time is chosen larger than the delay spread)

34

Bandwidth EfficiencyBandwidth Efficiencyyy

loz In a classical parallel system the channel is divided into p y N non-overlapping sub-channels to avoid inter-carrier interference (ICI)( )

loz The diagram for bandwidth efficiency of OFDM system is shown below

35

SummarySummaryyy

loz The advantage of the FFT-based OFDM system g yloz The use of IFFTFFT can reduce the computation complexityloz The orthogonality between the adjacent sub-carriers will make g y j

the use of transmission bandwidth more efficientloz The guard interval is used to resist the inter-symbol interference

(ISI)loz The main advantage of the OFDM transmission technique is its

high performance even in frequency selective channels

loz The drawbacks of the OFDM system loz It is highly vulnerable to synchronization errorsloz Peak to Average Power Ratio (PAPR) problems

36

SynchronizationSynchronization

Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offset (TO)loz Impact of timing offset (TO)loz Impact of carrier frequency offset (CFO)loz TO estimation algorithmloz CFO estimation algorithmg

38 Synchronization2010722

Introduction (13)Introduction (13)( )( )

loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods

loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead

loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the

cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of

computational complexity therefore it may be not be available in short-burst wireless communication

39 Synchronization2010722

Introduction (23)Introduction (23)( )( )

loz Transcevier

40 Synchronization2010722

Introduction (33)Introduction (33)( )( )

loz Synchronization erroryloz Symbol timing offset (TO)

loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)

loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect

41 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg

42 Synchronization2010722

Signal model (14)Signal model (14)g ( )g ( )

loz Time domain sample p1

2

0

1 Nj nk N

n kk

x X eN

πminus

minus= sum

Xk the frequency domain data at the k-th subcarrier0kN =

loz The multipath fading channel

( ) ( ) ( )1L

h t h t δminus

sum

loz Received time domain signal

( ) ( ) ( )0

l ll

h t h tτ τ δ τ τ=

= minussum

loz Received time-domain signal

( ) ( )1

L

n n ly h n l x w nminus

minus= +sum43

( ) ( )0l=sum

Synchronization2010722

Signal model (24)Signal model (24)g ( )g ( )

loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting

12

0

Nj kn N

k nn

Y y e πminus

minus

=

= sum

( )1 1

2

0

N Lj kn N

n l kn l

h n l x e Wπminus minus

minusminus

=

= +sumsum

loz The equation above could be expressed in matrix form

0n l

loz The equation above could be expressed in matrix form

= +Y HX W

44 Synchronization2010722

Signal model (34)Signal model (34)g ( )g ( )

H= = = = +Y Fy FCx FCF X HX W

loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal

y

H=H FCFcirculant matrix leading to be a diagonal matrix

=H FCF

( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0

h h L L hh h

⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥

( )( ) ( )

( ) ( )

1 10 1 0 0i

h Lh L h N L

⎢ ⎥⎢ ⎥minus minus⎢ ⎥

equiv= minus minus⎢ ⎥⎢ ⎥

C( ) ( )

( ) ( )

0 1 1 100

0 0 1 1 0

h L h N L

h N L L h N

⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

45

( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦

Synchronization2010722

Signal model (44)Signal model (44)g ( )g ( )

loz In a time-invariant channel the channel matrix H would be a diagonal matrix

k k k kY H X W= +

( )1 1

2

0 0

N Lj kl N

kn l

H h n l e πminus minus

minus

= =

= sumsum

loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI

k k k k kY H X I Wα= + +k k k k k

( ) ( )1 1 1

2 2 N N L

j m k Nj ml Nk mI X h n l e e ππ

minus minus minusminusminus⎛ ⎞

= ⎜ ⎟⎝ ⎠

sum sum sumICI

46

0 0 0m n lm k= = =ne

⎝ ⎠

Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

47 Synchronization2010722

Impact of timing offset Impact of timing offset p gp g

loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively

( )I f s

Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ

( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr

2 fj k Nk k k kY X H e Wπδminus= +

loz For integer part of timing offset k k k kY X H e W+

I sTδ ( ) 0g IN L δminus minus le leFFT

2 Ij k NY X H e Wπδminus= +

( ) ( ) 2 I sFFT

j f TI sh t T H f e π δδ minusminus rarr

48

k k k kY X H e W= +

Synchronization2010722

Impact of timing offset derivationImpact of timing offset derivationp gp g

49 Synchronization2010722

Discussion of symbol boundaryDiscussion of symbol boundaryy yy y

50 Synchronization2010722

Impact of timing offsetImpact of timing offsetp gp g

loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI

2 Ij k NIk k k k

NY X H e ISI W

Nπδδ minusminus

= + +

loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk

e e

p kloz Phase Shift can be resolved by Differential Encoding

Decoding or carrier recovery with pilots

51 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

52 Synchronization2010722

Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )

53 Synchronization2010722

Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )

( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s

i n t i N N T N T nTy y t e

= + + +=

( ) ( )( ) ( ) ( ) 12

sin

sin

g gI f f

I I

i N N N Nj jf N Ni k i k k

f

Y X H e eN

π ε ε π ε

ε ε

πεπε

+ + minus+

minus minus=⎛ ⎞⎜ ⎟

Faded signal attenuated and rotated by CFO

( )( ) ( ) ( ) ( )11 2

sin

sin g gI f I f

i N N N NN j j l kI f

NN

l k π ε ε π ε επ ε ε + + minusminus + + + minus

⎜ ⎟⎝ ⎠

+ + minussum

( )( )( )

( ) ( )2

0

sin

I f I f

I

j j l kI f N Ni l l

l l k I f

X H e el k

NN

π ε ε π ε ε

ε π ε ε

+ + +

= ne minus

+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠

sum

i k

N

W

⎜ ⎟⎝ ⎠

+ Inter-Carrier Interference (ICI)

54 Synchronization2010722

Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )

loz An easier representation of received signal with only p g yfractional CFO is

1N

( ) ( )1

0

0N

k k k l l kl l k

Y H X C H X C l k Wminus

= ne

= + minus +⎡ ⎤⎣ ⎦sum

where the ICI coefficient is given by

( )( )( ) ( )

sin 1exp 1fl k

C l k j l kπ ε

π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )

( )exp 1sin

f

f

C l k j l kNN l k

N

π επ ε

minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠

55 Synchronization2010722

Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )

loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality

loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift

56 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

57 Synchronization2010722

TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )

loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq

58 Synchronization2010722

TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )

loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below

loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and

rN2 can be expressed by

0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+

( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +

59

( )

Synchronization2010722

TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )

loz Without CFO the first half is identical to the second half (in time order)

loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below

( ) 12 in discretesft T f NT

NTφ π π π ε πε

⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟

⎝ ⎠

helliphellip

sNT⎝ ⎠

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)

60

Length=N2

Synchronization2010722

TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )

loz Received signal without CFOg0

0 0

jr a e θ=jθ 2 0

2 2 0

Nj j

N Nr a e a e

θ θ= =

loz Received signal with CFO0 0

0 0

j jr a e eθ φ= Extra phase rotation due to CFO0 0

2 2 20

2 2 0

N N Nj j jj

N Nr a e e a e e

θ φ φθ= =

due to CFO

Phase difference 2 2 0N N

( )2 02 20 2 0 0

Nj j T fN

r r a e a eφ φ πminus Δ= =

which contains the information about CFO

61

0 2 0 0Nr r a e a e

Synchronization2010722

TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )

loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be

loz Note that d is a time index corresponding to the first( ) ( ) 2 1

20

Nd m d m Nm

P d r rminus

+ + +== sum

loz Note that d is a time index corresponding to the first sample in a window of N samples

loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by

( )22 1

20

Nd m NR d rminus

+ += sumloz A timing metric can be defined as

))(()(

)( 2

2

dRdP

dM =

( ) 20 d m Nm + +=sum

62

))(( dR

Synchronization2010722

TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )

63 Synchronization2010722

TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )

loz If drsquo is the correct symbol timing offset

( )2

2 2

0 1 12

j T f j T f j T fN

P d a e a e a eπ π πΔ Δ Δ

minus= + + +

22

2 2

0 1 12

j T fN

e a a aπ Δ

minus

⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭

loz If drsquo falls behind the correct symbol timing offset 1 sample

2⎪ ⎪⎩ ⎭

64 Synchronization2010722

TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )

65 Synchronization2010722

TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )

loz Drawback Plateau effect

Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)

((N2) ((N2) ((N2)

helliphellip

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)

Sliding window (length = N)

66 Synchronization2010722

TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )

loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula

( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus

扣掉相加

( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +

loz The method also called delay correlator

67 Synchronization2010722

IEEE 80211aIEEE 80211a

68 Synchronization2010722

Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )

loz Minnrsquos Method [2][ ]B B －B －B

( )( )

( )

22

2Minn

P dM d =

1 4 1N minus

( )( )( )2

2

MinnR d

( ) ( ) ( ) ( )

1 4 1

2 2 2 40 0

N

d N m k d N m k Nm k

P d r r+ + + + += =

= sum sum

( ) ( ) ( )2

21 4 1

2 40 0

N

d N m k Nk

R d rminus

+ + += sum sum69

0 0m k= =

Synchronization2010722

Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )

loz Parkrsquos Method [3][ ]

where D is the symmetric version of CC D C D

where D is the symmetric version of C

( )( ) 2

3k

P dM d =( )

( )( )23

ParkM dR d

2N

( ) 2

30

N

d k d kk

P d r rminus +=

= sdotsum

( )3

22

0

N

d kk

R d r += sum70

0k=

Synchronization2010722

Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )

loz Renrsquos Method [4][ ]

AS AS

where S is the PN sequence weighted of the original bl 2preamble

( )( )( )( )

24

Re 24

n

P dM d

R d=

( )( )4R d

( ) 2 1

4 2 2

N

k k N d k d k NP d s s r rminus

+ + + += sum0k=sum

( )21

41 N

d kR d rminus

= sum71

( )402 d k

kR d r +

=sum

Synchronization2010722

Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )

72 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

73 Synchronization2010722

Fractional CFO estimation Fractional CFO estimation

loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected

11 Rminus⎛ ⎞sum

h L i th di t b t t id ti l bl k R i

0

12 d m d m L

ms

f r rLTπ + + +

=

⎛ ⎞= ang⎜ ⎟⎝ ⎠sum

where L is the distance between two identical block R is the block size

loz For exampleNN cp

L = N R = NL = N R = Ncp74 Synchronization2010722

Estimation rangeEstimation rangegg

loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval

f ( ) ( )s s⎣ ⎦

loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)

loz DVB-T (L=N) 802 11 (L N4)

[ ]05 05s sf fminus

[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)

(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus

loz Note that is the subcarrier spacing1sf NT=

75

sNT

Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz

plusmng q y

loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn

76 Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz

plusmng q y

loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for

subcarrier spacing is required

plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn

p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs

77 Synchronization2010722

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

OFDM System ModelOFDM System Modelyy

loz Multi-carrier Block Transmission

frequency

time

8T8

OFDM System ModelOFDM System Modelyy

loz OFDM A block modulation scheme that transmitsa block N source symbols in parallel by using subcarrierssubcarriersloz Sub-carriers are orthogonal in time but overlapped in

frequencyeque cyloz Frequency spacing 1

FFT

fT

Δ =

cos(2 )cos(2 ( ) ) 0FFTTf t f f t dtπ π + Δint 1 10

cos(2 )cos(2 ( ) ) 0f t f f t dtπ π + Δ =int

9

OFDM System ModelOFDM System Model

tNΔ

yy

( )tf02cos π

( )0a

( )0atT

tNΔ1+

( )tf2i

( )

( )0b

T

DataEncoder

tf s Δ=

1 ( ) ( ) njbna +

( )nd SP MUX ( )tDChannelInput

( )tf02sin π

( )tf N 12cos minusπbullbullbull

( )1minusNa

( )1minusNb( )0a1+

( )tf N 12sin minusπ

( )1Nb

tT

( )1a ( )1minusNa1minus

tΔ

Figure 1tNΔ

( )1a ( )1minusNa Figure 1

10

OFDM System ModelOFDM System Model

loz An OFDM system transmitter shown in Figure 1

yy

y gloz The transmitted waveform D(t) can be expressed as

)1( )2sin()()2cos()()(1

0summinus

=

+=N

nnn tfnbtfnatD ππ

where tNffnffn Δ=ΔΔ+=

1 and 0

loz Using a two-dimensional digital modulation format the data

tNΔ

g g symbols d(n) can be represented as a(n) + jb(n)loz a(n) in-phase componentloz b(n) quadrature component

11

OFDM System ModelOFDM System Model

loz The serial data elements spaced by are grouped and tΔ

yy

p y g pused to modulate N carriers Thus they are frequency division multiplexedp

loz The signaling interval is then increased to which tNΔmakes the system less susceptible to channel delay spread impairments

Small-scale fading(Based on multipath time delay spread)

Flat Fading1 BW of signal lt BW of channel

Frequency Selective Fading1 BW of signal gt BW of channel

2 Delay spread lt Symbol preiod 2 Delay spread gt Symbol preiod

12

OrthogonalityOrthogonalityg yg y

loz Consider a set of transmitted carriers as follows

(2) 1 1 0for )( 2 0

minus==⎟⎠⎞

⎜⎝⎛

Δ+

Nnett

tNnfj

n

πψ n

⎨⎧ =minus

intqpab

dtttb for )(

)()( ψψ⎩⎨ Δ=minusne

=int tNabqpdttt

a qp )( and for 0 )()( ψψ

13

OrthogonalityOrthogonality

b tqpjb

intint)(2π

g yg y

dtedtttajbj

b

atN

qpjb

a qp = Δminus

intint )()(

)(2)(2

)(2ψψπ

tNqpjee tN

qpjtN

qpj

Δminusminus

=Δ

minusΔ

minus

)(2

)(2)(2

π

ππ

ee

tNqpjba

tNqpj

tNbqpj

⎟⎟⎞

⎜⎜⎛minus

Δ

minusΔ

minusΔ

minus1

)(2)(1)(2)(2

π

ππ

tNqpj Δminus

⎟⎠

⎜⎝=

)(2πtNabqp Δ=minusne= )( and for 0

14

OrthogonalityOrthogonalityg yg y

tNΔ

15

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFTby using IDFTby using IDFT

loz According to the structure of Tx it must use Ng oscillators That increases the hardware complexity

loz The equivalent method is using IDFT (IFFT)

16

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

loz In general each carrier can be expressed as

by using IDFTby using IDFT

g p

( ) (6) )()( )(2 ttfjcc

ccetAtS φπ +=

loz We assume that there are N carriers in the OFDM signal Th th t t l l i l S (t) b t d bThen the total complex signal Ss(t) can be represented by

( ) (7))(1)(1

)(2etAtSN

ttfj nn φπ= summinus

+( )

where

(7) )()(

0

0

fnff

etAN

tSn

ns

Δ+=

= sum=

liihffrequency carrier phase amplitude are )( )( and

where 0

fttAfnff

nnn

n

φΔ+

lyrespective carrierth -of n17

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

loz Then we sample the signal at a sampling frequency tΔ1

by using IDFTby using IDFT

p g p g q y and An(t) and φn(t) becomes

(8))(t = φφ(9) )((8) )(

nn

nn

AtAt

== φφ

( ) (10) 1)(1

0

)(2 0summinus

=

+ΔΔ+=ΔN

n

tkfnfjns

neAN

tkS φπ

loz Then the sampled signal can be expressed as0=nN

( )( ) (11) 1)(1

0

22 0summinus

=

ΔΔ+Δ sdot=ΔN

n

tfnkjtkfjns eeA

NtkS n πφπ

0n

18

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

loz The inverse discrete Fourier transform (IDFT) is defined

by using IDFTby using IDFT

( )as the following

1 21

NnkjN

πsumminus

(12) )(1)( 2

0

Nnkj

nefnF

Ntkf πsum

=

Δ=Δ

loz Comparing eq(11) and eq(12) the condition must be satisfied in order to make eq(11) an inverse Fouriersatisfied in order to make eq(11) an inverse Fourier transform relationship

1 (13) 1tN

fΔ

=Δ

19

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

loz If eq(13) is satisfied

by using IDFTby using IDFT

q ( ) loz is the frequency domain signal loz is the time domain signal

( )ntkfjneA φπ +Δ02

)( tkSs Δ gloz is the sub-channel spacingloz is the symbol duration in each sub-channel

)(s

fΔtNΔloz s e sy bo du o e c sub c e

loz This outcome is the same as the result obtained in the

N

loz This outcome is the same as the result obtained in the system of Figure 1 Therefore IDFT can be used to generate an OFDM transmission signalgenerate an OFDM transmission signal

20

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

( )tf02cos π

( )0a

by using IDFTby using IDFT

( )nd ( )tDI( )tf02sin π

( )0a

( )0b

bull

tf s Δ=

1 ( ) ( ) njbna +

( )nd

tNΔffnffn Δ

=Δ+=1 0

( )tDChannelInput

( )tf N 12cos minusπ

( )1minusNa

bullbull

tNΔ

( )tf N 12sin minusπ

( )1minusNb

( )0d

( )nd ( )tDChannelInput

bull bull

( )( )1d

( )2d

tf s Δ=

1 ( ) ( ) njbna +

fNΔffnffn Δ

=Δ+=1 0

ChannelInput

bullbullbullbull

bullbullbullbull

( )1minusNd

21

Frequency Error Results in ICIFrequency Error Results in ICIq yq y

22

Synchronization Error Results in ICISynchronization Error Results in ICIyy

Not Orthogonal Any More

23

Cyclic PrefixCyclic Prefixyy

loz In multipath channel delayed replicas of previous OFDM signal lead to ISI between successive OFDM signals

loz Solution Insert a guard interval between successive OFDMloz Solution Insert a guard interval between successive OFDM signals

24

Cyclic PrefixCyclic Prefix

loz Guard interval leads to intercarrier interference (ICI) in OFDM d d l i

yy

demodulation

loz In DFT interval difference between two subcarriers does not maintain integer number of cycles loss of orthogonality

loz Delayed version of subcarrier 2 causes ICI in the process of d d l ti b i 1demodulating subcarrier 1

25

Cyclic PrefixCyclic Prefixloz Cyclic prefix (CP) A copy of the last part of OFDM signal is

attached to the front of itself

yy

attached to the front of itself

[ ]0d

[ ]1d

[ ]0D[ ]1D

[ ]nd[ ]kD~

symbols dataInput

bullbullbullbull

[ ]1d

[ ]2d

[ ][ ]2D

bullbullbullbullbull

y

bullbullbullbullbull

[ ]1minusNd [ ]1minusND

[ ]gNND minus

bullbullbull

26

Cyclic PrefixCyclic Prefix

loz All delayed replicas of subcarriers always have an

yy

y p yinteger number of cycles within DFT interval no ICI

27

Cyclic PrefixCyclic Prefix

loz Linear convolution vs circular convolution

yy

N

28

Cyclic PrefixCyclic Prefix

loz Channel effect with cyclic prefix

yy

Signal after removed CP

29

Cyclic PrefixCyclic Prefix

loz Time-Domain Explanation

yy

p

30

Cyclic PrefixCyclic Prefixyy

31

Cyclic PrefixCyclic Prefix

loz Spectrum of channel response with length (smaller [ ]nh hL

yy

than )h

gN[ ] nhFFTHk =

loz Received complete OFDM signal[ ] [ ] [ ] 20 ~~ minus++leleotimes= hg LNNnnhnDnr

loz Received useful part [ ]nr[ ] [ ] [ ]nhnDnr Notimes=

where is N-point circular convolution (due to CP)loz Received symbol at k-th subcarrier

[ ] [ ] [ ]N

Notimes

[ ] [ ] [ ] kkNk HXnhnDFFTnrFFTY =otimes== kYX =rArr ldquoUseful property for OFDM system to reduce

kk H

X =rArrcomplexity of channel equalizationrdquo

32

Cyclic Cyclic Prefix (OFDM Prefix (OFDM Receiver)Receiver)yy (( ))

bull bull( )( )nr~Symbols

dataOutput

bullbullbull

bullbullbull( )nr

bullbull

bullbull

33

Cyclic PrefixCyclic Prefix

loz One of the most important reasons to do OFDM is the

yy

pefficient way it deals with multipath delay spread

loz To eliminate inter-symbol interference (ISI) almostloz To eliminate inter symbol interference (ISI) almost completely a guard time is introduced for each OFDM symbolsymbol(The guard time is chosen larger than the delay spread)

34

Bandwidth EfficiencyBandwidth Efficiencyyy

loz In a classical parallel system the channel is divided into p y N non-overlapping sub-channels to avoid inter-carrier interference (ICI)( )

loz The diagram for bandwidth efficiency of OFDM system is shown below

35

SummarySummaryyy

loz The advantage of the FFT-based OFDM system g yloz The use of IFFTFFT can reduce the computation complexityloz The orthogonality between the adjacent sub-carriers will make g y j

the use of transmission bandwidth more efficientloz The guard interval is used to resist the inter-symbol interference

(ISI)loz The main advantage of the OFDM transmission technique is its

high performance even in frequency selective channels

loz The drawbacks of the OFDM system loz It is highly vulnerable to synchronization errorsloz Peak to Average Power Ratio (PAPR) problems

36

SynchronizationSynchronization

Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offset (TO)loz Impact of timing offset (TO)loz Impact of carrier frequency offset (CFO)loz TO estimation algorithmloz CFO estimation algorithmg

38 Synchronization2010722

Introduction (13)Introduction (13)( )( )

loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods

loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead

loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the

cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of

computational complexity therefore it may be not be available in short-burst wireless communication

39 Synchronization2010722

Introduction (23)Introduction (23)( )( )

loz Transcevier

40 Synchronization2010722

Introduction (33)Introduction (33)( )( )

loz Synchronization erroryloz Symbol timing offset (TO)

loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)

loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect

41 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg

42 Synchronization2010722

Signal model (14)Signal model (14)g ( )g ( )

loz Time domain sample p1

2

0

1 Nj nk N

n kk

x X eN

πminus

minus= sum

Xk the frequency domain data at the k-th subcarrier0kN =

loz The multipath fading channel

( ) ( ) ( )1L

h t h t δminus

sum

loz Received time domain signal

( ) ( ) ( )0

l ll

h t h tτ τ δ τ τ=

= minussum

loz Received time-domain signal

( ) ( )1

L

n n ly h n l x w nminus

minus= +sum43

( ) ( )0l=sum

Synchronization2010722

Signal model (24)Signal model (24)g ( )g ( )

loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting

12

0

Nj kn N

k nn

Y y e πminus

minus

=

= sum

( )1 1

2

0

N Lj kn N

n l kn l

h n l x e Wπminus minus

minusminus

=

= +sumsum

loz The equation above could be expressed in matrix form

0n l

loz The equation above could be expressed in matrix form

= +Y HX W

44 Synchronization2010722

Signal model (34)Signal model (34)g ( )g ( )

H= = = = +Y Fy FCx FCF X HX W

loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal

y

H=H FCFcirculant matrix leading to be a diagonal matrix

=H FCF

( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0

h h L L hh h

⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥

( )( ) ( )

( ) ( )

1 10 1 0 0i

h Lh L h N L

⎢ ⎥⎢ ⎥minus minus⎢ ⎥

equiv= minus minus⎢ ⎥⎢ ⎥

C( ) ( )

( ) ( )

0 1 1 100

0 0 1 1 0

h L h N L

h N L L h N

⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

45

( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦

Synchronization2010722

Signal model (44)Signal model (44)g ( )g ( )

loz In a time-invariant channel the channel matrix H would be a diagonal matrix

k k k kY H X W= +

( )1 1

2

0 0

N Lj kl N

kn l

H h n l e πminus minus

minus

= =

= sumsum

loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI

k k k k kY H X I Wα= + +k k k k k

( ) ( )1 1 1

2 2 N N L

j m k Nj ml Nk mI X h n l e e ππ

minus minus minusminusminus⎛ ⎞

= ⎜ ⎟⎝ ⎠

sum sum sumICI

46

0 0 0m n lm k= = =ne

⎝ ⎠

Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

47 Synchronization2010722

Impact of timing offset Impact of timing offset p gp g

loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively

( )I f s

Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ

( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr

2 fj k Nk k k kY X H e Wπδminus= +

loz For integer part of timing offset k k k kY X H e W+

I sTδ ( ) 0g IN L δminus minus le leFFT

2 Ij k NY X H e Wπδminus= +

( ) ( ) 2 I sFFT

j f TI sh t T H f e π δδ minusminus rarr

48

k k k kY X H e W= +

Synchronization2010722

Impact of timing offset derivationImpact of timing offset derivationp gp g

49 Synchronization2010722

Discussion of symbol boundaryDiscussion of symbol boundaryy yy y

50 Synchronization2010722

Impact of timing offsetImpact of timing offsetp gp g

loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI

2 Ij k NIk k k k

NY X H e ISI W

Nπδδ minusminus

= + +

loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk

e e

p kloz Phase Shift can be resolved by Differential Encoding

Decoding or carrier recovery with pilots

51 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

52 Synchronization2010722

Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )

53 Synchronization2010722

Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )

( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s

i n t i N N T N T nTy y t e

= + + +=

( ) ( )( ) ( ) ( ) 12

sin

sin

g gI f f

I I

i N N N Nj jf N Ni k i k k

f

Y X H e eN

π ε ε π ε

ε ε

πεπε

+ + minus+

minus minus=⎛ ⎞⎜ ⎟

Faded signal attenuated and rotated by CFO

( )( ) ( ) ( ) ( )11 2

sin

sin g gI f I f

i N N N NN j j l kI f

NN

l k π ε ε π ε επ ε ε + + minusminus + + + minus

⎜ ⎟⎝ ⎠

+ + minussum

( )( )( )

( ) ( )2

0

sin

I f I f

I

j j l kI f N Ni l l

l l k I f

X H e el k

NN

π ε ε π ε ε

ε π ε ε

+ + +

= ne minus

+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠

sum

i k

N

W

⎜ ⎟⎝ ⎠

+ Inter-Carrier Interference (ICI)

54 Synchronization2010722

Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )

loz An easier representation of received signal with only p g yfractional CFO is

1N

( ) ( )1

0

0N

k k k l l kl l k

Y H X C H X C l k Wminus

= ne

= + minus +⎡ ⎤⎣ ⎦sum

where the ICI coefficient is given by

( )( )( ) ( )

sin 1exp 1fl k

C l k j l kπ ε

π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )

( )exp 1sin

f

f

C l k j l kNN l k

N

π επ ε

minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠

55 Synchronization2010722

Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )

loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality

loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift

56 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

57 Synchronization2010722

TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )

loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq

58 Synchronization2010722

TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )

loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below

loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and

rN2 can be expressed by

0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+

( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +

59

( )

Synchronization2010722

TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )

loz Without CFO the first half is identical to the second half (in time order)

loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below

( ) 12 in discretesft T f NT

NTφ π π π ε πε

⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟

⎝ ⎠

helliphellip

sNT⎝ ⎠

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)

60

Length=N2

Synchronization2010722

TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )

loz Received signal without CFOg0

0 0

jr a e θ=jθ 2 0

2 2 0

Nj j

N Nr a e a e

θ θ= =

loz Received signal with CFO0 0

0 0

j jr a e eθ φ= Extra phase rotation due to CFO0 0

2 2 20

2 2 0

N N Nj j jj

N Nr a e e a e e

θ φ φθ= =

due to CFO

Phase difference 2 2 0N N

( )2 02 20 2 0 0

Nj j T fN

r r a e a eφ φ πminus Δ= =

which contains the information about CFO

61

0 2 0 0Nr r a e a e

Synchronization2010722

TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )

loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be

loz Note that d is a time index corresponding to the first( ) ( ) 2 1

20

Nd m d m Nm

P d r rminus

+ + +== sum

loz Note that d is a time index corresponding to the first sample in a window of N samples

loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by

( )22 1

20

Nd m NR d rminus

+ += sumloz A timing metric can be defined as

))(()(

)( 2

2

dRdP

dM =

( ) 20 d m Nm + +=sum

62

))(( dR

Synchronization2010722

TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )

63 Synchronization2010722

TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )

loz If drsquo is the correct symbol timing offset

( )2

2 2

0 1 12

j T f j T f j T fN

P d a e a e a eπ π πΔ Δ Δ

minus= + + +

22

2 2

0 1 12

j T fN

e a a aπ Δ

minus

⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭

loz If drsquo falls behind the correct symbol timing offset 1 sample

2⎪ ⎪⎩ ⎭

64 Synchronization2010722

TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )

65 Synchronization2010722

TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )

loz Drawback Plateau effect

Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)

((N2) ((N2) ((N2)

helliphellip

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)

Sliding window (length = N)

66 Synchronization2010722

TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )

loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula

( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus

扣掉相加

( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +

loz The method also called delay correlator

67 Synchronization2010722

IEEE 80211aIEEE 80211a

68 Synchronization2010722

Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )

loz Minnrsquos Method [2][ ]B B －B －B

( )( )

( )

22

2Minn

P dM d =

1 4 1N minus

( )( )( )2

2

MinnR d

( ) ( ) ( ) ( )

1 4 1

2 2 2 40 0

N

d N m k d N m k Nm k

P d r r+ + + + += =

= sum sum

( ) ( ) ( )2

21 4 1

2 40 0

N

d N m k Nk

R d rminus

+ + += sum sum69

0 0m k= =

Synchronization2010722

Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )

loz Parkrsquos Method [3][ ]

where D is the symmetric version of CC D C D

where D is the symmetric version of C

( )( ) 2

3k

P dM d =( )

( )( )23

ParkM dR d

2N

( ) 2

30

N

d k d kk

P d r rminus +=

= sdotsum

( )3

22

0

N

d kk

R d r += sum70

0k=

Synchronization2010722

Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )

loz Renrsquos Method [4][ ]

AS AS

where S is the PN sequence weighted of the original bl 2preamble

( )( )( )( )

24

Re 24

n

P dM d

R d=

( )( )4R d

( ) 2 1

4 2 2

N

k k N d k d k NP d s s r rminus

+ + + += sum0k=sum

( )21

41 N

d kR d rminus

= sum71

( )402 d k

kR d r +

=sum

Synchronization2010722

Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )

72 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

73 Synchronization2010722

Fractional CFO estimation Fractional CFO estimation

loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected

11 Rminus⎛ ⎞sum

h L i th di t b t t id ti l bl k R i

0

12 d m d m L

ms

f r rLTπ + + +

=

⎛ ⎞= ang⎜ ⎟⎝ ⎠sum

where L is the distance between two identical block R is the block size

loz For exampleNN cp

L = N R = NL = N R = Ncp74 Synchronization2010722

Estimation rangeEstimation rangegg

loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval

f ( ) ( )s s⎣ ⎦

loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)

loz DVB-T (L=N) 802 11 (L N4)

[ ]05 05s sf fminus

[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)

(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus

loz Note that is the subcarrier spacing1sf NT=

75

sNT

Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz

plusmng q y

loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn

76 Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz

plusmng q y

loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for

subcarrier spacing is required

plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn

p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs

77 Synchronization2010722

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

OFDM System ModelOFDM System Modelyy

loz OFDM A block modulation scheme that transmitsa block N source symbols in parallel by using subcarrierssubcarriersloz Sub-carriers are orthogonal in time but overlapped in

frequencyeque cyloz Frequency spacing 1

FFT

fT

Δ =

cos(2 )cos(2 ( ) ) 0FFTTf t f f t dtπ π + Δint 1 10

cos(2 )cos(2 ( ) ) 0f t f f t dtπ π + Δ =int

9

OFDM System ModelOFDM System Model

tNΔ

yy

( )tf02cos π

( )0a

( )0atT

tNΔ1+

( )tf2i

( )

( )0b

T

DataEncoder

tf s Δ=

1 ( ) ( ) njbna +

( )nd SP MUX ( )tDChannelInput

( )tf02sin π

( )tf N 12cos minusπbullbullbull

( )1minusNa

( )1minusNb( )0a1+

( )tf N 12sin minusπ

( )1Nb

tT

( )1a ( )1minusNa1minus

tΔ

Figure 1tNΔ

( )1a ( )1minusNa Figure 1

10

OFDM System ModelOFDM System Model

loz An OFDM system transmitter shown in Figure 1

yy

y gloz The transmitted waveform D(t) can be expressed as

)1( )2sin()()2cos()()(1

0summinus

=

+=N

nnn tfnbtfnatD ππ

where tNffnffn Δ=ΔΔ+=

1 and 0

loz Using a two-dimensional digital modulation format the data

tNΔ

g g symbols d(n) can be represented as a(n) + jb(n)loz a(n) in-phase componentloz b(n) quadrature component

11

OFDM System ModelOFDM System Model

loz The serial data elements spaced by are grouped and tΔ

yy

p y g pused to modulate N carriers Thus they are frequency division multiplexedp

loz The signaling interval is then increased to which tNΔmakes the system less susceptible to channel delay spread impairments

Small-scale fading(Based on multipath time delay spread)

Flat Fading1 BW of signal lt BW of channel

Frequency Selective Fading1 BW of signal gt BW of channel

2 Delay spread lt Symbol preiod 2 Delay spread gt Symbol preiod

12

OrthogonalityOrthogonalityg yg y

loz Consider a set of transmitted carriers as follows

(2) 1 1 0for )( 2 0

minus==⎟⎠⎞

⎜⎝⎛

Δ+

Nnett

tNnfj

n

πψ n

⎨⎧ =minus

intqpab

dtttb for )(

)()( ψψ⎩⎨ Δ=minusne

=int tNabqpdttt

a qp )( and for 0 )()( ψψ

13

OrthogonalityOrthogonality

b tqpjb

intint)(2π

g yg y

dtedtttajbj

b

atN

qpjb

a qp = Δminus

intint )()(

)(2)(2

)(2ψψπ

tNqpjee tN

qpjtN

qpj

Δminusminus

=Δ

minusΔ

minus

)(2

)(2)(2

π

ππ

ee

tNqpjba

tNqpj

tNbqpj

⎟⎟⎞

⎜⎜⎛minus

Δ

minusΔ

minusΔ

minus1

)(2)(1)(2)(2

π

ππ

tNqpj Δminus

⎟⎠

⎜⎝=

)(2πtNabqp Δ=minusne= )( and for 0

14

OrthogonalityOrthogonalityg yg y

tNΔ

15

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFTby using IDFTby using IDFT

loz According to the structure of Tx it must use Ng oscillators That increases the hardware complexity

loz The equivalent method is using IDFT (IFFT)

16

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

loz In general each carrier can be expressed as

by using IDFTby using IDFT

g p

( ) (6) )()( )(2 ttfjcc

ccetAtS φπ +=

loz We assume that there are N carriers in the OFDM signal Th th t t l l i l S (t) b t d bThen the total complex signal Ss(t) can be represented by

( ) (7))(1)(1

)(2etAtSN

ttfj nn φπ= summinus

+( )

where

(7) )()(

0

0

fnff

etAN

tSn

ns

Δ+=

= sum=

liihffrequency carrier phase amplitude are )( )( and

where 0

fttAfnff

nnn

n

φΔ+

lyrespective carrierth -of n17

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

loz Then we sample the signal at a sampling frequency tΔ1

by using IDFTby using IDFT

p g p g q y and An(t) and φn(t) becomes

(8))(t = φφ(9) )((8) )(

nn

nn

AtAt

== φφ

( ) (10) 1)(1

0

)(2 0summinus

=

+ΔΔ+=ΔN

n

tkfnfjns

neAN

tkS φπ

loz Then the sampled signal can be expressed as0=nN

( )( ) (11) 1)(1

0

22 0summinus

=

ΔΔ+Δ sdot=ΔN

n

tfnkjtkfjns eeA

NtkS n πφπ

0n

18

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

loz The inverse discrete Fourier transform (IDFT) is defined

by using IDFTby using IDFT

( )as the following

1 21

NnkjN

πsumminus

(12) )(1)( 2

0

Nnkj

nefnF

Ntkf πsum

=

Δ=Δ

loz Comparing eq(11) and eq(12) the condition must be satisfied in order to make eq(11) an inverse Fouriersatisfied in order to make eq(11) an inverse Fourier transform relationship

1 (13) 1tN

fΔ

=Δ

19

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

loz If eq(13) is satisfied

by using IDFTby using IDFT

q ( ) loz is the frequency domain signal loz is the time domain signal

( )ntkfjneA φπ +Δ02

)( tkSs Δ gloz is the sub-channel spacingloz is the symbol duration in each sub-channel

)(s

fΔtNΔloz s e sy bo du o e c sub c e

loz This outcome is the same as the result obtained in the

N

loz This outcome is the same as the result obtained in the system of Figure 1 Therefore IDFT can be used to generate an OFDM transmission signalgenerate an OFDM transmission signal

20

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

( )tf02cos π

( )0a

by using IDFTby using IDFT

( )nd ( )tDI( )tf02sin π

( )0a

( )0b

bull

tf s Δ=

1 ( ) ( ) njbna +

( )nd

tNΔffnffn Δ

=Δ+=1 0

( )tDChannelInput

( )tf N 12cos minusπ

( )1minusNa

bullbull

tNΔ

( )tf N 12sin minusπ

( )1minusNb

( )0d

( )nd ( )tDChannelInput

bull bull

( )( )1d

( )2d

tf s Δ=

1 ( ) ( ) njbna +

fNΔffnffn Δ

=Δ+=1 0

ChannelInput

bullbullbullbull

bullbullbullbull

( )1minusNd

21

Frequency Error Results in ICIFrequency Error Results in ICIq yq y

22

Synchronization Error Results in ICISynchronization Error Results in ICIyy

Not Orthogonal Any More

23

Cyclic PrefixCyclic Prefixyy

loz In multipath channel delayed replicas of previous OFDM signal lead to ISI between successive OFDM signals

loz Solution Insert a guard interval between successive OFDMloz Solution Insert a guard interval between successive OFDM signals

24

Cyclic PrefixCyclic Prefix

loz Guard interval leads to intercarrier interference (ICI) in OFDM d d l i

yy

demodulation

loz In DFT interval difference between two subcarriers does not maintain integer number of cycles loss of orthogonality

loz Delayed version of subcarrier 2 causes ICI in the process of d d l ti b i 1demodulating subcarrier 1

25

Cyclic PrefixCyclic Prefixloz Cyclic prefix (CP) A copy of the last part of OFDM signal is

attached to the front of itself

yy

attached to the front of itself

[ ]0d

[ ]1d

[ ]0D[ ]1D

[ ]nd[ ]kD~

symbols dataInput

bullbullbullbull

[ ]1d

[ ]2d

[ ][ ]2D

bullbullbullbullbull

y

bullbullbullbullbull

[ ]1minusNd [ ]1minusND

[ ]gNND minus

bullbullbull

26

Cyclic PrefixCyclic Prefix

loz All delayed replicas of subcarriers always have an

yy

y p yinteger number of cycles within DFT interval no ICI

27

Cyclic PrefixCyclic Prefix

loz Linear convolution vs circular convolution

yy

N

28

Cyclic PrefixCyclic Prefix

loz Channel effect with cyclic prefix

yy

Signal after removed CP

29

Cyclic PrefixCyclic Prefix

loz Time-Domain Explanation

yy

p

30

Cyclic PrefixCyclic Prefixyy

31

Cyclic PrefixCyclic Prefix

loz Spectrum of channel response with length (smaller [ ]nh hL

yy

than )h

gN[ ] nhFFTHk =

loz Received complete OFDM signal[ ] [ ] [ ] 20 ~~ minus++leleotimes= hg LNNnnhnDnr

loz Received useful part [ ]nr[ ] [ ] [ ]nhnDnr Notimes=

where is N-point circular convolution (due to CP)loz Received symbol at k-th subcarrier

[ ] [ ] [ ]N

Notimes

[ ] [ ] [ ] kkNk HXnhnDFFTnrFFTY =otimes== kYX =rArr ldquoUseful property for OFDM system to reduce

kk H

X =rArrcomplexity of channel equalizationrdquo

32

Cyclic Cyclic Prefix (OFDM Prefix (OFDM Receiver)Receiver)yy (( ))

bull bull( )( )nr~Symbols

dataOutput

bullbullbull

bullbullbull( )nr

bullbull

bullbull

33

Cyclic PrefixCyclic Prefix

loz One of the most important reasons to do OFDM is the

yy

pefficient way it deals with multipath delay spread

loz To eliminate inter-symbol interference (ISI) almostloz To eliminate inter symbol interference (ISI) almost completely a guard time is introduced for each OFDM symbolsymbol(The guard time is chosen larger than the delay spread)

34

Bandwidth EfficiencyBandwidth Efficiencyyy

loz In a classical parallel system the channel is divided into p y N non-overlapping sub-channels to avoid inter-carrier interference (ICI)( )

loz The diagram for bandwidth efficiency of OFDM system is shown below

35

SummarySummaryyy

loz The advantage of the FFT-based OFDM system g yloz The use of IFFTFFT can reduce the computation complexityloz The orthogonality between the adjacent sub-carriers will make g y j

the use of transmission bandwidth more efficientloz The guard interval is used to resist the inter-symbol interference

(ISI)loz The main advantage of the OFDM transmission technique is its

high performance even in frequency selective channels

loz The drawbacks of the OFDM system loz It is highly vulnerable to synchronization errorsloz Peak to Average Power Ratio (PAPR) problems

36

SynchronizationSynchronization

Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offset (TO)loz Impact of timing offset (TO)loz Impact of carrier frequency offset (CFO)loz TO estimation algorithmloz CFO estimation algorithmg

38 Synchronization2010722

Introduction (13)Introduction (13)( )( )

loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods

loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead

loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the

cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of

computational complexity therefore it may be not be available in short-burst wireless communication

39 Synchronization2010722

Introduction (23)Introduction (23)( )( )

loz Transcevier

40 Synchronization2010722

Introduction (33)Introduction (33)( )( )

loz Synchronization erroryloz Symbol timing offset (TO)

loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)

loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect

41 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg

42 Synchronization2010722

Signal model (14)Signal model (14)g ( )g ( )

loz Time domain sample p1

2

0

1 Nj nk N

n kk

x X eN

πminus

minus= sum

Xk the frequency domain data at the k-th subcarrier0kN =

loz The multipath fading channel

( ) ( ) ( )1L

h t h t δminus

sum

loz Received time domain signal

( ) ( ) ( )0

l ll

h t h tτ τ δ τ τ=

= minussum

loz Received time-domain signal

( ) ( )1

L

n n ly h n l x w nminus

minus= +sum43

( ) ( )0l=sum

Synchronization2010722

Signal model (24)Signal model (24)g ( )g ( )

loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting

12

0

Nj kn N

k nn

Y y e πminus

minus

=

= sum

( )1 1

2

0

N Lj kn N

n l kn l

h n l x e Wπminus minus

minusminus

=

= +sumsum

loz The equation above could be expressed in matrix form

0n l

loz The equation above could be expressed in matrix form

= +Y HX W

44 Synchronization2010722

Signal model (34)Signal model (34)g ( )g ( )

H= = = = +Y Fy FCx FCF X HX W

loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal

y

H=H FCFcirculant matrix leading to be a diagonal matrix

=H FCF

( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0

h h L L hh h

⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥

( )( ) ( )

( ) ( )

1 10 1 0 0i

h Lh L h N L

⎢ ⎥⎢ ⎥minus minus⎢ ⎥

equiv= minus minus⎢ ⎥⎢ ⎥

C( ) ( )

( ) ( )

0 1 1 100

0 0 1 1 0

h L h N L

h N L L h N

⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

45

( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦

Synchronization2010722

Signal model (44)Signal model (44)g ( )g ( )

loz In a time-invariant channel the channel matrix H would be a diagonal matrix

k k k kY H X W= +

( )1 1

2

0 0

N Lj kl N

kn l

H h n l e πminus minus

minus

= =

= sumsum

loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI

k k k k kY H X I Wα= + +k k k k k

( ) ( )1 1 1

2 2 N N L

j m k Nj ml Nk mI X h n l e e ππ

minus minus minusminusminus⎛ ⎞

= ⎜ ⎟⎝ ⎠

sum sum sumICI

46

0 0 0m n lm k= = =ne

⎝ ⎠

Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

47 Synchronization2010722

Impact of timing offset Impact of timing offset p gp g

loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively

( )I f s

Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ

( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr

2 fj k Nk k k kY X H e Wπδminus= +

loz For integer part of timing offset k k k kY X H e W+

I sTδ ( ) 0g IN L δminus minus le leFFT

2 Ij k NY X H e Wπδminus= +

( ) ( ) 2 I sFFT

j f TI sh t T H f e π δδ minusminus rarr

48

k k k kY X H e W= +

Synchronization2010722

Impact of timing offset derivationImpact of timing offset derivationp gp g

49 Synchronization2010722

Discussion of symbol boundaryDiscussion of symbol boundaryy yy y

50 Synchronization2010722

Impact of timing offsetImpact of timing offsetp gp g

loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI

2 Ij k NIk k k k

NY X H e ISI W

Nπδδ minusminus

= + +

loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk

e e

p kloz Phase Shift can be resolved by Differential Encoding

Decoding or carrier recovery with pilots

51 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

52 Synchronization2010722

Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )

53 Synchronization2010722

Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )

( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s

i n t i N N T N T nTy y t e

= + + +=

( ) ( )( ) ( ) ( ) 12

sin

sin

g gI f f

I I

i N N N Nj jf N Ni k i k k

f

Y X H e eN

π ε ε π ε

ε ε

πεπε

+ + minus+

minus minus=⎛ ⎞⎜ ⎟

Faded signal attenuated and rotated by CFO

( )( ) ( ) ( ) ( )11 2

sin

sin g gI f I f

i N N N NN j j l kI f

NN

l k π ε ε π ε επ ε ε + + minusminus + + + minus

⎜ ⎟⎝ ⎠

+ + minussum

( )( )( )

( ) ( )2

0

sin

I f I f

I

j j l kI f N Ni l l

l l k I f

X H e el k

NN

π ε ε π ε ε

ε π ε ε

+ + +

= ne minus

+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠

sum

i k

N

W

⎜ ⎟⎝ ⎠

+ Inter-Carrier Interference (ICI)

54 Synchronization2010722

Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )

loz An easier representation of received signal with only p g yfractional CFO is

1N

( ) ( )1

0

0N

k k k l l kl l k

Y H X C H X C l k Wminus

= ne

= + minus +⎡ ⎤⎣ ⎦sum

where the ICI coefficient is given by

( )( )( ) ( )

sin 1exp 1fl k

C l k j l kπ ε

π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )

( )exp 1sin

f

f

C l k j l kNN l k

N

π επ ε

minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠

55 Synchronization2010722

Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )

loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality

loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift

56 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

57 Synchronization2010722

TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )

loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq

58 Synchronization2010722

TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )

loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below

loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and

rN2 can be expressed by

0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+

( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +

59

( )

Synchronization2010722

TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )

loz Without CFO the first half is identical to the second half (in time order)

loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below

( ) 12 in discretesft T f NT

NTφ π π π ε πε

⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟

⎝ ⎠

helliphellip

sNT⎝ ⎠

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)

60

Length=N2

Synchronization2010722

TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )

loz Received signal without CFOg0

0 0

jr a e θ=jθ 2 0

2 2 0

Nj j

N Nr a e a e

θ θ= =

loz Received signal with CFO0 0

0 0

j jr a e eθ φ= Extra phase rotation due to CFO0 0

2 2 20

2 2 0

N N Nj j jj

N Nr a e e a e e

θ φ φθ= =

due to CFO

Phase difference 2 2 0N N

( )2 02 20 2 0 0

Nj j T fN

r r a e a eφ φ πminus Δ= =

which contains the information about CFO

61

0 2 0 0Nr r a e a e

Synchronization2010722

TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )

loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be

loz Note that d is a time index corresponding to the first( ) ( ) 2 1

20

Nd m d m Nm

P d r rminus

+ + +== sum

loz Note that d is a time index corresponding to the first sample in a window of N samples

loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by

( )22 1

20

Nd m NR d rminus

+ += sumloz A timing metric can be defined as

))(()(

)( 2

2

dRdP

dM =

( ) 20 d m Nm + +=sum

62

))(( dR

Synchronization2010722

TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )

63 Synchronization2010722

TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )

loz If drsquo is the correct symbol timing offset

( )2

2 2

0 1 12

j T f j T f j T fN

P d a e a e a eπ π πΔ Δ Δ

minus= + + +

22

2 2

0 1 12

j T fN

e a a aπ Δ

minus

⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭

loz If drsquo falls behind the correct symbol timing offset 1 sample

2⎪ ⎪⎩ ⎭

64 Synchronization2010722

TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )

65 Synchronization2010722

TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )

loz Drawback Plateau effect

Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)

((N2) ((N2) ((N2)

helliphellip

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)

Sliding window (length = N)

66 Synchronization2010722

TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )

loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula

( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus

扣掉相加

( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +

loz The method also called delay correlator

67 Synchronization2010722

IEEE 80211aIEEE 80211a

68 Synchronization2010722

Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )

loz Minnrsquos Method [2][ ]B B －B －B

( )( )

( )

22

2Minn

P dM d =

1 4 1N minus

( )( )( )2

2

MinnR d

( ) ( ) ( ) ( )

1 4 1

2 2 2 40 0

N

d N m k d N m k Nm k

P d r r+ + + + += =

= sum sum

( ) ( ) ( )2

21 4 1

2 40 0

N

d N m k Nk

R d rminus

+ + += sum sum69

0 0m k= =

Synchronization2010722

Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )

loz Parkrsquos Method [3][ ]

where D is the symmetric version of CC D C D

where D is the symmetric version of C

( )( ) 2

3k

P dM d =( )

( )( )23

ParkM dR d

2N

( ) 2

30

N

d k d kk

P d r rminus +=

= sdotsum

( )3

22

0

N

d kk

R d r += sum70

0k=

Synchronization2010722

Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )

loz Renrsquos Method [4][ ]

AS AS

where S is the PN sequence weighted of the original bl 2preamble

( )( )( )( )

24

Re 24

n

P dM d

R d=

( )( )4R d

( ) 2 1

4 2 2

N

k k N d k d k NP d s s r rminus

+ + + += sum0k=sum

( )21

41 N

d kR d rminus

= sum71

( )402 d k

kR d r +

=sum

Synchronization2010722

Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )

72 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

73 Synchronization2010722

Fractional CFO estimation Fractional CFO estimation

loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected

11 Rminus⎛ ⎞sum

h L i th di t b t t id ti l bl k R i

0

12 d m d m L

ms

f r rLTπ + + +

=

⎛ ⎞= ang⎜ ⎟⎝ ⎠sum

where L is the distance between two identical block R is the block size

loz For exampleNN cp

L = N R = NL = N R = Ncp74 Synchronization2010722

Estimation rangeEstimation rangegg

loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval

f ( ) ( )s s⎣ ⎦

loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)

loz DVB-T (L=N) 802 11 (L N4)

[ ]05 05s sf fminus

[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)

(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus

loz Note that is the subcarrier spacing1sf NT=

75

sNT

Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz

plusmng q y

loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn

76 Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz

plusmng q y

loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for

subcarrier spacing is required

plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn

p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs

77 Synchronization2010722

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

OFDM System ModelOFDM System Model

tNΔ

yy

( )tf02cos π

( )0a

( )0atT

tNΔ1+

( )tf2i

( )

( )0b

T

DataEncoder

tf s Δ=

1 ( ) ( ) njbna +

( )nd SP MUX ( )tDChannelInput

( )tf02sin π

( )tf N 12cos minusπbullbullbull

( )1minusNa

( )1minusNb( )0a1+

( )tf N 12sin minusπ

( )1Nb

tT

( )1a ( )1minusNa1minus

tΔ

Figure 1tNΔ

( )1a ( )1minusNa Figure 1

10

OFDM System ModelOFDM System Model

loz An OFDM system transmitter shown in Figure 1

yy

y gloz The transmitted waveform D(t) can be expressed as

)1( )2sin()()2cos()()(1

0summinus

=

+=N

nnn tfnbtfnatD ππ

where tNffnffn Δ=ΔΔ+=

1 and 0

loz Using a two-dimensional digital modulation format the data

tNΔ

g g symbols d(n) can be represented as a(n) + jb(n)loz a(n) in-phase componentloz b(n) quadrature component

11

OFDM System ModelOFDM System Model

loz The serial data elements spaced by are grouped and tΔ

yy

p y g pused to modulate N carriers Thus they are frequency division multiplexedp

loz The signaling interval is then increased to which tNΔmakes the system less susceptible to channel delay spread impairments

Small-scale fading(Based on multipath time delay spread)

Flat Fading1 BW of signal lt BW of channel

Frequency Selective Fading1 BW of signal gt BW of channel

2 Delay spread lt Symbol preiod 2 Delay spread gt Symbol preiod

12

OrthogonalityOrthogonalityg yg y

loz Consider a set of transmitted carriers as follows

(2) 1 1 0for )( 2 0

minus==⎟⎠⎞

⎜⎝⎛

Δ+

Nnett

tNnfj

n

πψ n

⎨⎧ =minus

intqpab

dtttb for )(

)()( ψψ⎩⎨ Δ=minusne

=int tNabqpdttt

a qp )( and for 0 )()( ψψ

13

OrthogonalityOrthogonality

b tqpjb

intint)(2π

g yg y

dtedtttajbj

b

atN

qpjb

a qp = Δminus

intint )()(

)(2)(2

)(2ψψπ

tNqpjee tN

qpjtN

qpj

Δminusminus

=Δ

minusΔ

minus

)(2

)(2)(2

π

ππ

ee

tNqpjba

tNqpj

tNbqpj

⎟⎟⎞

⎜⎜⎛minus

Δ

minusΔ

minusΔ

minus1

)(2)(1)(2)(2

π

ππ

tNqpj Δminus

⎟⎠

⎜⎝=

)(2πtNabqp Δ=minusne= )( and for 0

14

OrthogonalityOrthogonalityg yg y

tNΔ

15

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFTby using IDFTby using IDFT

loz According to the structure of Tx it must use Ng oscillators That increases the hardware complexity

loz The equivalent method is using IDFT (IFFT)

16

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

loz In general each carrier can be expressed as

by using IDFTby using IDFT

g p

( ) (6) )()( )(2 ttfjcc

ccetAtS φπ +=

loz We assume that there are N carriers in the OFDM signal Th th t t l l i l S (t) b t d bThen the total complex signal Ss(t) can be represented by

( ) (7))(1)(1

)(2etAtSN

ttfj nn φπ= summinus

+( )

where

(7) )()(

0

0

fnff

etAN

tSn

ns

Δ+=

= sum=

liihffrequency carrier phase amplitude are )( )( and

where 0

fttAfnff

nnn

n

φΔ+

lyrespective carrierth -of n17

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

loz Then we sample the signal at a sampling frequency tΔ1

by using IDFTby using IDFT

p g p g q y and An(t) and φn(t) becomes

(8))(t = φφ(9) )((8) )(

nn

nn

AtAt

== φφ

( ) (10) 1)(1

0

)(2 0summinus

=

+ΔΔ+=ΔN

n

tkfnfjns

neAN

tkS φπ

loz Then the sampled signal can be expressed as0=nN

( )( ) (11) 1)(1

0

22 0summinus

=

ΔΔ+Δ sdot=ΔN

n

tfnkjtkfjns eeA

NtkS n πφπ

0n

18

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

loz The inverse discrete Fourier transform (IDFT) is defined

by using IDFTby using IDFT

( )as the following

1 21

NnkjN

πsumminus

(12) )(1)( 2

0

Nnkj

nefnF

Ntkf πsum

=

Δ=Δ

loz Comparing eq(11) and eq(12) the condition must be satisfied in order to make eq(11) an inverse Fouriersatisfied in order to make eq(11) an inverse Fourier transform relationship

1 (13) 1tN

fΔ

=Δ

19

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

loz If eq(13) is satisfied

by using IDFTby using IDFT

q ( ) loz is the frequency domain signal loz is the time domain signal

( )ntkfjneA φπ +Δ02

)( tkSs Δ gloz is the sub-channel spacingloz is the symbol duration in each sub-channel

)(s

fΔtNΔloz s e sy bo du o e c sub c e

loz This outcome is the same as the result obtained in the

N

loz This outcome is the same as the result obtained in the system of Figure 1 Therefore IDFT can be used to generate an OFDM transmission signalgenerate an OFDM transmission signal

20

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

( )tf02cos π

( )0a

by using IDFTby using IDFT

( )nd ( )tDI( )tf02sin π

( )0a

( )0b

bull

tf s Δ=

1 ( ) ( ) njbna +

( )nd

tNΔffnffn Δ

=Δ+=1 0

( )tDChannelInput

( )tf N 12cos minusπ

( )1minusNa

bullbull

tNΔ

( )tf N 12sin minusπ

( )1minusNb

( )0d

( )nd ( )tDChannelInput

bull bull

( )( )1d

( )2d

tf s Δ=

1 ( ) ( ) njbna +

fNΔffnffn Δ

=Δ+=1 0

ChannelInput

bullbullbullbull

bullbullbullbull

( )1minusNd

21

Frequency Error Results in ICIFrequency Error Results in ICIq yq y

22

Synchronization Error Results in ICISynchronization Error Results in ICIyy

Not Orthogonal Any More

23

Cyclic PrefixCyclic Prefixyy

loz In multipath channel delayed replicas of previous OFDM signal lead to ISI between successive OFDM signals

loz Solution Insert a guard interval between successive OFDMloz Solution Insert a guard interval between successive OFDM signals

24

Cyclic PrefixCyclic Prefix

loz Guard interval leads to intercarrier interference (ICI) in OFDM d d l i

yy

demodulation

loz In DFT interval difference between two subcarriers does not maintain integer number of cycles loss of orthogonality

loz Delayed version of subcarrier 2 causes ICI in the process of d d l ti b i 1demodulating subcarrier 1

25

Cyclic PrefixCyclic Prefixloz Cyclic prefix (CP) A copy of the last part of OFDM signal is

attached to the front of itself

yy

attached to the front of itself

[ ]0d

[ ]1d

[ ]0D[ ]1D

[ ]nd[ ]kD~

symbols dataInput

bullbullbullbull

[ ]1d

[ ]2d

[ ][ ]2D

bullbullbullbullbull

y

bullbullbullbullbull

[ ]1minusNd [ ]1minusND

[ ]gNND minus

bullbullbull

26

Cyclic PrefixCyclic Prefix

loz All delayed replicas of subcarriers always have an

yy

y p yinteger number of cycles within DFT interval no ICI

27

Cyclic PrefixCyclic Prefix

loz Linear convolution vs circular convolution

yy

N

28

Cyclic PrefixCyclic Prefix

loz Channel effect with cyclic prefix

yy

Signal after removed CP

29

Cyclic PrefixCyclic Prefix

loz Time-Domain Explanation

yy

p

30

Cyclic PrefixCyclic Prefixyy

31

Cyclic PrefixCyclic Prefix

loz Spectrum of channel response with length (smaller [ ]nh hL

yy

than )h

gN[ ] nhFFTHk =

loz Received complete OFDM signal[ ] [ ] [ ] 20 ~~ minus++leleotimes= hg LNNnnhnDnr

loz Received useful part [ ]nr[ ] [ ] [ ]nhnDnr Notimes=

where is N-point circular convolution (due to CP)loz Received symbol at k-th subcarrier

[ ] [ ] [ ]N

Notimes

[ ] [ ] [ ] kkNk HXnhnDFFTnrFFTY =otimes== kYX =rArr ldquoUseful property for OFDM system to reduce

kk H

X =rArrcomplexity of channel equalizationrdquo

32

Cyclic Cyclic Prefix (OFDM Prefix (OFDM Receiver)Receiver)yy (( ))

bull bull( )( )nr~Symbols

dataOutput

bullbullbull

bullbullbull( )nr

bullbull

bullbull

33

Cyclic PrefixCyclic Prefix

loz One of the most important reasons to do OFDM is the

yy

pefficient way it deals with multipath delay spread

loz To eliminate inter-symbol interference (ISI) almostloz To eliminate inter symbol interference (ISI) almost completely a guard time is introduced for each OFDM symbolsymbol(The guard time is chosen larger than the delay spread)

34

Bandwidth EfficiencyBandwidth Efficiencyyy

loz In a classical parallel system the channel is divided into p y N non-overlapping sub-channels to avoid inter-carrier interference (ICI)( )

loz The diagram for bandwidth efficiency of OFDM system is shown below

35

SummarySummaryyy

loz The advantage of the FFT-based OFDM system g yloz The use of IFFTFFT can reduce the computation complexityloz The orthogonality between the adjacent sub-carriers will make g y j

the use of transmission bandwidth more efficientloz The guard interval is used to resist the inter-symbol interference

(ISI)loz The main advantage of the OFDM transmission technique is its

high performance even in frequency selective channels

loz The drawbacks of the OFDM system loz It is highly vulnerable to synchronization errorsloz Peak to Average Power Ratio (PAPR) problems

36

SynchronizationSynchronization

Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offset (TO)loz Impact of timing offset (TO)loz Impact of carrier frequency offset (CFO)loz TO estimation algorithmloz CFO estimation algorithmg

38 Synchronization2010722

Introduction (13)Introduction (13)( )( )

loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods

loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead

loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the

cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of

computational complexity therefore it may be not be available in short-burst wireless communication

39 Synchronization2010722

Introduction (23)Introduction (23)( )( )

loz Transcevier

40 Synchronization2010722

Introduction (33)Introduction (33)( )( )

loz Synchronization erroryloz Symbol timing offset (TO)

loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)

loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect

41 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg

42 Synchronization2010722

Signal model (14)Signal model (14)g ( )g ( )

loz Time domain sample p1

2

0

1 Nj nk N

n kk

x X eN

πminus

minus= sum

Xk the frequency domain data at the k-th subcarrier0kN =

loz The multipath fading channel

( ) ( ) ( )1L

h t h t δminus

sum

loz Received time domain signal

( ) ( ) ( )0

l ll

h t h tτ τ δ τ τ=

= minussum

loz Received time-domain signal

( ) ( )1

L

n n ly h n l x w nminus

minus= +sum43

( ) ( )0l=sum

Synchronization2010722

Signal model (24)Signal model (24)g ( )g ( )

loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting

12

0

Nj kn N

k nn

Y y e πminus

minus

=

= sum

( )1 1

2

0

N Lj kn N

n l kn l

h n l x e Wπminus minus

minusminus

=

= +sumsum

loz The equation above could be expressed in matrix form

0n l

loz The equation above could be expressed in matrix form

= +Y HX W

44 Synchronization2010722

Signal model (34)Signal model (34)g ( )g ( )

H= = = = +Y Fy FCx FCF X HX W

loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal

y

H=H FCFcirculant matrix leading to be a diagonal matrix

=H FCF

( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0

h h L L hh h

⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥

( )( ) ( )

( ) ( )

1 10 1 0 0i

h Lh L h N L

⎢ ⎥⎢ ⎥minus minus⎢ ⎥

equiv= minus minus⎢ ⎥⎢ ⎥

C( ) ( )

( ) ( )

0 1 1 100

0 0 1 1 0

h L h N L

h N L L h N

⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

45

( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦

Synchronization2010722

Signal model (44)Signal model (44)g ( )g ( )

loz In a time-invariant channel the channel matrix H would be a diagonal matrix

k k k kY H X W= +

( )1 1

2

0 0

N Lj kl N

kn l

H h n l e πminus minus

minus

= =

= sumsum

loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI

k k k k kY H X I Wα= + +k k k k k

( ) ( )1 1 1

2 2 N N L

j m k Nj ml Nk mI X h n l e e ππ

minus minus minusminusminus⎛ ⎞

= ⎜ ⎟⎝ ⎠

sum sum sumICI

46

0 0 0m n lm k= = =ne

⎝ ⎠

Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

47 Synchronization2010722

Impact of timing offset Impact of timing offset p gp g

loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively

( )I f s

Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ

( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr

2 fj k Nk k k kY X H e Wπδminus= +

loz For integer part of timing offset k k k kY X H e W+

I sTδ ( ) 0g IN L δminus minus le leFFT

2 Ij k NY X H e Wπδminus= +

( ) ( ) 2 I sFFT

j f TI sh t T H f e π δδ minusminus rarr

48

k k k kY X H e W= +

Synchronization2010722

Impact of timing offset derivationImpact of timing offset derivationp gp g

49 Synchronization2010722

Discussion of symbol boundaryDiscussion of symbol boundaryy yy y

50 Synchronization2010722

Impact of timing offsetImpact of timing offsetp gp g

loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI

2 Ij k NIk k k k

NY X H e ISI W

Nπδδ minusminus

= + +

loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk

e e

p kloz Phase Shift can be resolved by Differential Encoding

Decoding or carrier recovery with pilots

51 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

52 Synchronization2010722

Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )

53 Synchronization2010722

Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )

( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s

i n t i N N T N T nTy y t e

= + + +=

( ) ( )( ) ( ) ( ) 12

sin

sin

g gI f f

I I

i N N N Nj jf N Ni k i k k

f

Y X H e eN

π ε ε π ε

ε ε

πεπε

+ + minus+

minus minus=⎛ ⎞⎜ ⎟

Faded signal attenuated and rotated by CFO

( )( ) ( ) ( ) ( )11 2

sin

sin g gI f I f

i N N N NN j j l kI f

NN

l k π ε ε π ε επ ε ε + + minusminus + + + minus

⎜ ⎟⎝ ⎠

+ + minussum

( )( )( )

( ) ( )2

0

sin

I f I f

I

j j l kI f N Ni l l

l l k I f

X H e el k

NN

π ε ε π ε ε

ε π ε ε

+ + +

= ne minus

+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠

sum

i k

N

W

⎜ ⎟⎝ ⎠

+ Inter-Carrier Interference (ICI)

54 Synchronization2010722

Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )

loz An easier representation of received signal with only p g yfractional CFO is

1N

( ) ( )1

0

0N

k k k l l kl l k

Y H X C H X C l k Wminus

= ne

= + minus +⎡ ⎤⎣ ⎦sum

where the ICI coefficient is given by

( )( )( ) ( )

sin 1exp 1fl k

C l k j l kπ ε

π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )

( )exp 1sin

f

f

C l k j l kNN l k

N

π επ ε

minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠

55 Synchronization2010722

Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )

loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality

loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift

56 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

57 Synchronization2010722

TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )

loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq

58 Synchronization2010722

TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )

loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below

loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and

rN2 can be expressed by

0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+

( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +

59

( )

Synchronization2010722

TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )

loz Without CFO the first half is identical to the second half (in time order)

loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below

( ) 12 in discretesft T f NT

NTφ π π π ε πε

⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟

⎝ ⎠

helliphellip

sNT⎝ ⎠

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)

60

Length=N2

Synchronization2010722

TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )

loz Received signal without CFOg0

0 0

jr a e θ=jθ 2 0

2 2 0

Nj j

N Nr a e a e

θ θ= =

loz Received signal with CFO0 0

0 0

j jr a e eθ φ= Extra phase rotation due to CFO0 0

2 2 20

2 2 0

N N Nj j jj

N Nr a e e a e e

θ φ φθ= =

due to CFO

Phase difference 2 2 0N N

( )2 02 20 2 0 0

Nj j T fN

r r a e a eφ φ πminus Δ= =

which contains the information about CFO

61

0 2 0 0Nr r a e a e

Synchronization2010722

TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )

loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be

loz Note that d is a time index corresponding to the first( ) ( ) 2 1

20

Nd m d m Nm

P d r rminus

+ + +== sum

loz Note that d is a time index corresponding to the first sample in a window of N samples

loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by

( )22 1

20

Nd m NR d rminus

+ += sumloz A timing metric can be defined as

))(()(

)( 2

2

dRdP

dM =

( ) 20 d m Nm + +=sum

62

))(( dR

Synchronization2010722

TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )

63 Synchronization2010722

TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )

loz If drsquo is the correct symbol timing offset

( )2

2 2

0 1 12

j T f j T f j T fN

P d a e a e a eπ π πΔ Δ Δ

minus= + + +

22

2 2

0 1 12

j T fN

e a a aπ Δ

minus

⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭

loz If drsquo falls behind the correct symbol timing offset 1 sample

2⎪ ⎪⎩ ⎭

64 Synchronization2010722

TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )

65 Synchronization2010722

TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )

loz Drawback Plateau effect

Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)

((N2) ((N2) ((N2)

helliphellip

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)

Sliding window (length = N)

66 Synchronization2010722

TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )

loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula

( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus

扣掉相加

( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +

loz The method also called delay correlator

67 Synchronization2010722

IEEE 80211aIEEE 80211a

68 Synchronization2010722

Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )

loz Minnrsquos Method [2][ ]B B －B －B

( )( )

( )

22

2Minn

P dM d =

1 4 1N minus

( )( )( )2

2

MinnR d

( ) ( ) ( ) ( )

1 4 1

2 2 2 40 0

N

d N m k d N m k Nm k

P d r r+ + + + += =

= sum sum

( ) ( ) ( )2

21 4 1

2 40 0

N

d N m k Nk

R d rminus

+ + += sum sum69

0 0m k= =

Synchronization2010722

Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )

loz Parkrsquos Method [3][ ]

where D is the symmetric version of CC D C D

where D is the symmetric version of C

( )( ) 2

3k

P dM d =( )

( )( )23

ParkM dR d

2N

( ) 2

30

N

d k d kk

P d r rminus +=

= sdotsum

( )3

22

0

N

d kk

R d r += sum70

0k=

Synchronization2010722

Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )

loz Renrsquos Method [4][ ]

AS AS

where S is the PN sequence weighted of the original bl 2preamble

( )( )( )( )

24

Re 24

n

P dM d

R d=

( )( )4R d

( ) 2 1

4 2 2

N

k k N d k d k NP d s s r rminus

+ + + += sum0k=sum

( )21

41 N

d kR d rminus

= sum71

( )402 d k

kR d r +

=sum

Synchronization2010722

Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )

72 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

73 Synchronization2010722

Fractional CFO estimation Fractional CFO estimation

loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected

11 Rminus⎛ ⎞sum

h L i th di t b t t id ti l bl k R i

0

12 d m d m L

ms

f r rLTπ + + +

=

⎛ ⎞= ang⎜ ⎟⎝ ⎠sum

where L is the distance between two identical block R is the block size

loz For exampleNN cp

L = N R = NL = N R = Ncp74 Synchronization2010722

Estimation rangeEstimation rangegg

loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval

f ( ) ( )s s⎣ ⎦

loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)

loz DVB-T (L=N) 802 11 (L N4)

[ ]05 05s sf fminus

[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)

(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus

loz Note that is the subcarrier spacing1sf NT=

75

sNT

Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz

plusmng q y

loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn

76 Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz

plusmng q y

loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for

subcarrier spacing is required

plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn

p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs

77 Synchronization2010722

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

OFDM System ModelOFDM System Model

loz An OFDM system transmitter shown in Figure 1

yy

y gloz The transmitted waveform D(t) can be expressed as

)1( )2sin()()2cos()()(1

0summinus

=

+=N

nnn tfnbtfnatD ππ

where tNffnffn Δ=ΔΔ+=

1 and 0

loz Using a two-dimensional digital modulation format the data

tNΔ

g g symbols d(n) can be represented as a(n) + jb(n)loz a(n) in-phase componentloz b(n) quadrature component

11

OFDM System ModelOFDM System Model

loz The serial data elements spaced by are grouped and tΔ

yy

p y g pused to modulate N carriers Thus they are frequency division multiplexedp

loz The signaling interval is then increased to which tNΔmakes the system less susceptible to channel delay spread impairments

Small-scale fading(Based on multipath time delay spread)

Flat Fading1 BW of signal lt BW of channel

Frequency Selective Fading1 BW of signal gt BW of channel

2 Delay spread lt Symbol preiod 2 Delay spread gt Symbol preiod

12

OrthogonalityOrthogonalityg yg y

loz Consider a set of transmitted carriers as follows

(2) 1 1 0for )( 2 0

minus==⎟⎠⎞

⎜⎝⎛

Δ+

Nnett

tNnfj

n

πψ n

⎨⎧ =minus

intqpab

dtttb for )(

)()( ψψ⎩⎨ Δ=minusne

=int tNabqpdttt

a qp )( and for 0 )()( ψψ

13

OrthogonalityOrthogonality

b tqpjb

intint)(2π

g yg y

dtedtttajbj

b

atN

qpjb

a qp = Δminus

intint )()(

)(2)(2

)(2ψψπ

tNqpjee tN

qpjtN

qpj

Δminusminus

=Δ

minusΔ

minus

)(2

)(2)(2

π

ππ

ee

tNqpjba

tNqpj

tNbqpj

⎟⎟⎞

⎜⎜⎛minus

Δ

minusΔ

minusΔ

minus1

)(2)(1)(2)(2

π

ππ

tNqpj Δminus

⎟⎠

⎜⎝=

)(2πtNabqp Δ=minusne= )( and for 0

14

OrthogonalityOrthogonalityg yg y

tNΔ

15

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFTby using IDFTby using IDFT

loz According to the structure of Tx it must use Ng oscillators That increases the hardware complexity

loz The equivalent method is using IDFT (IFFT)

16

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

loz In general each carrier can be expressed as

by using IDFTby using IDFT

g p

( ) (6) )()( )(2 ttfjcc

ccetAtS φπ +=

loz We assume that there are N carriers in the OFDM signal Th th t t l l i l S (t) b t d bThen the total complex signal Ss(t) can be represented by

( ) (7))(1)(1

)(2etAtSN

ttfj nn φπ= summinus

+( )

where

(7) )()(

0

0

fnff

etAN

tSn

ns

Δ+=

= sum=

liihffrequency carrier phase amplitude are )( )( and

where 0

fttAfnff

nnn

n

φΔ+

lyrespective carrierth -of n17

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

loz Then we sample the signal at a sampling frequency tΔ1

by using IDFTby using IDFT

p g p g q y and An(t) and φn(t) becomes

(8))(t = φφ(9) )((8) )(

nn

nn

AtAt

== φφ

( ) (10) 1)(1

0

)(2 0summinus

=

+ΔΔ+=ΔN

n

tkfnfjns

neAN

tkS φπ

loz Then the sampled signal can be expressed as0=nN

( )( ) (11) 1)(1

0

22 0summinus

=

ΔΔ+Δ sdot=ΔN

n

tfnkjtkfjns eeA

NtkS n πφπ

0n

18

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

loz The inverse discrete Fourier transform (IDFT) is defined

by using IDFTby using IDFT

( )as the following

1 21

NnkjN

πsumminus

(12) )(1)( 2

0

Nnkj

nefnF

Ntkf πsum

=

Δ=Δ

loz Comparing eq(11) and eq(12) the condition must be satisfied in order to make eq(11) an inverse Fouriersatisfied in order to make eq(11) an inverse Fourier transform relationship

1 (13) 1tN

fΔ

=Δ

19

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

loz If eq(13) is satisfied

by using IDFTby using IDFT

q ( ) loz is the frequency domain signal loz is the time domain signal

( )ntkfjneA φπ +Δ02

)( tkSs Δ gloz is the sub-channel spacingloz is the symbol duration in each sub-channel

)(s

fΔtNΔloz s e sy bo du o e c sub c e

loz This outcome is the same as the result obtained in the

N

loz This outcome is the same as the result obtained in the system of Figure 1 Therefore IDFT can be used to generate an OFDM transmission signalgenerate an OFDM transmission signal

20

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

( )tf02cos π

( )0a

by using IDFTby using IDFT

( )nd ( )tDI( )tf02sin π

( )0a

( )0b

bull

tf s Δ=

1 ( ) ( ) njbna +

( )nd

tNΔffnffn Δ

=Δ+=1 0

( )tDChannelInput

( )tf N 12cos minusπ

( )1minusNa

bullbull

tNΔ

( )tf N 12sin minusπ

( )1minusNb

( )0d

( )nd ( )tDChannelInput

bull bull

( )( )1d

( )2d

tf s Δ=

1 ( ) ( ) njbna +

fNΔffnffn Δ

=Δ+=1 0

ChannelInput

bullbullbullbull

bullbullbullbull

( )1minusNd

21

Frequency Error Results in ICIFrequency Error Results in ICIq yq y

22

Synchronization Error Results in ICISynchronization Error Results in ICIyy

Not Orthogonal Any More

23

Cyclic PrefixCyclic Prefixyy

loz In multipath channel delayed replicas of previous OFDM signal lead to ISI between successive OFDM signals

loz Solution Insert a guard interval between successive OFDMloz Solution Insert a guard interval between successive OFDM signals

24

Cyclic PrefixCyclic Prefix

loz Guard interval leads to intercarrier interference (ICI) in OFDM d d l i

yy

demodulation

loz In DFT interval difference between two subcarriers does not maintain integer number of cycles loss of orthogonality

loz Delayed version of subcarrier 2 causes ICI in the process of d d l ti b i 1demodulating subcarrier 1

25

Cyclic PrefixCyclic Prefixloz Cyclic prefix (CP) A copy of the last part of OFDM signal is

attached to the front of itself

yy

attached to the front of itself

[ ]0d

[ ]1d

[ ]0D[ ]1D

[ ]nd[ ]kD~

symbols dataInput

bullbullbullbull

[ ]1d

[ ]2d

[ ][ ]2D

bullbullbullbullbull

y

bullbullbullbullbull

[ ]1minusNd [ ]1minusND

[ ]gNND minus

bullbullbull

26

Cyclic PrefixCyclic Prefix

loz All delayed replicas of subcarriers always have an

yy

y p yinteger number of cycles within DFT interval no ICI

27

Cyclic PrefixCyclic Prefix

loz Linear convolution vs circular convolution

yy

N

28

Cyclic PrefixCyclic Prefix

loz Channel effect with cyclic prefix

yy

Signal after removed CP

29

Cyclic PrefixCyclic Prefix

loz Time-Domain Explanation

yy

p

30

Cyclic PrefixCyclic Prefixyy

31

Cyclic PrefixCyclic Prefix

loz Spectrum of channel response with length (smaller [ ]nh hL

yy

than )h

gN[ ] nhFFTHk =

loz Received complete OFDM signal[ ] [ ] [ ] 20 ~~ minus++leleotimes= hg LNNnnhnDnr

loz Received useful part [ ]nr[ ] [ ] [ ]nhnDnr Notimes=

where is N-point circular convolution (due to CP)loz Received symbol at k-th subcarrier

[ ] [ ] [ ]N

Notimes

[ ] [ ] [ ] kkNk HXnhnDFFTnrFFTY =otimes== kYX =rArr ldquoUseful property for OFDM system to reduce

kk H

X =rArrcomplexity of channel equalizationrdquo

32

Cyclic Cyclic Prefix (OFDM Prefix (OFDM Receiver)Receiver)yy (( ))

bull bull( )( )nr~Symbols

dataOutput

bullbullbull

bullbullbull( )nr

bullbull

bullbull

33

Cyclic PrefixCyclic Prefix

loz One of the most important reasons to do OFDM is the

yy

pefficient way it deals with multipath delay spread

loz To eliminate inter-symbol interference (ISI) almostloz To eliminate inter symbol interference (ISI) almost completely a guard time is introduced for each OFDM symbolsymbol(The guard time is chosen larger than the delay spread)

34

Bandwidth EfficiencyBandwidth Efficiencyyy

loz In a classical parallel system the channel is divided into p y N non-overlapping sub-channels to avoid inter-carrier interference (ICI)( )

loz The diagram for bandwidth efficiency of OFDM system is shown below

35

SummarySummaryyy

loz The advantage of the FFT-based OFDM system g yloz The use of IFFTFFT can reduce the computation complexityloz The orthogonality between the adjacent sub-carriers will make g y j

the use of transmission bandwidth more efficientloz The guard interval is used to resist the inter-symbol interference

(ISI)loz The main advantage of the OFDM transmission technique is its

high performance even in frequency selective channels

loz The drawbacks of the OFDM system loz It is highly vulnerable to synchronization errorsloz Peak to Average Power Ratio (PAPR) problems

36

SynchronizationSynchronization

Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offset (TO)loz Impact of timing offset (TO)loz Impact of carrier frequency offset (CFO)loz TO estimation algorithmloz CFO estimation algorithmg

38 Synchronization2010722

Introduction (13)Introduction (13)( )( )

loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods

loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead

loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the

cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of

computational complexity therefore it may be not be available in short-burst wireless communication

39 Synchronization2010722

Introduction (23)Introduction (23)( )( )

loz Transcevier

40 Synchronization2010722

Introduction (33)Introduction (33)( )( )

loz Synchronization erroryloz Symbol timing offset (TO)

loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)

loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect

41 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg

42 Synchronization2010722

Signal model (14)Signal model (14)g ( )g ( )

loz Time domain sample p1

2

0

1 Nj nk N

n kk

x X eN

πminus

minus= sum

Xk the frequency domain data at the k-th subcarrier0kN =

loz The multipath fading channel

( ) ( ) ( )1L

h t h t δminus

sum

loz Received time domain signal

( ) ( ) ( )0

l ll

h t h tτ τ δ τ τ=

= minussum

loz Received time-domain signal

( ) ( )1

L

n n ly h n l x w nminus

minus= +sum43

( ) ( )0l=sum

Synchronization2010722

Signal model (24)Signal model (24)g ( )g ( )

loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting

12

0

Nj kn N

k nn

Y y e πminus

minus

=

= sum

( )1 1

2

0

N Lj kn N

n l kn l

h n l x e Wπminus minus

minusminus

=

= +sumsum

loz The equation above could be expressed in matrix form

0n l

loz The equation above could be expressed in matrix form

= +Y HX W

44 Synchronization2010722

Signal model (34)Signal model (34)g ( )g ( )

H= = = = +Y Fy FCx FCF X HX W

loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal

y

H=H FCFcirculant matrix leading to be a diagonal matrix

=H FCF

( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0

h h L L hh h

⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥

( )( ) ( )

( ) ( )

1 10 1 0 0i

h Lh L h N L

⎢ ⎥⎢ ⎥minus minus⎢ ⎥

equiv= minus minus⎢ ⎥⎢ ⎥

C( ) ( )

( ) ( )

0 1 1 100

0 0 1 1 0

h L h N L

h N L L h N

⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

45

( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦

Synchronization2010722

Signal model (44)Signal model (44)g ( )g ( )

loz In a time-invariant channel the channel matrix H would be a diagonal matrix

k k k kY H X W= +

( )1 1

2

0 0

N Lj kl N

kn l

H h n l e πminus minus

minus

= =

= sumsum

loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI

k k k k kY H X I Wα= + +k k k k k

( ) ( )1 1 1

2 2 N N L

j m k Nj ml Nk mI X h n l e e ππ

minus minus minusminusminus⎛ ⎞

= ⎜ ⎟⎝ ⎠

sum sum sumICI

46

0 0 0m n lm k= = =ne

⎝ ⎠

Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

47 Synchronization2010722

Impact of timing offset Impact of timing offset p gp g

loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively

( )I f s

Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ

( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr

2 fj k Nk k k kY X H e Wπδminus= +

loz For integer part of timing offset k k k kY X H e W+

I sTδ ( ) 0g IN L δminus minus le leFFT

2 Ij k NY X H e Wπδminus= +

( ) ( ) 2 I sFFT

j f TI sh t T H f e π δδ minusminus rarr

48

k k k kY X H e W= +

Synchronization2010722

Impact of timing offset derivationImpact of timing offset derivationp gp g

49 Synchronization2010722

Discussion of symbol boundaryDiscussion of symbol boundaryy yy y

50 Synchronization2010722

Impact of timing offsetImpact of timing offsetp gp g

loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI

2 Ij k NIk k k k

NY X H e ISI W

Nπδδ minusminus

= + +

loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk

e e

p kloz Phase Shift can be resolved by Differential Encoding

Decoding or carrier recovery with pilots

51 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

52 Synchronization2010722

Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )

53 Synchronization2010722

Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )

( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s

i n t i N N T N T nTy y t e

= + + +=

( ) ( )( ) ( ) ( ) 12

sin

sin

g gI f f

I I

i N N N Nj jf N Ni k i k k

f

Y X H e eN

π ε ε π ε

ε ε

πεπε

+ + minus+

minus minus=⎛ ⎞⎜ ⎟

Faded signal attenuated and rotated by CFO

( )( ) ( ) ( ) ( )11 2

sin

sin g gI f I f

i N N N NN j j l kI f

NN

l k π ε ε π ε επ ε ε + + minusminus + + + minus

⎜ ⎟⎝ ⎠

+ + minussum

( )( )( )

( ) ( )2

0

sin

I f I f

I

j j l kI f N Ni l l

l l k I f

X H e el k

NN

π ε ε π ε ε

ε π ε ε

+ + +

= ne minus

+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠

sum

i k

N

W

⎜ ⎟⎝ ⎠

+ Inter-Carrier Interference (ICI)

54 Synchronization2010722

Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )

loz An easier representation of received signal with only p g yfractional CFO is

1N

( ) ( )1

0

0N

k k k l l kl l k

Y H X C H X C l k Wminus

= ne

= + minus +⎡ ⎤⎣ ⎦sum

where the ICI coefficient is given by

( )( )( ) ( )

sin 1exp 1fl k

C l k j l kπ ε

π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )

( )exp 1sin

f

f

C l k j l kNN l k

N

π επ ε

minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠

55 Synchronization2010722

Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )

loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality

loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift

56 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

57 Synchronization2010722

TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )

loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq

58 Synchronization2010722

TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )

loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below

loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and

rN2 can be expressed by

0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+

( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +

59

( )

Synchronization2010722

TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )

loz Without CFO the first half is identical to the second half (in time order)

loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below

( ) 12 in discretesft T f NT

NTφ π π π ε πε

⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟

⎝ ⎠

helliphellip

sNT⎝ ⎠

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)

60

Length=N2

Synchronization2010722

TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )

loz Received signal without CFOg0

0 0

jr a e θ=jθ 2 0

2 2 0

Nj j

N Nr a e a e

θ θ= =

loz Received signal with CFO0 0

0 0

j jr a e eθ φ= Extra phase rotation due to CFO0 0

2 2 20

2 2 0

N N Nj j jj

N Nr a e e a e e

θ φ φθ= =

due to CFO

Phase difference 2 2 0N N

( )2 02 20 2 0 0

Nj j T fN

r r a e a eφ φ πminus Δ= =

which contains the information about CFO

61

0 2 0 0Nr r a e a e

Synchronization2010722

TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )

loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be

loz Note that d is a time index corresponding to the first( ) ( ) 2 1

20

Nd m d m Nm

P d r rminus

+ + +== sum

loz Note that d is a time index corresponding to the first sample in a window of N samples

loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by

( )22 1

20

Nd m NR d rminus

+ += sumloz A timing metric can be defined as

))(()(

)( 2

2

dRdP

dM =

( ) 20 d m Nm + +=sum

62

))(( dR

Synchronization2010722

TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )

63 Synchronization2010722

TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )

loz If drsquo is the correct symbol timing offset

( )2

2 2

0 1 12

j T f j T f j T fN

P d a e a e a eπ π πΔ Δ Δ

minus= + + +

22

2 2

0 1 12

j T fN

e a a aπ Δ

minus

⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭

loz If drsquo falls behind the correct symbol timing offset 1 sample

2⎪ ⎪⎩ ⎭

64 Synchronization2010722

TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )

65 Synchronization2010722

TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )

loz Drawback Plateau effect

Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)

((N2) ((N2) ((N2)

helliphellip

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)

Sliding window (length = N)

66 Synchronization2010722

TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )

loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula

( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus

扣掉相加

( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +

loz The method also called delay correlator

67 Synchronization2010722

IEEE 80211aIEEE 80211a

68 Synchronization2010722

Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )

loz Minnrsquos Method [2][ ]B B －B －B

( )( )

( )

22

2Minn

P dM d =

1 4 1N minus

( )( )( )2

2

MinnR d

( ) ( ) ( ) ( )

1 4 1

2 2 2 40 0

N

d N m k d N m k Nm k

P d r r+ + + + += =

= sum sum

( ) ( ) ( )2

21 4 1

2 40 0

N

d N m k Nk

R d rminus

+ + += sum sum69

0 0m k= =

Synchronization2010722

Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )

loz Parkrsquos Method [3][ ]

where D is the symmetric version of CC D C D

where D is the symmetric version of C

( )( ) 2

3k

P dM d =( )

( )( )23

ParkM dR d

2N

( ) 2

30

N

d k d kk

P d r rminus +=

= sdotsum

( )3

22

0

N

d kk

R d r += sum70

0k=

Synchronization2010722

Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )

loz Renrsquos Method [4][ ]

AS AS

where S is the PN sequence weighted of the original bl 2preamble

( )( )( )( )

24

Re 24

n

P dM d

R d=

( )( )4R d

( ) 2 1

4 2 2

N

k k N d k d k NP d s s r rminus

+ + + += sum0k=sum

( )21

41 N

d kR d rminus

= sum71

( )402 d k

kR d r +

=sum

Synchronization2010722

Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )

72 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

73 Synchronization2010722

Fractional CFO estimation Fractional CFO estimation

loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected

11 Rminus⎛ ⎞sum

h L i th di t b t t id ti l bl k R i

0

12 d m d m L

ms

f r rLTπ + + +

=

⎛ ⎞= ang⎜ ⎟⎝ ⎠sum

where L is the distance between two identical block R is the block size

loz For exampleNN cp

L = N R = NL = N R = Ncp74 Synchronization2010722

Estimation rangeEstimation rangegg

loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval

f ( ) ( )s s⎣ ⎦

loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)

loz DVB-T (L=N) 802 11 (L N4)

[ ]05 05s sf fminus

[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)

(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus

loz Note that is the subcarrier spacing1sf NT=

75

sNT

Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz

plusmng q y

loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn

76 Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz

plusmng q y

loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for

subcarrier spacing is required

plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn

p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs

77 Synchronization2010722

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

OFDM System ModelOFDM System Model

loz The serial data elements spaced by are grouped and tΔ

yy

p y g pused to modulate N carriers Thus they are frequency division multiplexedp

loz The signaling interval is then increased to which tNΔmakes the system less susceptible to channel delay spread impairments

Small-scale fading(Based on multipath time delay spread)

Flat Fading1 BW of signal lt BW of channel

Frequency Selective Fading1 BW of signal gt BW of channel

2 Delay spread lt Symbol preiod 2 Delay spread gt Symbol preiod

12

OrthogonalityOrthogonalityg yg y

loz Consider a set of transmitted carriers as follows

(2) 1 1 0for )( 2 0

minus==⎟⎠⎞

⎜⎝⎛

Δ+

Nnett

tNnfj

n

πψ n

⎨⎧ =minus

intqpab

dtttb for )(

)()( ψψ⎩⎨ Δ=minusne

=int tNabqpdttt

a qp )( and for 0 )()( ψψ

13

OrthogonalityOrthogonality

b tqpjb

intint)(2π

g yg y

dtedtttajbj

b

atN

qpjb

a qp = Δminus

intint )()(

)(2)(2

)(2ψψπ

tNqpjee tN

qpjtN

qpj

Δminusminus

=Δ

minusΔ

minus

)(2

)(2)(2

π

ππ

ee

tNqpjba

tNqpj

tNbqpj

⎟⎟⎞

⎜⎜⎛minus

Δ

minusΔ

minusΔ

minus1

)(2)(1)(2)(2

π

ππ

tNqpj Δminus

⎟⎠

⎜⎝=

)(2πtNabqp Δ=minusne= )( and for 0

14

OrthogonalityOrthogonalityg yg y

tNΔ

15

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFTby using IDFTby using IDFT

loz According to the structure of Tx it must use Ng oscillators That increases the hardware complexity

loz The equivalent method is using IDFT (IFFT)

16

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

loz In general each carrier can be expressed as

by using IDFTby using IDFT

g p

( ) (6) )()( )(2 ttfjcc

ccetAtS φπ +=

loz We assume that there are N carriers in the OFDM signal Th th t t l l i l S (t) b t d bThen the total complex signal Ss(t) can be represented by

( ) (7))(1)(1

)(2etAtSN

ttfj nn φπ= summinus

+( )

where

(7) )()(

0

0

fnff

etAN

tSn

ns

Δ+=

= sum=

liihffrequency carrier phase amplitude are )( )( and

where 0

fttAfnff

nnn

n

φΔ+

lyrespective carrierth -of n17

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

loz Then we sample the signal at a sampling frequency tΔ1

by using IDFTby using IDFT

p g p g q y and An(t) and φn(t) becomes

(8))(t = φφ(9) )((8) )(

nn

nn

AtAt

== φφ

( ) (10) 1)(1

0

)(2 0summinus

=

+ΔΔ+=ΔN

n

tkfnfjns

neAN

tkS φπ

loz Then the sampled signal can be expressed as0=nN

( )( ) (11) 1)(1

0

22 0summinus

=

ΔΔ+Δ sdot=ΔN

n

tfnkjtkfjns eeA

NtkS n πφπ

0n

18

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

loz The inverse discrete Fourier transform (IDFT) is defined

by using IDFTby using IDFT

( )as the following

1 21

NnkjN

πsumminus

(12) )(1)( 2

0

Nnkj

nefnF

Ntkf πsum

=

Δ=Δ

loz Comparing eq(11) and eq(12) the condition must be satisfied in order to make eq(11) an inverse Fouriersatisfied in order to make eq(11) an inverse Fourier transform relationship

1 (13) 1tN

fΔ

=Δ

19

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

loz If eq(13) is satisfied

by using IDFTby using IDFT

q ( ) loz is the frequency domain signal loz is the time domain signal

( )ntkfjneA φπ +Δ02

)( tkSs Δ gloz is the sub-channel spacingloz is the symbol duration in each sub-channel

)(s

fΔtNΔloz s e sy bo du o e c sub c e

loz This outcome is the same as the result obtained in the

N

loz This outcome is the same as the result obtained in the system of Figure 1 Therefore IDFT can be used to generate an OFDM transmission signalgenerate an OFDM transmission signal

20

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

( )tf02cos π

( )0a

by using IDFTby using IDFT

( )nd ( )tDI( )tf02sin π

( )0a

( )0b

bull

tf s Δ=

1 ( ) ( ) njbna +

( )nd

tNΔffnffn Δ

=Δ+=1 0

( )tDChannelInput

( )tf N 12cos minusπ

( )1minusNa

bullbull

tNΔ

( )tf N 12sin minusπ

( )1minusNb

( )0d

( )nd ( )tDChannelInput

bull bull

( )( )1d

( )2d

tf s Δ=

1 ( ) ( ) njbna +

fNΔffnffn Δ

=Δ+=1 0

ChannelInput

bullbullbullbull

bullbullbullbull

( )1minusNd

21

Frequency Error Results in ICIFrequency Error Results in ICIq yq y

22

Synchronization Error Results in ICISynchronization Error Results in ICIyy

Not Orthogonal Any More

23

Cyclic PrefixCyclic Prefixyy

loz In multipath channel delayed replicas of previous OFDM signal lead to ISI between successive OFDM signals

loz Solution Insert a guard interval between successive OFDMloz Solution Insert a guard interval between successive OFDM signals

24

Cyclic PrefixCyclic Prefix

loz Guard interval leads to intercarrier interference (ICI) in OFDM d d l i

yy

demodulation

loz In DFT interval difference between two subcarriers does not maintain integer number of cycles loss of orthogonality

loz Delayed version of subcarrier 2 causes ICI in the process of d d l ti b i 1demodulating subcarrier 1

25

Cyclic PrefixCyclic Prefixloz Cyclic prefix (CP) A copy of the last part of OFDM signal is

attached to the front of itself

yy

attached to the front of itself

[ ]0d

[ ]1d

[ ]0D[ ]1D

[ ]nd[ ]kD~

symbols dataInput

bullbullbullbull

[ ]1d

[ ]2d

[ ][ ]2D

bullbullbullbullbull

y

bullbullbullbullbull

[ ]1minusNd [ ]1minusND

[ ]gNND minus

bullbullbull

26

Cyclic PrefixCyclic Prefix

loz All delayed replicas of subcarriers always have an

yy

y p yinteger number of cycles within DFT interval no ICI

27

Cyclic PrefixCyclic Prefix

loz Linear convolution vs circular convolution

yy

N

28

Cyclic PrefixCyclic Prefix

loz Channel effect with cyclic prefix

yy

Signal after removed CP

29

Cyclic PrefixCyclic Prefix

loz Time-Domain Explanation

yy

p

30

Cyclic PrefixCyclic Prefixyy

31

Cyclic PrefixCyclic Prefix

loz Spectrum of channel response with length (smaller [ ]nh hL

yy

than )h

gN[ ] nhFFTHk =

loz Received complete OFDM signal[ ] [ ] [ ] 20 ~~ minus++leleotimes= hg LNNnnhnDnr

loz Received useful part [ ]nr[ ] [ ] [ ]nhnDnr Notimes=

where is N-point circular convolution (due to CP)loz Received symbol at k-th subcarrier

[ ] [ ] [ ]N

Notimes

[ ] [ ] [ ] kkNk HXnhnDFFTnrFFTY =otimes== kYX =rArr ldquoUseful property for OFDM system to reduce

kk H

X =rArrcomplexity of channel equalizationrdquo

32

Cyclic Cyclic Prefix (OFDM Prefix (OFDM Receiver)Receiver)yy (( ))

bull bull( )( )nr~Symbols

dataOutput

bullbullbull

bullbullbull( )nr

bullbull

bullbull

33

Cyclic PrefixCyclic Prefix

loz One of the most important reasons to do OFDM is the

yy

pefficient way it deals with multipath delay spread

loz To eliminate inter-symbol interference (ISI) almostloz To eliminate inter symbol interference (ISI) almost completely a guard time is introduced for each OFDM symbolsymbol(The guard time is chosen larger than the delay spread)

34

Bandwidth EfficiencyBandwidth Efficiencyyy

loz In a classical parallel system the channel is divided into p y N non-overlapping sub-channels to avoid inter-carrier interference (ICI)( )

loz The diagram for bandwidth efficiency of OFDM system is shown below

35

SummarySummaryyy

loz The advantage of the FFT-based OFDM system g yloz The use of IFFTFFT can reduce the computation complexityloz The orthogonality between the adjacent sub-carriers will make g y j

the use of transmission bandwidth more efficientloz The guard interval is used to resist the inter-symbol interference

(ISI)loz The main advantage of the OFDM transmission technique is its

high performance even in frequency selective channels

loz The drawbacks of the OFDM system loz It is highly vulnerable to synchronization errorsloz Peak to Average Power Ratio (PAPR) problems

36

SynchronizationSynchronization

Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offset (TO)loz Impact of timing offset (TO)loz Impact of carrier frequency offset (CFO)loz TO estimation algorithmloz CFO estimation algorithmg

38 Synchronization2010722

Introduction (13)Introduction (13)( )( )

loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods

loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead

loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the

cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of

computational complexity therefore it may be not be available in short-burst wireless communication

39 Synchronization2010722

Introduction (23)Introduction (23)( )( )

loz Transcevier

40 Synchronization2010722

Introduction (33)Introduction (33)( )( )

loz Synchronization erroryloz Symbol timing offset (TO)

loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)

loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect

41 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg

42 Synchronization2010722

Signal model (14)Signal model (14)g ( )g ( )

loz Time domain sample p1

2

0

1 Nj nk N

n kk

x X eN

πminus

minus= sum

Xk the frequency domain data at the k-th subcarrier0kN =

loz The multipath fading channel

( ) ( ) ( )1L

h t h t δminus

sum

loz Received time domain signal

( ) ( ) ( )0

l ll

h t h tτ τ δ τ τ=

= minussum

loz Received time-domain signal

( ) ( )1

L

n n ly h n l x w nminus

minus= +sum43

( ) ( )0l=sum

Synchronization2010722

Signal model (24)Signal model (24)g ( )g ( )

loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting

12

0

Nj kn N

k nn

Y y e πminus

minus

=

= sum

( )1 1

2

0

N Lj kn N

n l kn l

h n l x e Wπminus minus

minusminus

=

= +sumsum

loz The equation above could be expressed in matrix form

0n l

loz The equation above could be expressed in matrix form

= +Y HX W

44 Synchronization2010722

Signal model (34)Signal model (34)g ( )g ( )

H= = = = +Y Fy FCx FCF X HX W

loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal

y

H=H FCFcirculant matrix leading to be a diagonal matrix

=H FCF

( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0

h h L L hh h

⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥

( )( ) ( )

( ) ( )

1 10 1 0 0i

h Lh L h N L

⎢ ⎥⎢ ⎥minus minus⎢ ⎥

equiv= minus minus⎢ ⎥⎢ ⎥

C( ) ( )

( ) ( )

0 1 1 100

0 0 1 1 0

h L h N L

h N L L h N

⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

45

( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦

Synchronization2010722

Signal model (44)Signal model (44)g ( )g ( )

loz In a time-invariant channel the channel matrix H would be a diagonal matrix

k k k kY H X W= +

( )1 1

2

0 0

N Lj kl N

kn l

H h n l e πminus minus

minus

= =

= sumsum

loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI

k k k k kY H X I Wα= + +k k k k k

( ) ( )1 1 1

2 2 N N L

j m k Nj ml Nk mI X h n l e e ππ

minus minus minusminusminus⎛ ⎞

= ⎜ ⎟⎝ ⎠

sum sum sumICI

46

0 0 0m n lm k= = =ne

⎝ ⎠

Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

47 Synchronization2010722

Impact of timing offset Impact of timing offset p gp g

loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively

( )I f s

Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ

( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr

2 fj k Nk k k kY X H e Wπδminus= +

loz For integer part of timing offset k k k kY X H e W+

I sTδ ( ) 0g IN L δminus minus le leFFT

2 Ij k NY X H e Wπδminus= +

( ) ( ) 2 I sFFT

j f TI sh t T H f e π δδ minusminus rarr

48

k k k kY X H e W= +

Synchronization2010722

Impact of timing offset derivationImpact of timing offset derivationp gp g

49 Synchronization2010722

Discussion of symbol boundaryDiscussion of symbol boundaryy yy y

50 Synchronization2010722

Impact of timing offsetImpact of timing offsetp gp g

loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI

2 Ij k NIk k k k

NY X H e ISI W

Nπδδ minusminus

= + +

loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk

e e

p kloz Phase Shift can be resolved by Differential Encoding

Decoding or carrier recovery with pilots

51 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

52 Synchronization2010722

Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )

53 Synchronization2010722

Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )

( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s

i n t i N N T N T nTy y t e

= + + +=

( ) ( )( ) ( ) ( ) 12

sin

sin

g gI f f

I I

i N N N Nj jf N Ni k i k k

f

Y X H e eN

π ε ε π ε

ε ε

πεπε

+ + minus+

minus minus=⎛ ⎞⎜ ⎟

Faded signal attenuated and rotated by CFO

( )( ) ( ) ( ) ( )11 2

sin

sin g gI f I f

i N N N NN j j l kI f

NN

l k π ε ε π ε επ ε ε + + minusminus + + + minus

⎜ ⎟⎝ ⎠

+ + minussum

( )( )( )

( ) ( )2

0

sin

I f I f

I

j j l kI f N Ni l l

l l k I f

X H e el k

NN

π ε ε π ε ε

ε π ε ε

+ + +

= ne minus

+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠

sum

i k

N

W

⎜ ⎟⎝ ⎠

+ Inter-Carrier Interference (ICI)

54 Synchronization2010722

Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )

loz An easier representation of received signal with only p g yfractional CFO is

1N

( ) ( )1

0

0N

k k k l l kl l k

Y H X C H X C l k Wminus

= ne

= + minus +⎡ ⎤⎣ ⎦sum

where the ICI coefficient is given by

( )( )( ) ( )

sin 1exp 1fl k

C l k j l kπ ε

π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )

( )exp 1sin

f

f

C l k j l kNN l k

N

π επ ε

minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠

55 Synchronization2010722

Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )

loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality

loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift

56 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

57 Synchronization2010722

TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )

loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq

58 Synchronization2010722

TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )

loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below

loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and

rN2 can be expressed by

0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+

( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +

59

( )

Synchronization2010722

TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )

loz Without CFO the first half is identical to the second half (in time order)

loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below

( ) 12 in discretesft T f NT

NTφ π π π ε πε

⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟

⎝ ⎠

helliphellip

sNT⎝ ⎠

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)

60

Length=N2

Synchronization2010722

TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )

loz Received signal without CFOg0

0 0

jr a e θ=jθ 2 0

2 2 0

Nj j

N Nr a e a e

θ θ= =

loz Received signal with CFO0 0

0 0

j jr a e eθ φ= Extra phase rotation due to CFO0 0

2 2 20

2 2 0

N N Nj j jj

N Nr a e e a e e

θ φ φθ= =

due to CFO

Phase difference 2 2 0N N

( )2 02 20 2 0 0

Nj j T fN

r r a e a eφ φ πminus Δ= =

which contains the information about CFO

61

0 2 0 0Nr r a e a e

Synchronization2010722

TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )

loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be

loz Note that d is a time index corresponding to the first( ) ( ) 2 1

20

Nd m d m Nm

P d r rminus

+ + +== sum

loz Note that d is a time index corresponding to the first sample in a window of N samples

loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by

( )22 1

20

Nd m NR d rminus

+ += sumloz A timing metric can be defined as

))(()(

)( 2

2

dRdP

dM =

( ) 20 d m Nm + +=sum

62

))(( dR

Synchronization2010722

TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )

63 Synchronization2010722

TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )

loz If drsquo is the correct symbol timing offset

( )2

2 2

0 1 12

j T f j T f j T fN

P d a e a e a eπ π πΔ Δ Δ

minus= + + +

22

2 2

0 1 12

j T fN

e a a aπ Δ

minus

⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭

loz If drsquo falls behind the correct symbol timing offset 1 sample

2⎪ ⎪⎩ ⎭

64 Synchronization2010722

TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )

65 Synchronization2010722

TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )

loz Drawback Plateau effect

Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)

((N2) ((N2) ((N2)

helliphellip

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)

Sliding window (length = N)

66 Synchronization2010722

TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )

loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula

( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus

扣掉相加

( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +

loz The method also called delay correlator

67 Synchronization2010722

IEEE 80211aIEEE 80211a

68 Synchronization2010722

Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )

loz Minnrsquos Method [2][ ]B B －B －B

( )( )

( )

22

2Minn

P dM d =

1 4 1N minus

( )( )( )2

2

MinnR d

( ) ( ) ( ) ( )

1 4 1

2 2 2 40 0

N

d N m k d N m k Nm k

P d r r+ + + + += =

= sum sum

( ) ( ) ( )2

21 4 1

2 40 0

N

d N m k Nk

R d rminus

+ + += sum sum69

0 0m k= =

Synchronization2010722

Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )

loz Parkrsquos Method [3][ ]

where D is the symmetric version of CC D C D

where D is the symmetric version of C

( )( ) 2

3k

P dM d =( )

( )( )23

ParkM dR d

2N

( ) 2

30

N

d k d kk

P d r rminus +=

= sdotsum

( )3

22

0

N

d kk

R d r += sum70

0k=

Synchronization2010722

Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )

loz Renrsquos Method [4][ ]

AS AS

where S is the PN sequence weighted of the original bl 2preamble

( )( )( )( )

24

Re 24

n

P dM d

R d=

( )( )4R d

( ) 2 1

4 2 2

N

k k N d k d k NP d s s r rminus

+ + + += sum0k=sum

( )21

41 N

d kR d rminus

= sum71

( )402 d k

kR d r +

=sum

Synchronization2010722

Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )

72 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

73 Synchronization2010722

Fractional CFO estimation Fractional CFO estimation

loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected

11 Rminus⎛ ⎞sum

h L i th di t b t t id ti l bl k R i

0

12 d m d m L

ms

f r rLTπ + + +

=

⎛ ⎞= ang⎜ ⎟⎝ ⎠sum

where L is the distance between two identical block R is the block size

loz For exampleNN cp

L = N R = NL = N R = Ncp74 Synchronization2010722

Estimation rangeEstimation rangegg

loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval

f ( ) ( )s s⎣ ⎦

loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)

loz DVB-T (L=N) 802 11 (L N4)

[ ]05 05s sf fminus

[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)

(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus

loz Note that is the subcarrier spacing1sf NT=

75

sNT

Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz

plusmng q y

loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn

76 Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz

plusmng q y

loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for

subcarrier spacing is required

plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn

p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs

77 Synchronization2010722

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

OrthogonalityOrthogonalityg yg y

loz Consider a set of transmitted carriers as follows

(2) 1 1 0for )( 2 0

minus==⎟⎠⎞

⎜⎝⎛

Δ+

Nnett

tNnfj

n

πψ n

⎨⎧ =minus

intqpab

dtttb for )(

)()( ψψ⎩⎨ Δ=minusne

=int tNabqpdttt

a qp )( and for 0 )()( ψψ

13

OrthogonalityOrthogonality

b tqpjb

intint)(2π

g yg y

dtedtttajbj

b

atN

qpjb

a qp = Δminus

intint )()(

)(2)(2

)(2ψψπ

tNqpjee tN

qpjtN

qpj

Δminusminus

=Δ

minusΔ

minus

)(2

)(2)(2

π

ππ

ee

tNqpjba

tNqpj

tNbqpj

⎟⎟⎞

⎜⎜⎛minus

Δ

minusΔ

minusΔ

minus1

)(2)(1)(2)(2

π

ππ

tNqpj Δminus

⎟⎠

⎜⎝=

)(2πtNabqp Δ=minusne= )( and for 0

14

OrthogonalityOrthogonalityg yg y

tNΔ

15

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFTby using IDFTby using IDFT

loz According to the structure of Tx it must use Ng oscillators That increases the hardware complexity

loz The equivalent method is using IDFT (IFFT)

16

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

loz In general each carrier can be expressed as

by using IDFTby using IDFT

g p

( ) (6) )()( )(2 ttfjcc

ccetAtS φπ +=

loz We assume that there are N carriers in the OFDM signal Th th t t l l i l S (t) b t d bThen the total complex signal Ss(t) can be represented by

( ) (7))(1)(1

)(2etAtSN

ttfj nn φπ= summinus

+( )

where

(7) )()(

0

0

fnff

etAN

tSn

ns

Δ+=

= sum=

liihffrequency carrier phase amplitude are )( )( and

where 0

fttAfnff

nnn

n

φΔ+

lyrespective carrierth -of n17

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

loz Then we sample the signal at a sampling frequency tΔ1

by using IDFTby using IDFT

p g p g q y and An(t) and φn(t) becomes

(8))(t = φφ(9) )((8) )(

nn

nn

AtAt

== φφ

( ) (10) 1)(1

0

)(2 0summinus

=

+ΔΔ+=ΔN

n

tkfnfjns

neAN

tkS φπ

loz Then the sampled signal can be expressed as0=nN

( )( ) (11) 1)(1

0

22 0summinus

=

ΔΔ+Δ sdot=ΔN

n

tfnkjtkfjns eeA

NtkS n πφπ

0n

18

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

loz The inverse discrete Fourier transform (IDFT) is defined

by using IDFTby using IDFT

( )as the following

1 21

NnkjN

πsumminus

(12) )(1)( 2

0

Nnkj

nefnF

Ntkf πsum

=

Δ=Δ

loz Comparing eq(11) and eq(12) the condition must be satisfied in order to make eq(11) an inverse Fouriersatisfied in order to make eq(11) an inverse Fourier transform relationship

1 (13) 1tN

fΔ

=Δ

19

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

loz If eq(13) is satisfied

by using IDFTby using IDFT

q ( ) loz is the frequency domain signal loz is the time domain signal

( )ntkfjneA φπ +Δ02

)( tkSs Δ gloz is the sub-channel spacingloz is the symbol duration in each sub-channel

)(s

fΔtNΔloz s e sy bo du o e c sub c e

loz This outcome is the same as the result obtained in the

N

loz This outcome is the same as the result obtained in the system of Figure 1 Therefore IDFT can be used to generate an OFDM transmission signalgenerate an OFDM transmission signal

20

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

( )tf02cos π

( )0a

by using IDFTby using IDFT

( )nd ( )tDI( )tf02sin π

( )0a

( )0b

bull

tf s Δ=

1 ( ) ( ) njbna +

( )nd

tNΔffnffn Δ

=Δ+=1 0

( )tDChannelInput

( )tf N 12cos minusπ

( )1minusNa

bullbull

tNΔ

( )tf N 12sin minusπ

( )1minusNb

( )0d

( )nd ( )tDChannelInput

bull bull

( )( )1d

( )2d

tf s Δ=

1 ( ) ( ) njbna +

fNΔffnffn Δ

=Δ+=1 0

ChannelInput

bullbullbullbull

bullbullbullbull

( )1minusNd

21

Frequency Error Results in ICIFrequency Error Results in ICIq yq y

22

Synchronization Error Results in ICISynchronization Error Results in ICIyy

Not Orthogonal Any More

23

Cyclic PrefixCyclic Prefixyy

loz In multipath channel delayed replicas of previous OFDM signal lead to ISI between successive OFDM signals

loz Solution Insert a guard interval between successive OFDMloz Solution Insert a guard interval between successive OFDM signals

24

Cyclic PrefixCyclic Prefix

loz Guard interval leads to intercarrier interference (ICI) in OFDM d d l i

yy

demodulation

loz In DFT interval difference between two subcarriers does not maintain integer number of cycles loss of orthogonality

loz Delayed version of subcarrier 2 causes ICI in the process of d d l ti b i 1demodulating subcarrier 1

25

Cyclic PrefixCyclic Prefixloz Cyclic prefix (CP) A copy of the last part of OFDM signal is

attached to the front of itself

yy

attached to the front of itself

[ ]0d

[ ]1d

[ ]0D[ ]1D

[ ]nd[ ]kD~

symbols dataInput

bullbullbullbull

[ ]1d

[ ]2d

[ ][ ]2D

bullbullbullbullbull

y

bullbullbullbullbull

[ ]1minusNd [ ]1minusND

[ ]gNND minus

bullbullbull

26

Cyclic PrefixCyclic Prefix

loz All delayed replicas of subcarriers always have an

yy

y p yinteger number of cycles within DFT interval no ICI

27

Cyclic PrefixCyclic Prefix

loz Linear convolution vs circular convolution

yy

N

28

Cyclic PrefixCyclic Prefix

loz Channel effect with cyclic prefix

yy

Signal after removed CP

29

Cyclic PrefixCyclic Prefix

loz Time-Domain Explanation

yy

p

30

Cyclic PrefixCyclic Prefixyy

31

Cyclic PrefixCyclic Prefix

loz Spectrum of channel response with length (smaller [ ]nh hL

yy

than )h

gN[ ] nhFFTHk =

loz Received complete OFDM signal[ ] [ ] [ ] 20 ~~ minus++leleotimes= hg LNNnnhnDnr

loz Received useful part [ ]nr[ ] [ ] [ ]nhnDnr Notimes=

where is N-point circular convolution (due to CP)loz Received symbol at k-th subcarrier

[ ] [ ] [ ]N

Notimes

[ ] [ ] [ ] kkNk HXnhnDFFTnrFFTY =otimes== kYX =rArr ldquoUseful property for OFDM system to reduce

kk H

X =rArrcomplexity of channel equalizationrdquo

32

Cyclic Cyclic Prefix (OFDM Prefix (OFDM Receiver)Receiver)yy (( ))

bull bull( )( )nr~Symbols

dataOutput

bullbullbull

bullbullbull( )nr

bullbull

bullbull

33

Cyclic PrefixCyclic Prefix

loz One of the most important reasons to do OFDM is the

yy

pefficient way it deals with multipath delay spread

loz To eliminate inter-symbol interference (ISI) almostloz To eliminate inter symbol interference (ISI) almost completely a guard time is introduced for each OFDM symbolsymbol(The guard time is chosen larger than the delay spread)

34

Bandwidth EfficiencyBandwidth Efficiencyyy

loz In a classical parallel system the channel is divided into p y N non-overlapping sub-channels to avoid inter-carrier interference (ICI)( )

loz The diagram for bandwidth efficiency of OFDM system is shown below

35

SummarySummaryyy

loz The advantage of the FFT-based OFDM system g yloz The use of IFFTFFT can reduce the computation complexityloz The orthogonality between the adjacent sub-carriers will make g y j

the use of transmission bandwidth more efficientloz The guard interval is used to resist the inter-symbol interference

(ISI)loz The main advantage of the OFDM transmission technique is its

high performance even in frequency selective channels

loz The drawbacks of the OFDM system loz It is highly vulnerable to synchronization errorsloz Peak to Average Power Ratio (PAPR) problems

36

SynchronizationSynchronization

Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offset (TO)loz Impact of timing offset (TO)loz Impact of carrier frequency offset (CFO)loz TO estimation algorithmloz CFO estimation algorithmg

38 Synchronization2010722

Introduction (13)Introduction (13)( )( )

loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods

loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead

loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the

cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of

computational complexity therefore it may be not be available in short-burst wireless communication

39 Synchronization2010722

Introduction (23)Introduction (23)( )( )

loz Transcevier

40 Synchronization2010722

Introduction (33)Introduction (33)( )( )

loz Synchronization erroryloz Symbol timing offset (TO)

loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)

loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect

41 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg

42 Synchronization2010722

Signal model (14)Signal model (14)g ( )g ( )

loz Time domain sample p1

2

0

1 Nj nk N

n kk

x X eN

πminus

minus= sum

Xk the frequency domain data at the k-th subcarrier0kN =

loz The multipath fading channel

( ) ( ) ( )1L

h t h t δminus

sum

loz Received time domain signal

( ) ( ) ( )0

l ll

h t h tτ τ δ τ τ=

= minussum

loz Received time-domain signal

( ) ( )1

L

n n ly h n l x w nminus

minus= +sum43

( ) ( )0l=sum

Synchronization2010722

Signal model (24)Signal model (24)g ( )g ( )

loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting

12

0

Nj kn N

k nn

Y y e πminus

minus

=

= sum

( )1 1

2

0

N Lj kn N

n l kn l

h n l x e Wπminus minus

minusminus

=

= +sumsum

loz The equation above could be expressed in matrix form

0n l

loz The equation above could be expressed in matrix form

= +Y HX W

44 Synchronization2010722

Signal model (34)Signal model (34)g ( )g ( )

H= = = = +Y Fy FCx FCF X HX W

loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal

y

H=H FCFcirculant matrix leading to be a diagonal matrix

=H FCF

( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0

h h L L hh h

⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥

( )( ) ( )

( ) ( )

1 10 1 0 0i

h Lh L h N L

⎢ ⎥⎢ ⎥minus minus⎢ ⎥

equiv= minus minus⎢ ⎥⎢ ⎥

C( ) ( )

( ) ( )

0 1 1 100

0 0 1 1 0

h L h N L

h N L L h N

⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

45

( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦

Synchronization2010722

Signal model (44)Signal model (44)g ( )g ( )

loz In a time-invariant channel the channel matrix H would be a diagonal matrix

k k k kY H X W= +

( )1 1

2

0 0

N Lj kl N

kn l

H h n l e πminus minus

minus

= =

= sumsum

loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI

k k k k kY H X I Wα= + +k k k k k

( ) ( )1 1 1

2 2 N N L

j m k Nj ml Nk mI X h n l e e ππ

minus minus minusminusminus⎛ ⎞

= ⎜ ⎟⎝ ⎠

sum sum sumICI

46

0 0 0m n lm k= = =ne

⎝ ⎠

Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

47 Synchronization2010722

Impact of timing offset Impact of timing offset p gp g

loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively

( )I f s

Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ

( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr

2 fj k Nk k k kY X H e Wπδminus= +

loz For integer part of timing offset k k k kY X H e W+

I sTδ ( ) 0g IN L δminus minus le leFFT

2 Ij k NY X H e Wπδminus= +

( ) ( ) 2 I sFFT

j f TI sh t T H f e π δδ minusminus rarr

48

k k k kY X H e W= +

Synchronization2010722

Impact of timing offset derivationImpact of timing offset derivationp gp g

49 Synchronization2010722

Discussion of symbol boundaryDiscussion of symbol boundaryy yy y

50 Synchronization2010722

Impact of timing offsetImpact of timing offsetp gp g

loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI

2 Ij k NIk k k k

NY X H e ISI W

Nπδδ minusminus

= + +

loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk

e e

p kloz Phase Shift can be resolved by Differential Encoding

Decoding or carrier recovery with pilots

51 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

52 Synchronization2010722

Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )

53 Synchronization2010722

Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )

( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s

i n t i N N T N T nTy y t e

= + + +=

( ) ( )( ) ( ) ( ) 12

sin

sin

g gI f f

I I

i N N N Nj jf N Ni k i k k

f

Y X H e eN

π ε ε π ε

ε ε

πεπε

+ + minus+

minus minus=⎛ ⎞⎜ ⎟

Faded signal attenuated and rotated by CFO

( )( ) ( ) ( ) ( )11 2

sin

sin g gI f I f

i N N N NN j j l kI f

NN

l k π ε ε π ε επ ε ε + + minusminus + + + minus

⎜ ⎟⎝ ⎠

+ + minussum

( )( )( )

( ) ( )2

0

sin

I f I f

I

j j l kI f N Ni l l

l l k I f

X H e el k

NN

π ε ε π ε ε

ε π ε ε

+ + +

= ne minus

+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠

sum

i k

N

W

⎜ ⎟⎝ ⎠

+ Inter-Carrier Interference (ICI)

54 Synchronization2010722

Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )

loz An easier representation of received signal with only p g yfractional CFO is

1N

( ) ( )1

0

0N

k k k l l kl l k

Y H X C H X C l k Wminus

= ne

= + minus +⎡ ⎤⎣ ⎦sum

where the ICI coefficient is given by

( )( )( ) ( )

sin 1exp 1fl k

C l k j l kπ ε

π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )

( )exp 1sin

f

f

C l k j l kNN l k

N

π επ ε

minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠

55 Synchronization2010722

Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )

loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality

loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift

56 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

57 Synchronization2010722

TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )

loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq

58 Synchronization2010722

TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )

loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below

loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and

rN2 can be expressed by

0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+

( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +

59

( )

Synchronization2010722

TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )

loz Without CFO the first half is identical to the second half (in time order)

loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below

( ) 12 in discretesft T f NT

NTφ π π π ε πε

⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟

⎝ ⎠

helliphellip

sNT⎝ ⎠

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)

60

Length=N2

Synchronization2010722

TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )

loz Received signal without CFOg0

0 0

jr a e θ=jθ 2 0

2 2 0

Nj j

N Nr a e a e

θ θ= =

loz Received signal with CFO0 0

0 0

j jr a e eθ φ= Extra phase rotation due to CFO0 0

2 2 20

2 2 0

N N Nj j jj

N Nr a e e a e e

θ φ φθ= =

due to CFO

Phase difference 2 2 0N N

( )2 02 20 2 0 0

Nj j T fN

r r a e a eφ φ πminus Δ= =

which contains the information about CFO

61

0 2 0 0Nr r a e a e

Synchronization2010722

TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )

loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be

loz Note that d is a time index corresponding to the first( ) ( ) 2 1

20

Nd m d m Nm

P d r rminus

+ + +== sum

loz Note that d is a time index corresponding to the first sample in a window of N samples

loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by

( )22 1

20

Nd m NR d rminus

+ += sumloz A timing metric can be defined as

))(()(

)( 2

2

dRdP

dM =

( ) 20 d m Nm + +=sum

62

))(( dR

Synchronization2010722

TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )

63 Synchronization2010722

TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )

loz If drsquo is the correct symbol timing offset

( )2

2 2

0 1 12

j T f j T f j T fN

P d a e a e a eπ π πΔ Δ Δ

minus= + + +

22

2 2

0 1 12

j T fN

e a a aπ Δ

minus

⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭

loz If drsquo falls behind the correct symbol timing offset 1 sample

2⎪ ⎪⎩ ⎭

64 Synchronization2010722

TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )

65 Synchronization2010722

TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )

loz Drawback Plateau effect

Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)

((N2) ((N2) ((N2)

helliphellip

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)

Sliding window (length = N)

66 Synchronization2010722

TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )

loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula

( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus

扣掉相加

( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +

loz The method also called delay correlator

67 Synchronization2010722

IEEE 80211aIEEE 80211a

68 Synchronization2010722

Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )

loz Minnrsquos Method [2][ ]B B －B －B

( )( )

( )

22

2Minn

P dM d =

1 4 1N minus

( )( )( )2

2

MinnR d

( ) ( ) ( ) ( )

1 4 1

2 2 2 40 0

N

d N m k d N m k Nm k

P d r r+ + + + += =

= sum sum

( ) ( ) ( )2

21 4 1

2 40 0

N

d N m k Nk

R d rminus

+ + += sum sum69

0 0m k= =

Synchronization2010722

Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )

loz Parkrsquos Method [3][ ]

where D is the symmetric version of CC D C D

where D is the symmetric version of C

( )( ) 2

3k

P dM d =( )

( )( )23

ParkM dR d

2N

( ) 2

30

N

d k d kk

P d r rminus +=

= sdotsum

( )3

22

0

N

d kk

R d r += sum70

0k=

Synchronization2010722

Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )

loz Renrsquos Method [4][ ]

AS AS

where S is the PN sequence weighted of the original bl 2preamble

( )( )( )( )

24

Re 24

n

P dM d

R d=

( )( )4R d

( ) 2 1

4 2 2

N

k k N d k d k NP d s s r rminus

+ + + += sum0k=sum

( )21

41 N

d kR d rminus

= sum71

( )402 d k

kR d r +

=sum

Synchronization2010722

Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )

72 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

73 Synchronization2010722

Fractional CFO estimation Fractional CFO estimation

loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected

11 Rminus⎛ ⎞sum

h L i th di t b t t id ti l bl k R i

0

12 d m d m L

ms

f r rLTπ + + +

=

⎛ ⎞= ang⎜ ⎟⎝ ⎠sum

where L is the distance between two identical block R is the block size

loz For exampleNN cp

L = N R = NL = N R = Ncp74 Synchronization2010722

Estimation rangeEstimation rangegg

loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval

f ( ) ( )s s⎣ ⎦

loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)

loz DVB-T (L=N) 802 11 (L N4)

[ ]05 05s sf fminus

[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)

(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus

loz Note that is the subcarrier spacing1sf NT=

75

sNT

Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz

plusmng q y

loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn

76 Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz

plusmng q y

loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for

subcarrier spacing is required

plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn

p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs

77 Synchronization2010722

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

OrthogonalityOrthogonality

b tqpjb

intint)(2π

g yg y

dtedtttajbj

b

atN

qpjb

a qp = Δminus

intint )()(

)(2)(2

)(2ψψπ

tNqpjee tN

qpjtN

qpj

Δminusminus

=Δ

minusΔ

minus

)(2

)(2)(2

π

ππ

ee

tNqpjba

tNqpj

tNbqpj

⎟⎟⎞

⎜⎜⎛minus

Δ

minusΔ

minusΔ

minus1

)(2)(1)(2)(2

π

ππ

tNqpj Δminus

⎟⎠

⎜⎝=

)(2πtNabqp Δ=minusne= )( and for 0

14

OrthogonalityOrthogonalityg yg y

tNΔ

15

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFTby using IDFTby using IDFT

loz According to the structure of Tx it must use Ng oscillators That increases the hardware complexity

loz The equivalent method is using IDFT (IFFT)

16

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

loz In general each carrier can be expressed as

by using IDFTby using IDFT

g p

( ) (6) )()( )(2 ttfjcc

ccetAtS φπ +=

loz We assume that there are N carriers in the OFDM signal Th th t t l l i l S (t) b t d bThen the total complex signal Ss(t) can be represented by

( ) (7))(1)(1

)(2etAtSN

ttfj nn φπ= summinus

+( )

where

(7) )()(

0

0

fnff

etAN

tSn

ns

Δ+=

= sum=

liihffrequency carrier phase amplitude are )( )( and

where 0

fttAfnff

nnn

n

φΔ+

lyrespective carrierth -of n17

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

loz Then we sample the signal at a sampling frequency tΔ1

by using IDFTby using IDFT

p g p g q y and An(t) and φn(t) becomes

(8))(t = φφ(9) )((8) )(

nn

nn

AtAt

== φφ

( ) (10) 1)(1

0

)(2 0summinus

=

+ΔΔ+=ΔN

n

tkfnfjns

neAN

tkS φπ

loz Then the sampled signal can be expressed as0=nN

( )( ) (11) 1)(1

0

22 0summinus

=

ΔΔ+Δ sdot=ΔN

n

tfnkjtkfjns eeA

NtkS n πφπ

0n

18

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

loz The inverse discrete Fourier transform (IDFT) is defined

by using IDFTby using IDFT

( )as the following

1 21

NnkjN

πsumminus

(12) )(1)( 2

0

Nnkj

nefnF

Ntkf πsum

=

Δ=Δ

loz Comparing eq(11) and eq(12) the condition must be satisfied in order to make eq(11) an inverse Fouriersatisfied in order to make eq(11) an inverse Fourier transform relationship

1 (13) 1tN

fΔ

=Δ

19

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

loz If eq(13) is satisfied

by using IDFTby using IDFT

q ( ) loz is the frequency domain signal loz is the time domain signal

( )ntkfjneA φπ +Δ02

)( tkSs Δ gloz is the sub-channel spacingloz is the symbol duration in each sub-channel

)(s

fΔtNΔloz s e sy bo du o e c sub c e

loz This outcome is the same as the result obtained in the

N

loz This outcome is the same as the result obtained in the system of Figure 1 Therefore IDFT can be used to generate an OFDM transmission signalgenerate an OFDM transmission signal

20

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

( )tf02cos π

( )0a

by using IDFTby using IDFT

( )nd ( )tDI( )tf02sin π

( )0a

( )0b

bull

tf s Δ=

1 ( ) ( ) njbna +

( )nd

tNΔffnffn Δ

=Δ+=1 0

( )tDChannelInput

( )tf N 12cos minusπ

( )1minusNa

bullbull

tNΔ

( )tf N 12sin minusπ

( )1minusNb

( )0d

( )nd ( )tDChannelInput

bull bull

( )( )1d

( )2d

tf s Δ=

1 ( ) ( ) njbna +

fNΔffnffn Δ

=Δ+=1 0

ChannelInput

bullbullbullbull

bullbullbullbull

( )1minusNd

21

Frequency Error Results in ICIFrequency Error Results in ICIq yq y

22

Synchronization Error Results in ICISynchronization Error Results in ICIyy

Not Orthogonal Any More

23

Cyclic PrefixCyclic Prefixyy

loz In multipath channel delayed replicas of previous OFDM signal lead to ISI between successive OFDM signals

loz Solution Insert a guard interval between successive OFDMloz Solution Insert a guard interval between successive OFDM signals

24

Cyclic PrefixCyclic Prefix

loz Guard interval leads to intercarrier interference (ICI) in OFDM d d l i

yy

demodulation

loz In DFT interval difference between two subcarriers does not maintain integer number of cycles loss of orthogonality

loz Delayed version of subcarrier 2 causes ICI in the process of d d l ti b i 1demodulating subcarrier 1

25

Cyclic PrefixCyclic Prefixloz Cyclic prefix (CP) A copy of the last part of OFDM signal is

attached to the front of itself

yy

attached to the front of itself

[ ]0d

[ ]1d

[ ]0D[ ]1D

[ ]nd[ ]kD~

symbols dataInput

bullbullbullbull

[ ]1d

[ ]2d

[ ][ ]2D

bullbullbullbullbull

y

bullbullbullbullbull

[ ]1minusNd [ ]1minusND

[ ]gNND minus

bullbullbull

26

Cyclic PrefixCyclic Prefix

loz All delayed replicas of subcarriers always have an

yy

y p yinteger number of cycles within DFT interval no ICI

27

Cyclic PrefixCyclic Prefix

loz Linear convolution vs circular convolution

yy

N

28

Cyclic PrefixCyclic Prefix

loz Channel effect with cyclic prefix

yy

Signal after removed CP

29

Cyclic PrefixCyclic Prefix

loz Time-Domain Explanation

yy

p

30

Cyclic PrefixCyclic Prefixyy

31

Cyclic PrefixCyclic Prefix

loz Spectrum of channel response with length (smaller [ ]nh hL

yy

than )h

gN[ ] nhFFTHk =

loz Received complete OFDM signal[ ] [ ] [ ] 20 ~~ minus++leleotimes= hg LNNnnhnDnr

loz Received useful part [ ]nr[ ] [ ] [ ]nhnDnr Notimes=

where is N-point circular convolution (due to CP)loz Received symbol at k-th subcarrier

[ ] [ ] [ ]N

Notimes

[ ] [ ] [ ] kkNk HXnhnDFFTnrFFTY =otimes== kYX =rArr ldquoUseful property for OFDM system to reduce

kk H

X =rArrcomplexity of channel equalizationrdquo

32

Cyclic Cyclic Prefix (OFDM Prefix (OFDM Receiver)Receiver)yy (( ))

bull bull( )( )nr~Symbols

dataOutput

bullbullbull

bullbullbull( )nr

bullbull

bullbull

33

Cyclic PrefixCyclic Prefix

loz One of the most important reasons to do OFDM is the

yy

pefficient way it deals with multipath delay spread

loz To eliminate inter-symbol interference (ISI) almostloz To eliminate inter symbol interference (ISI) almost completely a guard time is introduced for each OFDM symbolsymbol(The guard time is chosen larger than the delay spread)

34

Bandwidth EfficiencyBandwidth Efficiencyyy

loz In a classical parallel system the channel is divided into p y N non-overlapping sub-channels to avoid inter-carrier interference (ICI)( )

loz The diagram for bandwidth efficiency of OFDM system is shown below

35

SummarySummaryyy

loz The advantage of the FFT-based OFDM system g yloz The use of IFFTFFT can reduce the computation complexityloz The orthogonality between the adjacent sub-carriers will make g y j

the use of transmission bandwidth more efficientloz The guard interval is used to resist the inter-symbol interference

(ISI)loz The main advantage of the OFDM transmission technique is its

high performance even in frequency selective channels

loz The drawbacks of the OFDM system loz It is highly vulnerable to synchronization errorsloz Peak to Average Power Ratio (PAPR) problems

36

SynchronizationSynchronization

Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offset (TO)loz Impact of timing offset (TO)loz Impact of carrier frequency offset (CFO)loz TO estimation algorithmloz CFO estimation algorithmg

38 Synchronization2010722

Introduction (13)Introduction (13)( )( )

loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods

loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead

loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the

cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of

computational complexity therefore it may be not be available in short-burst wireless communication

39 Synchronization2010722

Introduction (23)Introduction (23)( )( )

loz Transcevier

40 Synchronization2010722

Introduction (33)Introduction (33)( )( )

loz Synchronization erroryloz Symbol timing offset (TO)

loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)

loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect

41 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg

42 Synchronization2010722

Signal model (14)Signal model (14)g ( )g ( )

loz Time domain sample p1

2

0

1 Nj nk N

n kk

x X eN

πminus

minus= sum

Xk the frequency domain data at the k-th subcarrier0kN =

loz The multipath fading channel

( ) ( ) ( )1L

h t h t δminus

sum

loz Received time domain signal

( ) ( ) ( )0

l ll

h t h tτ τ δ τ τ=

= minussum

loz Received time-domain signal

( ) ( )1

L

n n ly h n l x w nminus

minus= +sum43

( ) ( )0l=sum

Synchronization2010722

Signal model (24)Signal model (24)g ( )g ( )

loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting

12

0

Nj kn N

k nn

Y y e πminus

minus

=

= sum

( )1 1

2

0

N Lj kn N

n l kn l

h n l x e Wπminus minus

minusminus

=

= +sumsum

loz The equation above could be expressed in matrix form

0n l

loz The equation above could be expressed in matrix form

= +Y HX W

44 Synchronization2010722

Signal model (34)Signal model (34)g ( )g ( )

H= = = = +Y Fy FCx FCF X HX W

loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal

y

H=H FCFcirculant matrix leading to be a diagonal matrix

=H FCF

( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0

h h L L hh h

⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥

( )( ) ( )

( ) ( )

1 10 1 0 0i

h Lh L h N L

⎢ ⎥⎢ ⎥minus minus⎢ ⎥

equiv= minus minus⎢ ⎥⎢ ⎥

C( ) ( )

( ) ( )

0 1 1 100

0 0 1 1 0

h L h N L

h N L L h N

⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

45

( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦

Synchronization2010722

Signal model (44)Signal model (44)g ( )g ( )

loz In a time-invariant channel the channel matrix H would be a diagonal matrix

k k k kY H X W= +

( )1 1

2

0 0

N Lj kl N

kn l

H h n l e πminus minus

minus

= =

= sumsum

loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI

k k k k kY H X I Wα= + +k k k k k

( ) ( )1 1 1

2 2 N N L

j m k Nj ml Nk mI X h n l e e ππ

minus minus minusminusminus⎛ ⎞

= ⎜ ⎟⎝ ⎠

sum sum sumICI

46

0 0 0m n lm k= = =ne

⎝ ⎠

Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

47 Synchronization2010722

Impact of timing offset Impact of timing offset p gp g

loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively

( )I f s

Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ

( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr

2 fj k Nk k k kY X H e Wπδminus= +

loz For integer part of timing offset k k k kY X H e W+

I sTδ ( ) 0g IN L δminus minus le leFFT

2 Ij k NY X H e Wπδminus= +

( ) ( ) 2 I sFFT

j f TI sh t T H f e π δδ minusminus rarr

48

k k k kY X H e W= +

Synchronization2010722

Impact of timing offset derivationImpact of timing offset derivationp gp g

49 Synchronization2010722

Discussion of symbol boundaryDiscussion of symbol boundaryy yy y

50 Synchronization2010722

Impact of timing offsetImpact of timing offsetp gp g

loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI

2 Ij k NIk k k k

NY X H e ISI W

Nπδδ minusminus

= + +

loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk

e e

p kloz Phase Shift can be resolved by Differential Encoding

Decoding or carrier recovery with pilots

51 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

52 Synchronization2010722

Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )

53 Synchronization2010722

Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )

( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s

i n t i N N T N T nTy y t e

= + + +=

( ) ( )( ) ( ) ( ) 12

sin

sin

g gI f f

I I

i N N N Nj jf N Ni k i k k

f

Y X H e eN

π ε ε π ε

ε ε

πεπε

+ + minus+

minus minus=⎛ ⎞⎜ ⎟

Faded signal attenuated and rotated by CFO

( )( ) ( ) ( ) ( )11 2

sin

sin g gI f I f

i N N N NN j j l kI f

NN

l k π ε ε π ε επ ε ε + + minusminus + + + minus

⎜ ⎟⎝ ⎠

+ + minussum

( )( )( )

( ) ( )2

0

sin

I f I f

I

j j l kI f N Ni l l

l l k I f

X H e el k

NN

π ε ε π ε ε

ε π ε ε

+ + +

= ne minus

+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠

sum

i k

N

W

⎜ ⎟⎝ ⎠

+ Inter-Carrier Interference (ICI)

54 Synchronization2010722

Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )

loz An easier representation of received signal with only p g yfractional CFO is

1N

( ) ( )1

0

0N

k k k l l kl l k

Y H X C H X C l k Wminus

= ne

= + minus +⎡ ⎤⎣ ⎦sum

where the ICI coefficient is given by

( )( )( ) ( )

sin 1exp 1fl k

C l k j l kπ ε

π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )

( )exp 1sin

f

f

C l k j l kNN l k

N

π επ ε

minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠

55 Synchronization2010722

Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )

loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality

loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift

56 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

57 Synchronization2010722

TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )

loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq

58 Synchronization2010722

TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )

loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below

loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and

rN2 can be expressed by

0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+

( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +

59

( )

Synchronization2010722

TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )

loz Without CFO the first half is identical to the second half (in time order)

loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below

( ) 12 in discretesft T f NT

NTφ π π π ε πε

⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟

⎝ ⎠

helliphellip

sNT⎝ ⎠

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)

60

Length=N2

Synchronization2010722

TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )

loz Received signal without CFOg0

0 0

jr a e θ=jθ 2 0

2 2 0

Nj j

N Nr a e a e

θ θ= =

loz Received signal with CFO0 0

0 0

j jr a e eθ φ= Extra phase rotation due to CFO0 0

2 2 20

2 2 0

N N Nj j jj

N Nr a e e a e e

θ φ φθ= =

due to CFO

Phase difference 2 2 0N N

( )2 02 20 2 0 0

Nj j T fN

r r a e a eφ φ πminus Δ= =

which contains the information about CFO

61

0 2 0 0Nr r a e a e

Synchronization2010722

TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )

loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be

loz Note that d is a time index corresponding to the first( ) ( ) 2 1

20

Nd m d m Nm

P d r rminus

+ + +== sum

loz Note that d is a time index corresponding to the first sample in a window of N samples

loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by

( )22 1

20

Nd m NR d rminus

+ += sumloz A timing metric can be defined as

))(()(

)( 2

2

dRdP

dM =

( ) 20 d m Nm + +=sum

62

))(( dR

Synchronization2010722

TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )

63 Synchronization2010722

TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )

loz If drsquo is the correct symbol timing offset

( )2

2 2

0 1 12

j T f j T f j T fN

P d a e a e a eπ π πΔ Δ Δ

minus= + + +

22

2 2

0 1 12

j T fN

e a a aπ Δ

minus

⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭

loz If drsquo falls behind the correct symbol timing offset 1 sample

2⎪ ⎪⎩ ⎭

64 Synchronization2010722

TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )

65 Synchronization2010722

TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )

loz Drawback Plateau effect

Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)

((N2) ((N2) ((N2)

helliphellip

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)

Sliding window (length = N)

66 Synchronization2010722

TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )

loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula

( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus

扣掉相加

( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +

loz The method also called delay correlator

67 Synchronization2010722

IEEE 80211aIEEE 80211a

68 Synchronization2010722

Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )

loz Minnrsquos Method [2][ ]B B －B －B

( )( )

( )

22

2Minn

P dM d =

1 4 1N minus

( )( )( )2

2

MinnR d

( ) ( ) ( ) ( )

1 4 1

2 2 2 40 0

N

d N m k d N m k Nm k

P d r r+ + + + += =

= sum sum

( ) ( ) ( )2

21 4 1

2 40 0

N

d N m k Nk

R d rminus

+ + += sum sum69

0 0m k= =

Synchronization2010722

Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )

loz Parkrsquos Method [3][ ]

where D is the symmetric version of CC D C D

where D is the symmetric version of C

( )( ) 2

3k

P dM d =( )

( )( )23

ParkM dR d

2N

( ) 2

30

N

d k d kk

P d r rminus +=

= sdotsum

( )3

22

0

N

d kk

R d r += sum70

0k=

Synchronization2010722

Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )

loz Renrsquos Method [4][ ]

AS AS

where S is the PN sequence weighted of the original bl 2preamble

( )( )( )( )

24

Re 24

n

P dM d

R d=

( )( )4R d

( ) 2 1

4 2 2

N

k k N d k d k NP d s s r rminus

+ + + += sum0k=sum

( )21

41 N

d kR d rminus

= sum71

( )402 d k

kR d r +

=sum

Synchronization2010722

Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )

72 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

73 Synchronization2010722

Fractional CFO estimation Fractional CFO estimation

loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected

11 Rminus⎛ ⎞sum

h L i th di t b t t id ti l bl k R i

0

12 d m d m L

ms

f r rLTπ + + +

=

⎛ ⎞= ang⎜ ⎟⎝ ⎠sum

where L is the distance between two identical block R is the block size

loz For exampleNN cp

L = N R = NL = N R = Ncp74 Synchronization2010722

Estimation rangeEstimation rangegg

loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval

f ( ) ( )s s⎣ ⎦

loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)

loz DVB-T (L=N) 802 11 (L N4)

[ ]05 05s sf fminus

[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)

(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus

loz Note that is the subcarrier spacing1sf NT=

75

sNT

Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz

plusmng q y

loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn

76 Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz

plusmng q y

loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for

subcarrier spacing is required

plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn

p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs

77 Synchronization2010722

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

OrthogonalityOrthogonalityg yg y

tNΔ

15

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFTby using IDFTby using IDFT

loz According to the structure of Tx it must use Ng oscillators That increases the hardware complexity

loz The equivalent method is using IDFT (IFFT)

16

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

loz In general each carrier can be expressed as

by using IDFTby using IDFT

g p

( ) (6) )()( )(2 ttfjcc

ccetAtS φπ +=

loz We assume that there are N carriers in the OFDM signal Th th t t l l i l S (t) b t d bThen the total complex signal Ss(t) can be represented by

( ) (7))(1)(1

)(2etAtSN

ttfj nn φπ= summinus

+( )

where

(7) )()(

0

0

fnff

etAN

tSn

ns

Δ+=

= sum=

liihffrequency carrier phase amplitude are )( )( and

where 0

fttAfnff

nnn

n

φΔ+

lyrespective carrierth -of n17

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

loz Then we sample the signal at a sampling frequency tΔ1

by using IDFTby using IDFT

p g p g q y and An(t) and φn(t) becomes

(8))(t = φφ(9) )((8) )(

nn

nn

AtAt

== φφ

( ) (10) 1)(1

0

)(2 0summinus

=

+ΔΔ+=ΔN

n

tkfnfjns

neAN

tkS φπ

loz Then the sampled signal can be expressed as0=nN

( )( ) (11) 1)(1

0

22 0summinus

=

ΔΔ+Δ sdot=ΔN

n

tfnkjtkfjns eeA

NtkS n πφπ

0n

18

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

loz The inverse discrete Fourier transform (IDFT) is defined

by using IDFTby using IDFT

( )as the following

1 21

NnkjN

πsumminus

(12) )(1)( 2

0

Nnkj

nefnF

Ntkf πsum

=

Δ=Δ

loz Comparing eq(11) and eq(12) the condition must be satisfied in order to make eq(11) an inverse Fouriersatisfied in order to make eq(11) an inverse Fourier transform relationship

1 (13) 1tN

fΔ

=Δ

19

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

loz If eq(13) is satisfied

by using IDFTby using IDFT

q ( ) loz is the frequency domain signal loz is the time domain signal

( )ntkfjneA φπ +Δ02

)( tkSs Δ gloz is the sub-channel spacingloz is the symbol duration in each sub-channel

)(s

fΔtNΔloz s e sy bo du o e c sub c e

loz This outcome is the same as the result obtained in the

N

loz This outcome is the same as the result obtained in the system of Figure 1 Therefore IDFT can be used to generate an OFDM transmission signalgenerate an OFDM transmission signal

20

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

( )tf02cos π

( )0a

by using IDFTby using IDFT

( )nd ( )tDI( )tf02sin π

( )0a

( )0b

bull

tf s Δ=

1 ( ) ( ) njbna +

( )nd

tNΔffnffn Δ

=Δ+=1 0

( )tDChannelInput

( )tf N 12cos minusπ

( )1minusNa

bullbull

tNΔ

( )tf N 12sin minusπ

( )1minusNb

( )0d

( )nd ( )tDChannelInput

bull bull

( )( )1d

( )2d

tf s Δ=

1 ( ) ( ) njbna +

fNΔffnffn Δ

=Δ+=1 0

ChannelInput

bullbullbullbull

bullbullbullbull

( )1minusNd

21

Frequency Error Results in ICIFrequency Error Results in ICIq yq y

22

Synchronization Error Results in ICISynchronization Error Results in ICIyy

Not Orthogonal Any More

23

Cyclic PrefixCyclic Prefixyy

loz In multipath channel delayed replicas of previous OFDM signal lead to ISI between successive OFDM signals

loz Solution Insert a guard interval between successive OFDMloz Solution Insert a guard interval between successive OFDM signals

24

Cyclic PrefixCyclic Prefix

loz Guard interval leads to intercarrier interference (ICI) in OFDM d d l i

yy

demodulation

loz In DFT interval difference between two subcarriers does not maintain integer number of cycles loss of orthogonality

loz Delayed version of subcarrier 2 causes ICI in the process of d d l ti b i 1demodulating subcarrier 1

25

Cyclic PrefixCyclic Prefixloz Cyclic prefix (CP) A copy of the last part of OFDM signal is

attached to the front of itself

yy

attached to the front of itself

[ ]0d

[ ]1d

[ ]0D[ ]1D

[ ]nd[ ]kD~

symbols dataInput

bullbullbullbull

[ ]1d

[ ]2d

[ ][ ]2D

bullbullbullbullbull

y

bullbullbullbullbull

[ ]1minusNd [ ]1minusND

[ ]gNND minus

bullbullbull

26

Cyclic PrefixCyclic Prefix

loz All delayed replicas of subcarriers always have an

yy

y p yinteger number of cycles within DFT interval no ICI

27

Cyclic PrefixCyclic Prefix

loz Linear convolution vs circular convolution

yy

N

28

Cyclic PrefixCyclic Prefix

loz Channel effect with cyclic prefix

yy

Signal after removed CP

29

Cyclic PrefixCyclic Prefix

loz Time-Domain Explanation

yy

p

30

Cyclic PrefixCyclic Prefixyy

31

Cyclic PrefixCyclic Prefix

loz Spectrum of channel response with length (smaller [ ]nh hL

yy

than )h

gN[ ] nhFFTHk =

loz Received complete OFDM signal[ ] [ ] [ ] 20 ~~ minus++leleotimes= hg LNNnnhnDnr

loz Received useful part [ ]nr[ ] [ ] [ ]nhnDnr Notimes=

where is N-point circular convolution (due to CP)loz Received symbol at k-th subcarrier

[ ] [ ] [ ]N

Notimes

[ ] [ ] [ ] kkNk HXnhnDFFTnrFFTY =otimes== kYX =rArr ldquoUseful property for OFDM system to reduce

kk H

X =rArrcomplexity of channel equalizationrdquo

32

Cyclic Cyclic Prefix (OFDM Prefix (OFDM Receiver)Receiver)yy (( ))

bull bull( )( )nr~Symbols

dataOutput

bullbullbull

bullbullbull( )nr

bullbull

bullbull

33

Cyclic PrefixCyclic Prefix

loz One of the most important reasons to do OFDM is the

yy

pefficient way it deals with multipath delay spread

loz To eliminate inter-symbol interference (ISI) almostloz To eliminate inter symbol interference (ISI) almost completely a guard time is introduced for each OFDM symbolsymbol(The guard time is chosen larger than the delay spread)

34

Bandwidth EfficiencyBandwidth Efficiencyyy

loz In a classical parallel system the channel is divided into p y N non-overlapping sub-channels to avoid inter-carrier interference (ICI)( )

loz The diagram for bandwidth efficiency of OFDM system is shown below

35

SummarySummaryyy

loz The advantage of the FFT-based OFDM system g yloz The use of IFFTFFT can reduce the computation complexityloz The orthogonality between the adjacent sub-carriers will make g y j

the use of transmission bandwidth more efficientloz The guard interval is used to resist the inter-symbol interference

(ISI)loz The main advantage of the OFDM transmission technique is its

high performance even in frequency selective channels

loz The drawbacks of the OFDM system loz It is highly vulnerable to synchronization errorsloz Peak to Average Power Ratio (PAPR) problems

36

SynchronizationSynchronization

Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offset (TO)loz Impact of timing offset (TO)loz Impact of carrier frequency offset (CFO)loz TO estimation algorithmloz CFO estimation algorithmg

38 Synchronization2010722

Introduction (13)Introduction (13)( )( )

loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods

loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead

loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the

cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of

computational complexity therefore it may be not be available in short-burst wireless communication

39 Synchronization2010722

Introduction (23)Introduction (23)( )( )

loz Transcevier

40 Synchronization2010722

Introduction (33)Introduction (33)( )( )

loz Synchronization erroryloz Symbol timing offset (TO)

loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)

loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect

41 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg

42 Synchronization2010722

Signal model (14)Signal model (14)g ( )g ( )

loz Time domain sample p1

2

0

1 Nj nk N

n kk

x X eN

πminus

minus= sum

Xk the frequency domain data at the k-th subcarrier0kN =

loz The multipath fading channel

( ) ( ) ( )1L

h t h t δminus

sum

loz Received time domain signal

( ) ( ) ( )0

l ll

h t h tτ τ δ τ τ=

= minussum

loz Received time-domain signal

( ) ( )1

L

n n ly h n l x w nminus

minus= +sum43

( ) ( )0l=sum

Synchronization2010722

Signal model (24)Signal model (24)g ( )g ( )

loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting

12

0

Nj kn N

k nn

Y y e πminus

minus

=

= sum

( )1 1

2

0

N Lj kn N

n l kn l

h n l x e Wπminus minus

minusminus

=

= +sumsum

loz The equation above could be expressed in matrix form

0n l

loz The equation above could be expressed in matrix form

= +Y HX W

44 Synchronization2010722

Signal model (34)Signal model (34)g ( )g ( )

H= = = = +Y Fy FCx FCF X HX W

loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal

y

H=H FCFcirculant matrix leading to be a diagonal matrix

=H FCF

( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0

h h L L hh h

⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥

( )( ) ( )

( ) ( )

1 10 1 0 0i

h Lh L h N L

⎢ ⎥⎢ ⎥minus minus⎢ ⎥

equiv= minus minus⎢ ⎥⎢ ⎥

C( ) ( )

( ) ( )

0 1 1 100

0 0 1 1 0

h L h N L

h N L L h N

⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

45

( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦

Synchronization2010722

Signal model (44)Signal model (44)g ( )g ( )

loz In a time-invariant channel the channel matrix H would be a diagonal matrix

k k k kY H X W= +

( )1 1

2

0 0

N Lj kl N

kn l

H h n l e πminus minus

minus

= =

= sumsum

loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI

k k k k kY H X I Wα= + +k k k k k

( ) ( )1 1 1

2 2 N N L

j m k Nj ml Nk mI X h n l e e ππ

minus minus minusminusminus⎛ ⎞

= ⎜ ⎟⎝ ⎠

sum sum sumICI

46

0 0 0m n lm k= = =ne

⎝ ⎠

Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

47 Synchronization2010722

Impact of timing offset Impact of timing offset p gp g

loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively

( )I f s

Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ

( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr

2 fj k Nk k k kY X H e Wπδminus= +

loz For integer part of timing offset k k k kY X H e W+

I sTδ ( ) 0g IN L δminus minus le leFFT

2 Ij k NY X H e Wπδminus= +

( ) ( ) 2 I sFFT

j f TI sh t T H f e π δδ minusminus rarr

48

k k k kY X H e W= +

Synchronization2010722

Impact of timing offset derivationImpact of timing offset derivationp gp g

49 Synchronization2010722

Discussion of symbol boundaryDiscussion of symbol boundaryy yy y

50 Synchronization2010722

Impact of timing offsetImpact of timing offsetp gp g

loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI

2 Ij k NIk k k k

NY X H e ISI W

Nπδδ minusminus

= + +

loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk

e e

p kloz Phase Shift can be resolved by Differential Encoding

Decoding or carrier recovery with pilots

51 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

52 Synchronization2010722

Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )

53 Synchronization2010722

Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )

( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s

i n t i N N T N T nTy y t e

= + + +=

( ) ( )( ) ( ) ( ) 12

sin

sin

g gI f f

I I

i N N N Nj jf N Ni k i k k

f

Y X H e eN

π ε ε π ε

ε ε

πεπε

+ + minus+

minus minus=⎛ ⎞⎜ ⎟

Faded signal attenuated and rotated by CFO

( )( ) ( ) ( ) ( )11 2

sin

sin g gI f I f

i N N N NN j j l kI f

NN

l k π ε ε π ε επ ε ε + + minusminus + + + minus

⎜ ⎟⎝ ⎠

+ + minussum

( )( )( )

( ) ( )2

0

sin

I f I f

I

j j l kI f N Ni l l

l l k I f

X H e el k

NN

π ε ε π ε ε

ε π ε ε

+ + +

= ne minus

+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠

sum

i k

N

W

⎜ ⎟⎝ ⎠

+ Inter-Carrier Interference (ICI)

54 Synchronization2010722

Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )

loz An easier representation of received signal with only p g yfractional CFO is

1N

( ) ( )1

0

0N

k k k l l kl l k

Y H X C H X C l k Wminus

= ne

= + minus +⎡ ⎤⎣ ⎦sum

where the ICI coefficient is given by

( )( )( ) ( )

sin 1exp 1fl k

C l k j l kπ ε

π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )

( )exp 1sin

f

f

C l k j l kNN l k

N

π επ ε

minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠

55 Synchronization2010722

Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )

loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality

loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift

56 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

57 Synchronization2010722

TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )

loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq

58 Synchronization2010722

TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )

loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below

loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and

rN2 can be expressed by

0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+

( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +

59

( )

Synchronization2010722

TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )

loz Without CFO the first half is identical to the second half (in time order)

loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below

( ) 12 in discretesft T f NT

NTφ π π π ε πε

⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟

⎝ ⎠

helliphellip

sNT⎝ ⎠

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)

60

Length=N2

Synchronization2010722

TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )

loz Received signal without CFOg0

0 0

jr a e θ=jθ 2 0

2 2 0

Nj j

N Nr a e a e

θ θ= =

loz Received signal with CFO0 0

0 0

j jr a e eθ φ= Extra phase rotation due to CFO0 0

2 2 20

2 2 0

N N Nj j jj

N Nr a e e a e e

θ φ φθ= =

due to CFO

Phase difference 2 2 0N N

( )2 02 20 2 0 0

Nj j T fN

r r a e a eφ φ πminus Δ= =

which contains the information about CFO

61

0 2 0 0Nr r a e a e

Synchronization2010722

TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )

loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be

loz Note that d is a time index corresponding to the first( ) ( ) 2 1

20

Nd m d m Nm

P d r rminus

+ + +== sum

loz Note that d is a time index corresponding to the first sample in a window of N samples

loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by

( )22 1

20

Nd m NR d rminus

+ += sumloz A timing metric can be defined as

))(()(

)( 2

2

dRdP

dM =

( ) 20 d m Nm + +=sum

62

))(( dR

Synchronization2010722

TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )

63 Synchronization2010722

TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )

loz If drsquo is the correct symbol timing offset

( )2

2 2

0 1 12

j T f j T f j T fN

P d a e a e a eπ π πΔ Δ Δ

minus= + + +

22

2 2

0 1 12

j T fN

e a a aπ Δ

minus

⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭

loz If drsquo falls behind the correct symbol timing offset 1 sample

2⎪ ⎪⎩ ⎭

64 Synchronization2010722

TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )

65 Synchronization2010722

TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )

loz Drawback Plateau effect

Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)

((N2) ((N2) ((N2)

helliphellip

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)

Sliding window (length = N)

66 Synchronization2010722

TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )

loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula

( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus

扣掉相加

( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +

loz The method also called delay correlator

67 Synchronization2010722

IEEE 80211aIEEE 80211a

68 Synchronization2010722

Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )

loz Minnrsquos Method [2][ ]B B －B －B

( )( )

( )

22

2Minn

P dM d =

1 4 1N minus

( )( )( )2

2

MinnR d

( ) ( ) ( ) ( )

1 4 1

2 2 2 40 0

N

d N m k d N m k Nm k

P d r r+ + + + += =

= sum sum

( ) ( ) ( )2

21 4 1

2 40 0

N

d N m k Nk

R d rminus

+ + += sum sum69

0 0m k= =

Synchronization2010722

Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )

loz Parkrsquos Method [3][ ]

where D is the symmetric version of CC D C D

where D is the symmetric version of C

( )( ) 2

3k

P dM d =( )

( )( )23

ParkM dR d

2N

( ) 2

30

N

d k d kk

P d r rminus +=

= sdotsum

( )3

22

0

N

d kk

R d r += sum70

0k=

Synchronization2010722

Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )

loz Renrsquos Method [4][ ]

AS AS

where S is the PN sequence weighted of the original bl 2preamble

( )( )( )( )

24

Re 24

n

P dM d

R d=

( )( )4R d

( ) 2 1

4 2 2

N

k k N d k d k NP d s s r rminus

+ + + += sum0k=sum

( )21

41 N

d kR d rminus

= sum71

( )402 d k

kR d r +

=sum

Synchronization2010722

Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )

72 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

73 Synchronization2010722

Fractional CFO estimation Fractional CFO estimation

loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected

11 Rminus⎛ ⎞sum

h L i th di t b t t id ti l bl k R i

0

12 d m d m L

ms

f r rLTπ + + +

=

⎛ ⎞= ang⎜ ⎟⎝ ⎠sum

where L is the distance between two identical block R is the block size

loz For exampleNN cp

L = N R = NL = N R = Ncp74 Synchronization2010722

Estimation rangeEstimation rangegg

loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval

f ( ) ( )s s⎣ ⎦

loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)

loz DVB-T (L=N) 802 11 (L N4)

[ ]05 05s sf fminus

[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)

(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus

loz Note that is the subcarrier spacing1sf NT=

75

sNT

Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz

plusmng q y

loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn

76 Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz

plusmng q y

loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for

subcarrier spacing is required

plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn

p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs

77 Synchronization2010722

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFTby using IDFTby using IDFT

loz According to the structure of Tx it must use Ng oscillators That increases the hardware complexity

loz The equivalent method is using IDFT (IFFT)

16

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

loz In general each carrier can be expressed as

by using IDFTby using IDFT

g p

( ) (6) )()( )(2 ttfjcc

ccetAtS φπ +=

loz We assume that there are N carriers in the OFDM signal Th th t t l l i l S (t) b t d bThen the total complex signal Ss(t) can be represented by

( ) (7))(1)(1

)(2etAtSN

ttfj nn φπ= summinus

+( )

where

(7) )()(

0

0

fnff

etAN

tSn

ns

Δ+=

= sum=

liihffrequency carrier phase amplitude are )( )( and

where 0

fttAfnff

nnn

n

φΔ+

lyrespective carrierth -of n17

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

loz Then we sample the signal at a sampling frequency tΔ1

by using IDFTby using IDFT

p g p g q y and An(t) and φn(t) becomes

(8))(t = φφ(9) )((8) )(

nn

nn

AtAt

== φφ

( ) (10) 1)(1

0

)(2 0summinus

=

+ΔΔ+=ΔN

n

tkfnfjns

neAN

tkS φπ

loz Then the sampled signal can be expressed as0=nN

( )( ) (11) 1)(1

0

22 0summinus

=

ΔΔ+Δ sdot=ΔN

n

tfnkjtkfjns eeA

NtkS n πφπ

0n

18

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

loz The inverse discrete Fourier transform (IDFT) is defined

by using IDFTby using IDFT

( )as the following

1 21

NnkjN

πsumminus

(12) )(1)( 2

0

Nnkj

nefnF

Ntkf πsum

=

Δ=Δ

loz Comparing eq(11) and eq(12) the condition must be satisfied in order to make eq(11) an inverse Fouriersatisfied in order to make eq(11) an inverse Fourier transform relationship

1 (13) 1tN

fΔ

=Δ

19

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

loz If eq(13) is satisfied

by using IDFTby using IDFT

q ( ) loz is the frequency domain signal loz is the time domain signal

( )ntkfjneA φπ +Δ02

)( tkSs Δ gloz is the sub-channel spacingloz is the symbol duration in each sub-channel

)(s

fΔtNΔloz s e sy bo du o e c sub c e

loz This outcome is the same as the result obtained in the

N

loz This outcome is the same as the result obtained in the system of Figure 1 Therefore IDFT can be used to generate an OFDM transmission signalgenerate an OFDM transmission signal

20

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

( )tf02cos π

( )0a

by using IDFTby using IDFT

( )nd ( )tDI( )tf02sin π

( )0a

( )0b

bull

tf s Δ=

1 ( ) ( ) njbna +

( )nd

tNΔffnffn Δ

=Δ+=1 0

( )tDChannelInput

( )tf N 12cos minusπ

( )1minusNa

bullbull

tNΔ

( )tf N 12sin minusπ

( )1minusNb

( )0d

( )nd ( )tDChannelInput

bull bull

( )( )1d

( )2d

tf s Δ=

1 ( ) ( ) njbna +

fNΔffnffn Δ

=Δ+=1 0

ChannelInput

bullbullbullbull

bullbullbullbull

( )1minusNd

21

Frequency Error Results in ICIFrequency Error Results in ICIq yq y

22

Synchronization Error Results in ICISynchronization Error Results in ICIyy

Not Orthogonal Any More

23

Cyclic PrefixCyclic Prefixyy

loz In multipath channel delayed replicas of previous OFDM signal lead to ISI between successive OFDM signals

loz Solution Insert a guard interval between successive OFDMloz Solution Insert a guard interval between successive OFDM signals

24

Cyclic PrefixCyclic Prefix

loz Guard interval leads to intercarrier interference (ICI) in OFDM d d l i

yy

demodulation

loz In DFT interval difference between two subcarriers does not maintain integer number of cycles loss of orthogonality

loz Delayed version of subcarrier 2 causes ICI in the process of d d l ti b i 1demodulating subcarrier 1

25

Cyclic PrefixCyclic Prefixloz Cyclic prefix (CP) A copy of the last part of OFDM signal is

attached to the front of itself

yy

attached to the front of itself

[ ]0d

[ ]1d

[ ]0D[ ]1D

[ ]nd[ ]kD~

symbols dataInput

bullbullbullbull

[ ]1d

[ ]2d

[ ][ ]2D

bullbullbullbullbull

y

bullbullbullbullbull

[ ]1minusNd [ ]1minusND

[ ]gNND minus

bullbullbull

26

Cyclic PrefixCyclic Prefix

loz All delayed replicas of subcarriers always have an

yy

y p yinteger number of cycles within DFT interval no ICI

27

Cyclic PrefixCyclic Prefix

loz Linear convolution vs circular convolution

yy

N

28

Cyclic PrefixCyclic Prefix

loz Channel effect with cyclic prefix

yy

Signal after removed CP

29

Cyclic PrefixCyclic Prefix

loz Time-Domain Explanation

yy

p

30

Cyclic PrefixCyclic Prefixyy

31

Cyclic PrefixCyclic Prefix

loz Spectrum of channel response with length (smaller [ ]nh hL

yy

than )h

gN[ ] nhFFTHk =

loz Received complete OFDM signal[ ] [ ] [ ] 20 ~~ minus++leleotimes= hg LNNnnhnDnr

loz Received useful part [ ]nr[ ] [ ] [ ]nhnDnr Notimes=

where is N-point circular convolution (due to CP)loz Received symbol at k-th subcarrier

[ ] [ ] [ ]N

Notimes

[ ] [ ] [ ] kkNk HXnhnDFFTnrFFTY =otimes== kYX =rArr ldquoUseful property for OFDM system to reduce

kk H

X =rArrcomplexity of channel equalizationrdquo

32

Cyclic Cyclic Prefix (OFDM Prefix (OFDM Receiver)Receiver)yy (( ))

bull bull( )( )nr~Symbols

dataOutput

bullbullbull

bullbullbull( )nr

bullbull

bullbull

33

Cyclic PrefixCyclic Prefix

loz One of the most important reasons to do OFDM is the

yy

pefficient way it deals with multipath delay spread

loz To eliminate inter-symbol interference (ISI) almostloz To eliminate inter symbol interference (ISI) almost completely a guard time is introduced for each OFDM symbolsymbol(The guard time is chosen larger than the delay spread)

34

Bandwidth EfficiencyBandwidth Efficiencyyy

loz In a classical parallel system the channel is divided into p y N non-overlapping sub-channels to avoid inter-carrier interference (ICI)( )

loz The diagram for bandwidth efficiency of OFDM system is shown below

35

SummarySummaryyy

loz The advantage of the FFT-based OFDM system g yloz The use of IFFTFFT can reduce the computation complexityloz The orthogonality between the adjacent sub-carriers will make g y j

the use of transmission bandwidth more efficientloz The guard interval is used to resist the inter-symbol interference

(ISI)loz The main advantage of the OFDM transmission technique is its

high performance even in frequency selective channels

loz The drawbacks of the OFDM system loz It is highly vulnerable to synchronization errorsloz Peak to Average Power Ratio (PAPR) problems

36

SynchronizationSynchronization

Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offset (TO)loz Impact of timing offset (TO)loz Impact of carrier frequency offset (CFO)loz TO estimation algorithmloz CFO estimation algorithmg

38 Synchronization2010722

Introduction (13)Introduction (13)( )( )

loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods

loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead

loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the

cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of

computational complexity therefore it may be not be available in short-burst wireless communication

39 Synchronization2010722

Introduction (23)Introduction (23)( )( )

loz Transcevier

40 Synchronization2010722

Introduction (33)Introduction (33)( )( )

loz Synchronization erroryloz Symbol timing offset (TO)

loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)

loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect

41 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg

42 Synchronization2010722

Signal model (14)Signal model (14)g ( )g ( )

loz Time domain sample p1

2

0

1 Nj nk N

n kk

x X eN

πminus

minus= sum

Xk the frequency domain data at the k-th subcarrier0kN =

loz The multipath fading channel

( ) ( ) ( )1L

h t h t δminus

sum

loz Received time domain signal

( ) ( ) ( )0

l ll

h t h tτ τ δ τ τ=

= minussum

loz Received time-domain signal

( ) ( )1

L

n n ly h n l x w nminus

minus= +sum43

( ) ( )0l=sum

Synchronization2010722

Signal model (24)Signal model (24)g ( )g ( )

loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting

12

0

Nj kn N

k nn

Y y e πminus

minus

=

= sum

( )1 1

2

0

N Lj kn N

n l kn l

h n l x e Wπminus minus

minusminus

=

= +sumsum

loz The equation above could be expressed in matrix form

0n l

loz The equation above could be expressed in matrix form

= +Y HX W

44 Synchronization2010722

Signal model (34)Signal model (34)g ( )g ( )

H= = = = +Y Fy FCx FCF X HX W

loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal

y

H=H FCFcirculant matrix leading to be a diagonal matrix

=H FCF

( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0

h h L L hh h

⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥

( )( ) ( )

( ) ( )

1 10 1 0 0i

h Lh L h N L

⎢ ⎥⎢ ⎥minus minus⎢ ⎥

equiv= minus minus⎢ ⎥⎢ ⎥

C( ) ( )

( ) ( )

0 1 1 100

0 0 1 1 0

h L h N L

h N L L h N

⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

45

( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦

Synchronization2010722

Signal model (44)Signal model (44)g ( )g ( )

loz In a time-invariant channel the channel matrix H would be a diagonal matrix

k k k kY H X W= +

( )1 1

2

0 0

N Lj kl N

kn l

H h n l e πminus minus

minus

= =

= sumsum

loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI

k k k k kY H X I Wα= + +k k k k k

( ) ( )1 1 1

2 2 N N L

j m k Nj ml Nk mI X h n l e e ππ

minus minus minusminusminus⎛ ⎞

= ⎜ ⎟⎝ ⎠

sum sum sumICI

46

0 0 0m n lm k= = =ne

⎝ ⎠

Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

47 Synchronization2010722

Impact of timing offset Impact of timing offset p gp g

loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively

( )I f s

Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ

( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr

2 fj k Nk k k kY X H e Wπδminus= +

loz For integer part of timing offset k k k kY X H e W+

I sTδ ( ) 0g IN L δminus minus le leFFT

2 Ij k NY X H e Wπδminus= +

( ) ( ) 2 I sFFT

j f TI sh t T H f e π δδ minusminus rarr

48

k k k kY X H e W= +

Synchronization2010722

Impact of timing offset derivationImpact of timing offset derivationp gp g

49 Synchronization2010722

Discussion of symbol boundaryDiscussion of symbol boundaryy yy y

50 Synchronization2010722

Impact of timing offsetImpact of timing offsetp gp g

loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI

2 Ij k NIk k k k

NY X H e ISI W

Nπδδ minusminus

= + +

loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk

e e

p kloz Phase Shift can be resolved by Differential Encoding

Decoding or carrier recovery with pilots

51 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

52 Synchronization2010722

Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )

53 Synchronization2010722

Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )

( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s

i n t i N N T N T nTy y t e

= + + +=

( ) ( )( ) ( ) ( ) 12

sin

sin

g gI f f

I I

i N N N Nj jf N Ni k i k k

f

Y X H e eN

π ε ε π ε

ε ε

πεπε

+ + minus+

minus minus=⎛ ⎞⎜ ⎟

Faded signal attenuated and rotated by CFO

( )( ) ( ) ( ) ( )11 2

sin

sin g gI f I f

i N N N NN j j l kI f

NN

l k π ε ε π ε επ ε ε + + minusminus + + + minus

⎜ ⎟⎝ ⎠

+ + minussum

( )( )( )

( ) ( )2

0

sin

I f I f

I

j j l kI f N Ni l l

l l k I f

X H e el k

NN

π ε ε π ε ε

ε π ε ε

+ + +

= ne minus

+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠

sum

i k

N

W

⎜ ⎟⎝ ⎠

+ Inter-Carrier Interference (ICI)

54 Synchronization2010722

Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )

loz An easier representation of received signal with only p g yfractional CFO is

1N

( ) ( )1

0

0N

k k k l l kl l k

Y H X C H X C l k Wminus

= ne

= + minus +⎡ ⎤⎣ ⎦sum

where the ICI coefficient is given by

( )( )( ) ( )

sin 1exp 1fl k

C l k j l kπ ε

π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )

( )exp 1sin

f

f

C l k j l kNN l k

N

π επ ε

minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠

55 Synchronization2010722

Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )

loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality

loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift

56 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

57 Synchronization2010722

TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )

loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq

58 Synchronization2010722

TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )

loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below

loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and

rN2 can be expressed by

0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+

( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +

59

( )

Synchronization2010722

TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )

loz Without CFO the first half is identical to the second half (in time order)

loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below

( ) 12 in discretesft T f NT

NTφ π π π ε πε

⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟

⎝ ⎠

helliphellip

sNT⎝ ⎠

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)

60

Length=N2

Synchronization2010722

TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )

loz Received signal without CFOg0

0 0

jr a e θ=jθ 2 0

2 2 0

Nj j

N Nr a e a e

θ θ= =

loz Received signal with CFO0 0

0 0

j jr a e eθ φ= Extra phase rotation due to CFO0 0

2 2 20

2 2 0

N N Nj j jj

N Nr a e e a e e

θ φ φθ= =

due to CFO

Phase difference 2 2 0N N

( )2 02 20 2 0 0

Nj j T fN

r r a e a eφ φ πminus Δ= =

which contains the information about CFO

61

0 2 0 0Nr r a e a e

Synchronization2010722

TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )

loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be

loz Note that d is a time index corresponding to the first( ) ( ) 2 1

20

Nd m d m Nm

P d r rminus

+ + +== sum

loz Note that d is a time index corresponding to the first sample in a window of N samples

loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by

( )22 1

20

Nd m NR d rminus

+ += sumloz A timing metric can be defined as

))(()(

)( 2

2

dRdP

dM =

( ) 20 d m Nm + +=sum

62

))(( dR

Synchronization2010722

TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )

63 Synchronization2010722

TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )

loz If drsquo is the correct symbol timing offset

( )2

2 2

0 1 12

j T f j T f j T fN

P d a e a e a eπ π πΔ Δ Δ

minus= + + +

22

2 2

0 1 12

j T fN

e a a aπ Δ

minus

⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭

loz If drsquo falls behind the correct symbol timing offset 1 sample

2⎪ ⎪⎩ ⎭

64 Synchronization2010722

TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )

65 Synchronization2010722

TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )

loz Drawback Plateau effect

Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)

((N2) ((N2) ((N2)

helliphellip

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)

Sliding window (length = N)

66 Synchronization2010722

TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )

loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula

( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus

扣掉相加

( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +

loz The method also called delay correlator

67 Synchronization2010722

IEEE 80211aIEEE 80211a

68 Synchronization2010722

Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )

loz Minnrsquos Method [2][ ]B B －B －B

( )( )

( )

22

2Minn

P dM d =

1 4 1N minus

( )( )( )2

2

MinnR d

( ) ( ) ( ) ( )

1 4 1

2 2 2 40 0

N

d N m k d N m k Nm k

P d r r+ + + + += =

= sum sum

( ) ( ) ( )2

21 4 1

2 40 0

N

d N m k Nk

R d rminus

+ + += sum sum69

0 0m k= =

Synchronization2010722

Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )

loz Parkrsquos Method [3][ ]

where D is the symmetric version of CC D C D

where D is the symmetric version of C

( )( ) 2

3k

P dM d =( )

( )( )23

ParkM dR d

2N

( ) 2

30

N

d k d kk

P d r rminus +=

= sdotsum

( )3

22

0

N

d kk

R d r += sum70

0k=

Synchronization2010722

Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )

loz Renrsquos Method [4][ ]

AS AS

where S is the PN sequence weighted of the original bl 2preamble

( )( )( )( )

24

Re 24

n

P dM d

R d=

( )( )4R d

( ) 2 1

4 2 2

N

k k N d k d k NP d s s r rminus

+ + + += sum0k=sum

( )21

41 N

d kR d rminus

= sum71

( )402 d k

kR d r +

=sum

Synchronization2010722

Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )

72 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

73 Synchronization2010722

Fractional CFO estimation Fractional CFO estimation

loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected

11 Rminus⎛ ⎞sum

h L i th di t b t t id ti l bl k R i

0

12 d m d m L

ms

f r rLTπ + + +

=

⎛ ⎞= ang⎜ ⎟⎝ ⎠sum

where L is the distance between two identical block R is the block size

loz For exampleNN cp

L = N R = NL = N R = Ncp74 Synchronization2010722

Estimation rangeEstimation rangegg

loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval

f ( ) ( )s s⎣ ⎦

loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)

loz DVB-T (L=N) 802 11 (L N4)

[ ]05 05s sf fminus

[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)

(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus

loz Note that is the subcarrier spacing1sf NT=

75

sNT

Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz

plusmng q y

loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn

76 Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz

plusmng q y

loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for

subcarrier spacing is required

plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn

p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs

77 Synchronization2010722

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

loz In general each carrier can be expressed as

by using IDFTby using IDFT

g p

( ) (6) )()( )(2 ttfjcc

ccetAtS φπ +=

loz We assume that there are N carriers in the OFDM signal Th th t t l l i l S (t) b t d bThen the total complex signal Ss(t) can be represented by

( ) (7))(1)(1

)(2etAtSN

ttfj nn φπ= summinus

+( )

where

(7) )()(

0

0

fnff

etAN

tSn

ns

Δ+=

= sum=

liihffrequency carrier phase amplitude are )( )( and

where 0

fttAfnff

nnn

n

φΔ+

lyrespective carrierth -of n17

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

loz Then we sample the signal at a sampling frequency tΔ1

by using IDFTby using IDFT

p g p g q y and An(t) and φn(t) becomes

(8))(t = φφ(9) )((8) )(

nn

nn

AtAt

== φφ

( ) (10) 1)(1

0

)(2 0summinus

=

+ΔΔ+=ΔN

n

tkfnfjns

neAN

tkS φπ

loz Then the sampled signal can be expressed as0=nN

( )( ) (11) 1)(1

0

22 0summinus

=

ΔΔ+Δ sdot=ΔN

n

tfnkjtkfjns eeA

NtkS n πφπ

0n

18

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

loz The inverse discrete Fourier transform (IDFT) is defined

by using IDFTby using IDFT

( )as the following

1 21

NnkjN

πsumminus

(12) )(1)( 2

0

Nnkj

nefnF

Ntkf πsum

=

Δ=Δ

loz Comparing eq(11) and eq(12) the condition must be satisfied in order to make eq(11) an inverse Fouriersatisfied in order to make eq(11) an inverse Fourier transform relationship

1 (13) 1tN

fΔ

=Δ

19

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

loz If eq(13) is satisfied

by using IDFTby using IDFT

q ( ) loz is the frequency domain signal loz is the time domain signal

( )ntkfjneA φπ +Δ02

)( tkSs Δ gloz is the sub-channel spacingloz is the symbol duration in each sub-channel

)(s

fΔtNΔloz s e sy bo du o e c sub c e

loz This outcome is the same as the result obtained in the

N

loz This outcome is the same as the result obtained in the system of Figure 1 Therefore IDFT can be used to generate an OFDM transmission signalgenerate an OFDM transmission signal

20

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

( )tf02cos π

( )0a

by using IDFTby using IDFT

( )nd ( )tDI( )tf02sin π

( )0a

( )0b

bull

tf s Δ=

1 ( ) ( ) njbna +

( )nd

tNΔffnffn Δ

=Δ+=1 0

( )tDChannelInput

( )tf N 12cos minusπ

( )1minusNa

bullbull

tNΔ

( )tf N 12sin minusπ

( )1minusNb

( )0d

( )nd ( )tDChannelInput

bull bull

( )( )1d

( )2d

tf s Δ=

1 ( ) ( ) njbna +

fNΔffnffn Δ

=Δ+=1 0

ChannelInput

bullbullbullbull

bullbullbullbull

( )1minusNd

21

Frequency Error Results in ICIFrequency Error Results in ICIq yq y

22

Synchronization Error Results in ICISynchronization Error Results in ICIyy

Not Orthogonal Any More

23

Cyclic PrefixCyclic Prefixyy

loz In multipath channel delayed replicas of previous OFDM signal lead to ISI between successive OFDM signals

loz Solution Insert a guard interval between successive OFDMloz Solution Insert a guard interval between successive OFDM signals

24

Cyclic PrefixCyclic Prefix

loz Guard interval leads to intercarrier interference (ICI) in OFDM d d l i

yy

demodulation

loz In DFT interval difference between two subcarriers does not maintain integer number of cycles loss of orthogonality

loz Delayed version of subcarrier 2 causes ICI in the process of d d l ti b i 1demodulating subcarrier 1

25

Cyclic PrefixCyclic Prefixloz Cyclic prefix (CP) A copy of the last part of OFDM signal is

attached to the front of itself

yy

attached to the front of itself

[ ]0d

[ ]1d

[ ]0D[ ]1D

[ ]nd[ ]kD~

symbols dataInput

bullbullbullbull

[ ]1d

[ ]2d

[ ][ ]2D

bullbullbullbullbull

y

bullbullbullbullbull

[ ]1minusNd [ ]1minusND

[ ]gNND minus

bullbullbull

26

Cyclic PrefixCyclic Prefix

loz All delayed replicas of subcarriers always have an

yy

y p yinteger number of cycles within DFT interval no ICI

27

Cyclic PrefixCyclic Prefix

loz Linear convolution vs circular convolution

yy

N

28

Cyclic PrefixCyclic Prefix

loz Channel effect with cyclic prefix

yy

Signal after removed CP

29

Cyclic PrefixCyclic Prefix

loz Time-Domain Explanation

yy

p

30

Cyclic PrefixCyclic Prefixyy

31

Cyclic PrefixCyclic Prefix

loz Spectrum of channel response with length (smaller [ ]nh hL

yy

than )h

gN[ ] nhFFTHk =

loz Received complete OFDM signal[ ] [ ] [ ] 20 ~~ minus++leleotimes= hg LNNnnhnDnr

loz Received useful part [ ]nr[ ] [ ] [ ]nhnDnr Notimes=

where is N-point circular convolution (due to CP)loz Received symbol at k-th subcarrier

[ ] [ ] [ ]N

Notimes

[ ] [ ] [ ] kkNk HXnhnDFFTnrFFTY =otimes== kYX =rArr ldquoUseful property for OFDM system to reduce

kk H

X =rArrcomplexity of channel equalizationrdquo

32

Cyclic Cyclic Prefix (OFDM Prefix (OFDM Receiver)Receiver)yy (( ))

bull bull( )( )nr~Symbols

dataOutput

bullbullbull

bullbullbull( )nr

bullbull

bullbull

33

Cyclic PrefixCyclic Prefix

loz One of the most important reasons to do OFDM is the

yy

pefficient way it deals with multipath delay spread

loz To eliminate inter-symbol interference (ISI) almostloz To eliminate inter symbol interference (ISI) almost completely a guard time is introduced for each OFDM symbolsymbol(The guard time is chosen larger than the delay spread)

34

Bandwidth EfficiencyBandwidth Efficiencyyy

loz In a classical parallel system the channel is divided into p y N non-overlapping sub-channels to avoid inter-carrier interference (ICI)( )

loz The diagram for bandwidth efficiency of OFDM system is shown below

35

SummarySummaryyy

loz The advantage of the FFT-based OFDM system g yloz The use of IFFTFFT can reduce the computation complexityloz The orthogonality between the adjacent sub-carriers will make g y j

the use of transmission bandwidth more efficientloz The guard interval is used to resist the inter-symbol interference

(ISI)loz The main advantage of the OFDM transmission technique is its

high performance even in frequency selective channels

loz The drawbacks of the OFDM system loz It is highly vulnerable to synchronization errorsloz Peak to Average Power Ratio (PAPR) problems

36

SynchronizationSynchronization

Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offset (TO)loz Impact of timing offset (TO)loz Impact of carrier frequency offset (CFO)loz TO estimation algorithmloz CFO estimation algorithmg

38 Synchronization2010722

Introduction (13)Introduction (13)( )( )

loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods

loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead

loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the

cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of

computational complexity therefore it may be not be available in short-burst wireless communication

39 Synchronization2010722

Introduction (23)Introduction (23)( )( )

loz Transcevier

40 Synchronization2010722

Introduction (33)Introduction (33)( )( )

loz Synchronization erroryloz Symbol timing offset (TO)

loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)

loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect

41 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg

42 Synchronization2010722

Signal model (14)Signal model (14)g ( )g ( )

loz Time domain sample p1

2

0

1 Nj nk N

n kk

x X eN

πminus

minus= sum

Xk the frequency domain data at the k-th subcarrier0kN =

loz The multipath fading channel

( ) ( ) ( )1L

h t h t δminus

sum

loz Received time domain signal

( ) ( ) ( )0

l ll

h t h tτ τ δ τ τ=

= minussum

loz Received time-domain signal

( ) ( )1

L

n n ly h n l x w nminus

minus= +sum43

( ) ( )0l=sum

Synchronization2010722

Signal model (24)Signal model (24)g ( )g ( )

loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting

12

0

Nj kn N

k nn

Y y e πminus

minus

=

= sum

( )1 1

2

0

N Lj kn N

n l kn l

h n l x e Wπminus minus

minusminus

=

= +sumsum

loz The equation above could be expressed in matrix form

0n l

loz The equation above could be expressed in matrix form

= +Y HX W

44 Synchronization2010722

Signal model (34)Signal model (34)g ( )g ( )

H= = = = +Y Fy FCx FCF X HX W

loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal

y

H=H FCFcirculant matrix leading to be a diagonal matrix

=H FCF

( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0

h h L L hh h

⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥

( )( ) ( )

( ) ( )

1 10 1 0 0i

h Lh L h N L

⎢ ⎥⎢ ⎥minus minus⎢ ⎥

equiv= minus minus⎢ ⎥⎢ ⎥

C( ) ( )

( ) ( )

0 1 1 100

0 0 1 1 0

h L h N L

h N L L h N

⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

45

( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦

Synchronization2010722

Signal model (44)Signal model (44)g ( )g ( )

loz In a time-invariant channel the channel matrix H would be a diagonal matrix

k k k kY H X W= +

( )1 1

2

0 0

N Lj kl N

kn l

H h n l e πminus minus

minus

= =

= sumsum

loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI

k k k k kY H X I Wα= + +k k k k k

( ) ( )1 1 1

2 2 N N L

j m k Nj ml Nk mI X h n l e e ππ

minus minus minusminusminus⎛ ⎞

= ⎜ ⎟⎝ ⎠

sum sum sumICI

46

0 0 0m n lm k= = =ne

⎝ ⎠

Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

47 Synchronization2010722

Impact of timing offset Impact of timing offset p gp g

loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively

( )I f s

Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ

( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr

2 fj k Nk k k kY X H e Wπδminus= +

loz For integer part of timing offset k k k kY X H e W+

I sTδ ( ) 0g IN L δminus minus le leFFT

2 Ij k NY X H e Wπδminus= +

( ) ( ) 2 I sFFT

j f TI sh t T H f e π δδ minusminus rarr

48

k k k kY X H e W= +

Synchronization2010722

Impact of timing offset derivationImpact of timing offset derivationp gp g

49 Synchronization2010722

Discussion of symbol boundaryDiscussion of symbol boundaryy yy y

50 Synchronization2010722

Impact of timing offsetImpact of timing offsetp gp g

loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI

2 Ij k NIk k k k

NY X H e ISI W

Nπδδ minusminus

= + +

loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk

e e

p kloz Phase Shift can be resolved by Differential Encoding

Decoding or carrier recovery with pilots

51 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

52 Synchronization2010722

Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )

53 Synchronization2010722

Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )

( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s

i n t i N N T N T nTy y t e

= + + +=

( ) ( )( ) ( ) ( ) 12

sin

sin

g gI f f

I I

i N N N Nj jf N Ni k i k k

f

Y X H e eN

π ε ε π ε

ε ε

πεπε

+ + minus+

minus minus=⎛ ⎞⎜ ⎟

Faded signal attenuated and rotated by CFO

( )( ) ( ) ( ) ( )11 2

sin

sin g gI f I f

i N N N NN j j l kI f

NN

l k π ε ε π ε επ ε ε + + minusminus + + + minus

⎜ ⎟⎝ ⎠

+ + minussum

( )( )( )

( ) ( )2

0

sin

I f I f

I

j j l kI f N Ni l l

l l k I f

X H e el k

NN

π ε ε π ε ε

ε π ε ε

+ + +

= ne minus

+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠

sum

i k

N

W

⎜ ⎟⎝ ⎠

+ Inter-Carrier Interference (ICI)

54 Synchronization2010722

Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )

loz An easier representation of received signal with only p g yfractional CFO is

1N

( ) ( )1

0

0N

k k k l l kl l k

Y H X C H X C l k Wminus

= ne

= + minus +⎡ ⎤⎣ ⎦sum

where the ICI coefficient is given by

( )( )( ) ( )

sin 1exp 1fl k

C l k j l kπ ε

π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )

( )exp 1sin

f

f

C l k j l kNN l k

N

π επ ε

minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠

55 Synchronization2010722

Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )

loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality

loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift

56 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

57 Synchronization2010722

TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )

loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq

58 Synchronization2010722

TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )

loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below

loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and

rN2 can be expressed by

0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+

( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +

59

( )

Synchronization2010722

TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )

loz Without CFO the first half is identical to the second half (in time order)

loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below

( ) 12 in discretesft T f NT

NTφ π π π ε πε

⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟

⎝ ⎠

helliphellip

sNT⎝ ⎠

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)

60

Length=N2

Synchronization2010722

TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )

loz Received signal without CFOg0

0 0

jr a e θ=jθ 2 0

2 2 0

Nj j

N Nr a e a e

θ θ= =

loz Received signal with CFO0 0

0 0

j jr a e eθ φ= Extra phase rotation due to CFO0 0

2 2 20

2 2 0

N N Nj j jj

N Nr a e e a e e

θ φ φθ= =

due to CFO

Phase difference 2 2 0N N

( )2 02 20 2 0 0

Nj j T fN

r r a e a eφ φ πminus Δ= =

which contains the information about CFO

61

0 2 0 0Nr r a e a e

Synchronization2010722

TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )

loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be

loz Note that d is a time index corresponding to the first( ) ( ) 2 1

20

Nd m d m Nm

P d r rminus

+ + +== sum

loz Note that d is a time index corresponding to the first sample in a window of N samples

loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by

( )22 1

20

Nd m NR d rminus

+ += sumloz A timing metric can be defined as

))(()(

)( 2

2

dRdP

dM =

( ) 20 d m Nm + +=sum

62

))(( dR

Synchronization2010722

TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )

63 Synchronization2010722

TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )

loz If drsquo is the correct symbol timing offset

( )2

2 2

0 1 12

j T f j T f j T fN

P d a e a e a eπ π πΔ Δ Δ

minus= + + +

22

2 2

0 1 12

j T fN

e a a aπ Δ

minus

⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭

loz If drsquo falls behind the correct symbol timing offset 1 sample

2⎪ ⎪⎩ ⎭

64 Synchronization2010722

TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )

65 Synchronization2010722

TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )

loz Drawback Plateau effect

Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)

((N2) ((N2) ((N2)

helliphellip

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)

Sliding window (length = N)

66 Synchronization2010722

TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )

loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula

( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus

扣掉相加

( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +

loz The method also called delay correlator

67 Synchronization2010722

IEEE 80211aIEEE 80211a

68 Synchronization2010722

Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )

loz Minnrsquos Method [2][ ]B B －B －B

( )( )

( )

22

2Minn

P dM d =

1 4 1N minus

( )( )( )2

2

MinnR d

( ) ( ) ( ) ( )

1 4 1

2 2 2 40 0

N

d N m k d N m k Nm k

P d r r+ + + + += =

= sum sum

( ) ( ) ( )2

21 4 1

2 40 0

N

d N m k Nk

R d rminus

+ + += sum sum69

0 0m k= =

Synchronization2010722

Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )

loz Parkrsquos Method [3][ ]

where D is the symmetric version of CC D C D

where D is the symmetric version of C

( )( ) 2

3k

P dM d =( )

( )( )23

ParkM dR d

2N

( ) 2

30

N

d k d kk

P d r rminus +=

= sdotsum

( )3

22

0

N

d kk

R d r += sum70

0k=

Synchronization2010722

Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )

loz Renrsquos Method [4][ ]

AS AS

where S is the PN sequence weighted of the original bl 2preamble

( )( )( )( )

24

Re 24

n

P dM d

R d=

( )( )4R d

( ) 2 1

4 2 2

N

k k N d k d k NP d s s r rminus

+ + + += sum0k=sum

( )21

41 N

d kR d rminus

= sum71

( )402 d k

kR d r +

=sum

Synchronization2010722

Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )

72 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

73 Synchronization2010722

Fractional CFO estimation Fractional CFO estimation

loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected

11 Rminus⎛ ⎞sum

h L i th di t b t t id ti l bl k R i

0

12 d m d m L

ms

f r rLTπ + + +

=

⎛ ⎞= ang⎜ ⎟⎝ ⎠sum

where L is the distance between two identical block R is the block size

loz For exampleNN cp

L = N R = NL = N R = Ncp74 Synchronization2010722

Estimation rangeEstimation rangegg

loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval

f ( ) ( )s s⎣ ⎦

loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)

loz DVB-T (L=N) 802 11 (L N4)

[ ]05 05s sf fminus

[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)

(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus

loz Note that is the subcarrier spacing1sf NT=

75

sNT

Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz

plusmng q y

loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn

76 Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz

plusmng q y

loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for

subcarrier spacing is required

plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn

p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs

77 Synchronization2010722

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

loz Then we sample the signal at a sampling frequency tΔ1

by using IDFTby using IDFT

p g p g q y and An(t) and φn(t) becomes

(8))(t = φφ(9) )((8) )(

nn

nn

AtAt

== φφ

( ) (10) 1)(1

0

)(2 0summinus

=

+ΔΔ+=ΔN

n

tkfnfjns

neAN

tkS φπ

loz Then the sampled signal can be expressed as0=nN

( )( ) (11) 1)(1

0

22 0summinus

=

ΔΔ+Δ sdot=ΔN

n

tfnkjtkfjns eeA

NtkS n πφπ

0n

18

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

loz The inverse discrete Fourier transform (IDFT) is defined

by using IDFTby using IDFT

( )as the following

1 21

NnkjN

πsumminus

(12) )(1)( 2

0

Nnkj

nefnF

Ntkf πsum

=

Δ=Δ

loz Comparing eq(11) and eq(12) the condition must be satisfied in order to make eq(11) an inverse Fouriersatisfied in order to make eq(11) an inverse Fourier transform relationship

1 (13) 1tN

fΔ

=Δ

19

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

loz If eq(13) is satisfied

by using IDFTby using IDFT

q ( ) loz is the frequency domain signal loz is the time domain signal

( )ntkfjneA φπ +Δ02

)( tkSs Δ gloz is the sub-channel spacingloz is the symbol duration in each sub-channel

)(s

fΔtNΔloz s e sy bo du o e c sub c e

loz This outcome is the same as the result obtained in the

N

loz This outcome is the same as the result obtained in the system of Figure 1 Therefore IDFT can be used to generate an OFDM transmission signalgenerate an OFDM transmission signal

20

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

( )tf02cos π

( )0a

by using IDFTby using IDFT

( )nd ( )tDI( )tf02sin π

( )0a

( )0b

bull

tf s Δ=

1 ( ) ( ) njbna +

( )nd

tNΔffnffn Δ

=Δ+=1 0

( )tDChannelInput

( )tf N 12cos minusπ

( )1minusNa

bullbull

tNΔ

( )tf N 12sin minusπ

( )1minusNb

( )0d

( )nd ( )tDChannelInput

bull bull

( )( )1d

( )2d

tf s Δ=

1 ( ) ( ) njbna +

fNΔffnffn Δ

=Δ+=1 0

ChannelInput

bullbullbullbull

bullbullbullbull

( )1minusNd

21

Frequency Error Results in ICIFrequency Error Results in ICIq yq y

22

Synchronization Error Results in ICISynchronization Error Results in ICIyy

Not Orthogonal Any More

23

Cyclic PrefixCyclic Prefixyy

loz In multipath channel delayed replicas of previous OFDM signal lead to ISI between successive OFDM signals

loz Solution Insert a guard interval between successive OFDMloz Solution Insert a guard interval between successive OFDM signals

24

Cyclic PrefixCyclic Prefix

loz Guard interval leads to intercarrier interference (ICI) in OFDM d d l i

yy

demodulation

loz In DFT interval difference between two subcarriers does not maintain integer number of cycles loss of orthogonality

loz Delayed version of subcarrier 2 causes ICI in the process of d d l ti b i 1demodulating subcarrier 1

25

Cyclic PrefixCyclic Prefixloz Cyclic prefix (CP) A copy of the last part of OFDM signal is

attached to the front of itself

yy

attached to the front of itself

[ ]0d

[ ]1d

[ ]0D[ ]1D

[ ]nd[ ]kD~

symbols dataInput

bullbullbullbull

[ ]1d

[ ]2d

[ ][ ]2D

bullbullbullbullbull

y

bullbullbullbullbull

[ ]1minusNd [ ]1minusND

[ ]gNND minus

bullbullbull

26

Cyclic PrefixCyclic Prefix

loz All delayed replicas of subcarriers always have an

yy

y p yinteger number of cycles within DFT interval no ICI

27

Cyclic PrefixCyclic Prefix

loz Linear convolution vs circular convolution

yy

N

28

Cyclic PrefixCyclic Prefix

loz Channel effect with cyclic prefix

yy

Signal after removed CP

29

Cyclic PrefixCyclic Prefix

loz Time-Domain Explanation

yy

p

30

Cyclic PrefixCyclic Prefixyy

31

Cyclic PrefixCyclic Prefix

loz Spectrum of channel response with length (smaller [ ]nh hL

yy

than )h

gN[ ] nhFFTHk =

loz Received complete OFDM signal[ ] [ ] [ ] 20 ~~ minus++leleotimes= hg LNNnnhnDnr

loz Received useful part [ ]nr[ ] [ ] [ ]nhnDnr Notimes=

where is N-point circular convolution (due to CP)loz Received symbol at k-th subcarrier

[ ] [ ] [ ]N

Notimes

[ ] [ ] [ ] kkNk HXnhnDFFTnrFFTY =otimes== kYX =rArr ldquoUseful property for OFDM system to reduce

kk H

X =rArrcomplexity of channel equalizationrdquo

32

Cyclic Cyclic Prefix (OFDM Prefix (OFDM Receiver)Receiver)yy (( ))

bull bull( )( )nr~Symbols

dataOutput

bullbullbull

bullbullbull( )nr

bullbull

bullbull

33

Cyclic PrefixCyclic Prefix

loz One of the most important reasons to do OFDM is the

yy

pefficient way it deals with multipath delay spread

loz To eliminate inter-symbol interference (ISI) almostloz To eliminate inter symbol interference (ISI) almost completely a guard time is introduced for each OFDM symbolsymbol(The guard time is chosen larger than the delay spread)

34

Bandwidth EfficiencyBandwidth Efficiencyyy

loz In a classical parallel system the channel is divided into p y N non-overlapping sub-channels to avoid inter-carrier interference (ICI)( )

loz The diagram for bandwidth efficiency of OFDM system is shown below

35

SummarySummaryyy

loz The advantage of the FFT-based OFDM system g yloz The use of IFFTFFT can reduce the computation complexityloz The orthogonality between the adjacent sub-carriers will make g y j

the use of transmission bandwidth more efficientloz The guard interval is used to resist the inter-symbol interference

(ISI)loz The main advantage of the OFDM transmission technique is its

high performance even in frequency selective channels

loz The drawbacks of the OFDM system loz It is highly vulnerable to synchronization errorsloz Peak to Average Power Ratio (PAPR) problems

36

SynchronizationSynchronization

Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offset (TO)loz Impact of timing offset (TO)loz Impact of carrier frequency offset (CFO)loz TO estimation algorithmloz CFO estimation algorithmg

38 Synchronization2010722

Introduction (13)Introduction (13)( )( )

loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods

loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead

loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the

cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of

computational complexity therefore it may be not be available in short-burst wireless communication

39 Synchronization2010722

Introduction (23)Introduction (23)( )( )

loz Transcevier

40 Synchronization2010722

Introduction (33)Introduction (33)( )( )

loz Synchronization erroryloz Symbol timing offset (TO)

loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)

loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect

41 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg

42 Synchronization2010722

Signal model (14)Signal model (14)g ( )g ( )

loz Time domain sample p1

2

0

1 Nj nk N

n kk

x X eN

πminus

minus= sum

Xk the frequency domain data at the k-th subcarrier0kN =

loz The multipath fading channel

( ) ( ) ( )1L

h t h t δminus

sum

loz Received time domain signal

( ) ( ) ( )0

l ll

h t h tτ τ δ τ τ=

= minussum

loz Received time-domain signal

( ) ( )1

L

n n ly h n l x w nminus

minus= +sum43

( ) ( )0l=sum

Synchronization2010722

Signal model (24)Signal model (24)g ( )g ( )

loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting

12

0

Nj kn N

k nn

Y y e πminus

minus

=

= sum

( )1 1

2

0

N Lj kn N

n l kn l

h n l x e Wπminus minus

minusminus

=

= +sumsum

loz The equation above could be expressed in matrix form

0n l

loz The equation above could be expressed in matrix form

= +Y HX W

44 Synchronization2010722

Signal model (34)Signal model (34)g ( )g ( )

H= = = = +Y Fy FCx FCF X HX W

loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal

y

H=H FCFcirculant matrix leading to be a diagonal matrix

=H FCF

( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0

h h L L hh h

⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥

( )( ) ( )

( ) ( )

1 10 1 0 0i

h Lh L h N L

⎢ ⎥⎢ ⎥minus minus⎢ ⎥

equiv= minus minus⎢ ⎥⎢ ⎥

C( ) ( )

( ) ( )

0 1 1 100

0 0 1 1 0

h L h N L

h N L L h N

⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

45

( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦

Synchronization2010722

Signal model (44)Signal model (44)g ( )g ( )

loz In a time-invariant channel the channel matrix H would be a diagonal matrix

k k k kY H X W= +

( )1 1

2

0 0

N Lj kl N

kn l

H h n l e πminus minus

minus

= =

= sumsum

loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI

k k k k kY H X I Wα= + +k k k k k

( ) ( )1 1 1

2 2 N N L

j m k Nj ml Nk mI X h n l e e ππ

minus minus minusminusminus⎛ ⎞

= ⎜ ⎟⎝ ⎠

sum sum sumICI

46

0 0 0m n lm k= = =ne

⎝ ⎠

Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

47 Synchronization2010722

Impact of timing offset Impact of timing offset p gp g

loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively

( )I f s

Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ

( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr

2 fj k Nk k k kY X H e Wπδminus= +

loz For integer part of timing offset k k k kY X H e W+

I sTδ ( ) 0g IN L δminus minus le leFFT

2 Ij k NY X H e Wπδminus= +

( ) ( ) 2 I sFFT

j f TI sh t T H f e π δδ minusminus rarr

48

k k k kY X H e W= +

Synchronization2010722

Impact of timing offset derivationImpact of timing offset derivationp gp g

49 Synchronization2010722

Discussion of symbol boundaryDiscussion of symbol boundaryy yy y

50 Synchronization2010722

Impact of timing offsetImpact of timing offsetp gp g

loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI

2 Ij k NIk k k k

NY X H e ISI W

Nπδδ minusminus

= + +

loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk

e e

p kloz Phase Shift can be resolved by Differential Encoding

Decoding or carrier recovery with pilots

51 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

52 Synchronization2010722

Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )

53 Synchronization2010722

Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )

( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s

i n t i N N T N T nTy y t e

= + + +=

( ) ( )( ) ( ) ( ) 12

sin

sin

g gI f f

I I

i N N N Nj jf N Ni k i k k

f

Y X H e eN

π ε ε π ε

ε ε

πεπε

+ + minus+

minus minus=⎛ ⎞⎜ ⎟

Faded signal attenuated and rotated by CFO

( )( ) ( ) ( ) ( )11 2

sin

sin g gI f I f

i N N N NN j j l kI f

NN

l k π ε ε π ε επ ε ε + + minusminus + + + minus

⎜ ⎟⎝ ⎠

+ + minussum

( )( )( )

( ) ( )2

0

sin

I f I f

I

j j l kI f N Ni l l

l l k I f

X H e el k

NN

π ε ε π ε ε

ε π ε ε

+ + +

= ne minus

+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠

sum

i k

N

W

⎜ ⎟⎝ ⎠

+ Inter-Carrier Interference (ICI)

54 Synchronization2010722

Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )

loz An easier representation of received signal with only p g yfractional CFO is

1N

( ) ( )1

0

0N

k k k l l kl l k

Y H X C H X C l k Wminus

= ne

= + minus +⎡ ⎤⎣ ⎦sum

where the ICI coefficient is given by

( )( )( ) ( )

sin 1exp 1fl k

C l k j l kπ ε

π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )

( )exp 1sin

f

f

C l k j l kNN l k

N

π επ ε

minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠

55 Synchronization2010722

Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )

loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality

loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift

56 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

57 Synchronization2010722

TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )

loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq

58 Synchronization2010722

TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )

loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below

loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and

rN2 can be expressed by

0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+

( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +

59

( )

Synchronization2010722

TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )

loz Without CFO the first half is identical to the second half (in time order)

loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below

( ) 12 in discretesft T f NT

NTφ π π π ε πε

⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟

⎝ ⎠

helliphellip

sNT⎝ ⎠

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)

60

Length=N2

Synchronization2010722

TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )

loz Received signal without CFOg0

0 0

jr a e θ=jθ 2 0

2 2 0

Nj j

N Nr a e a e

θ θ= =

loz Received signal with CFO0 0

0 0

j jr a e eθ φ= Extra phase rotation due to CFO0 0

2 2 20

2 2 0

N N Nj j jj

N Nr a e e a e e

θ φ φθ= =

due to CFO

Phase difference 2 2 0N N

( )2 02 20 2 0 0

Nj j T fN

r r a e a eφ φ πminus Δ= =

which contains the information about CFO

61

0 2 0 0Nr r a e a e

Synchronization2010722

TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )

loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be

loz Note that d is a time index corresponding to the first( ) ( ) 2 1

20

Nd m d m Nm

P d r rminus

+ + +== sum

loz Note that d is a time index corresponding to the first sample in a window of N samples

loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by

( )22 1

20

Nd m NR d rminus

+ += sumloz A timing metric can be defined as

))(()(

)( 2

2

dRdP

dM =

( ) 20 d m Nm + +=sum

62

))(( dR

Synchronization2010722

TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )

63 Synchronization2010722

TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )

loz If drsquo is the correct symbol timing offset

( )2

2 2

0 1 12

j T f j T f j T fN

P d a e a e a eπ π πΔ Δ Δ

minus= + + +

22

2 2

0 1 12

j T fN

e a a aπ Δ

minus

⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭

loz If drsquo falls behind the correct symbol timing offset 1 sample

2⎪ ⎪⎩ ⎭

64 Synchronization2010722

TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )

65 Synchronization2010722

TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )

loz Drawback Plateau effect

Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)

((N2) ((N2) ((N2)

helliphellip

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)

Sliding window (length = N)

66 Synchronization2010722

TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )

loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula

( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus

扣掉相加

( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +

loz The method also called delay correlator

67 Synchronization2010722

IEEE 80211aIEEE 80211a

68 Synchronization2010722

Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )

loz Minnrsquos Method [2][ ]B B －B －B

( )( )

( )

22

2Minn

P dM d =

1 4 1N minus

( )( )( )2

2

MinnR d

( ) ( ) ( ) ( )

1 4 1

2 2 2 40 0

N

d N m k d N m k Nm k

P d r r+ + + + += =

= sum sum

( ) ( ) ( )2

21 4 1

2 40 0

N

d N m k Nk

R d rminus

+ + += sum sum69

0 0m k= =

Synchronization2010722

Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )

loz Parkrsquos Method [3][ ]

where D is the symmetric version of CC D C D

where D is the symmetric version of C

( )( ) 2

3k

P dM d =( )

( )( )23

ParkM dR d

2N

( ) 2

30

N

d k d kk

P d r rminus +=

= sdotsum

( )3

22

0

N

d kk

R d r += sum70

0k=

Synchronization2010722

Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )

loz Renrsquos Method [4][ ]

AS AS

where S is the PN sequence weighted of the original bl 2preamble

( )( )( )( )

24

Re 24

n

P dM d

R d=

( )( )4R d

( ) 2 1

4 2 2

N

k k N d k d k NP d s s r rminus

+ + + += sum0k=sum

( )21

41 N

d kR d rminus

= sum71

( )402 d k

kR d r +

=sum

Synchronization2010722

Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )

72 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

73 Synchronization2010722

Fractional CFO estimation Fractional CFO estimation

loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected

11 Rminus⎛ ⎞sum

h L i th di t b t t id ti l bl k R i

0

12 d m d m L

ms

f r rLTπ + + +

=

⎛ ⎞= ang⎜ ⎟⎝ ⎠sum

where L is the distance between two identical block R is the block size

loz For exampleNN cp

L = N R = NL = N R = Ncp74 Synchronization2010722

Estimation rangeEstimation rangegg

loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval

f ( ) ( )s s⎣ ⎦

loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)

loz DVB-T (L=N) 802 11 (L N4)

[ ]05 05s sf fminus

[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)

(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus

loz Note that is the subcarrier spacing1sf NT=

75

sNT

Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz

plusmng q y

loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn

76 Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz

plusmng q y

loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for

subcarrier spacing is required

plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn

p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs

77 Synchronization2010722

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

loz The inverse discrete Fourier transform (IDFT) is defined

by using IDFTby using IDFT

( )as the following

1 21

NnkjN

πsumminus

(12) )(1)( 2

0

Nnkj

nefnF

Ntkf πsum

=

Δ=Δ

loz Comparing eq(11) and eq(12) the condition must be satisfied in order to make eq(11) an inverse Fouriersatisfied in order to make eq(11) an inverse Fourier transform relationship

1 (13) 1tN

fΔ

=Δ

19

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

loz If eq(13) is satisfied

by using IDFTby using IDFT

q ( ) loz is the frequency domain signal loz is the time domain signal

( )ntkfjneA φπ +Δ02

)( tkSs Δ gloz is the sub-channel spacingloz is the symbol duration in each sub-channel

)(s

fΔtNΔloz s e sy bo du o e c sub c e

loz This outcome is the same as the result obtained in the

N

loz This outcome is the same as the result obtained in the system of Figure 1 Therefore IDFT can be used to generate an OFDM transmission signalgenerate an OFDM transmission signal

20

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

( )tf02cos π

( )0a

by using IDFTby using IDFT

( )nd ( )tDI( )tf02sin π

( )0a

( )0b

bull

tf s Δ=

1 ( ) ( ) njbna +

( )nd

tNΔffnffn Δ

=Δ+=1 0

( )tDChannelInput

( )tf N 12cos minusπ

( )1minusNa

bullbull

tNΔ

( )tf N 12sin minusπ

( )1minusNb

( )0d

( )nd ( )tDChannelInput

bull bull

( )( )1d

( )2d

tf s Δ=

1 ( ) ( ) njbna +

fNΔffnffn Δ

=Δ+=1 0

ChannelInput

bullbullbullbull

bullbullbullbull

( )1minusNd

21

Frequency Error Results in ICIFrequency Error Results in ICIq yq y

22

Synchronization Error Results in ICISynchronization Error Results in ICIyy

Not Orthogonal Any More

23

Cyclic PrefixCyclic Prefixyy

loz In multipath channel delayed replicas of previous OFDM signal lead to ISI between successive OFDM signals

loz Solution Insert a guard interval between successive OFDMloz Solution Insert a guard interval between successive OFDM signals

24

Cyclic PrefixCyclic Prefix

loz Guard interval leads to intercarrier interference (ICI) in OFDM d d l i

yy

demodulation

loz In DFT interval difference between two subcarriers does not maintain integer number of cycles loss of orthogonality

loz Delayed version of subcarrier 2 causes ICI in the process of d d l ti b i 1demodulating subcarrier 1

25

Cyclic PrefixCyclic Prefixloz Cyclic prefix (CP) A copy of the last part of OFDM signal is

attached to the front of itself

yy

attached to the front of itself

[ ]0d

[ ]1d

[ ]0D[ ]1D

[ ]nd[ ]kD~

symbols dataInput

bullbullbullbull

[ ]1d

[ ]2d

[ ][ ]2D

bullbullbullbullbull

y

bullbullbullbullbull

[ ]1minusNd [ ]1minusND

[ ]gNND minus

bullbullbull

26

Cyclic PrefixCyclic Prefix

loz All delayed replicas of subcarriers always have an

yy

y p yinteger number of cycles within DFT interval no ICI

27

Cyclic PrefixCyclic Prefix

loz Linear convolution vs circular convolution

yy

N

28

Cyclic PrefixCyclic Prefix

loz Channel effect with cyclic prefix

yy

Signal after removed CP

29

Cyclic PrefixCyclic Prefix

loz Time-Domain Explanation

yy

p

30

Cyclic PrefixCyclic Prefixyy

31

Cyclic PrefixCyclic Prefix

loz Spectrum of channel response with length (smaller [ ]nh hL

yy

than )h

gN[ ] nhFFTHk =

loz Received complete OFDM signal[ ] [ ] [ ] 20 ~~ minus++leleotimes= hg LNNnnhnDnr

loz Received useful part [ ]nr[ ] [ ] [ ]nhnDnr Notimes=

where is N-point circular convolution (due to CP)loz Received symbol at k-th subcarrier

[ ] [ ] [ ]N

Notimes

[ ] [ ] [ ] kkNk HXnhnDFFTnrFFTY =otimes== kYX =rArr ldquoUseful property for OFDM system to reduce

kk H

X =rArrcomplexity of channel equalizationrdquo

32

Cyclic Cyclic Prefix (OFDM Prefix (OFDM Receiver)Receiver)yy (( ))

bull bull( )( )nr~Symbols

dataOutput

bullbullbull

bullbullbull( )nr

bullbull

bullbull

33

Cyclic PrefixCyclic Prefix

loz One of the most important reasons to do OFDM is the

yy

pefficient way it deals with multipath delay spread

loz To eliminate inter-symbol interference (ISI) almostloz To eliminate inter symbol interference (ISI) almost completely a guard time is introduced for each OFDM symbolsymbol(The guard time is chosen larger than the delay spread)

34

Bandwidth EfficiencyBandwidth Efficiencyyy

loz In a classical parallel system the channel is divided into p y N non-overlapping sub-channels to avoid inter-carrier interference (ICI)( )

loz The diagram for bandwidth efficiency of OFDM system is shown below

35

SummarySummaryyy

loz The advantage of the FFT-based OFDM system g yloz The use of IFFTFFT can reduce the computation complexityloz The orthogonality between the adjacent sub-carriers will make g y j

the use of transmission bandwidth more efficientloz The guard interval is used to resist the inter-symbol interference

(ISI)loz The main advantage of the OFDM transmission technique is its

high performance even in frequency selective channels

loz The drawbacks of the OFDM system loz It is highly vulnerable to synchronization errorsloz Peak to Average Power Ratio (PAPR) problems

36

SynchronizationSynchronization

Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offset (TO)loz Impact of timing offset (TO)loz Impact of carrier frequency offset (CFO)loz TO estimation algorithmloz CFO estimation algorithmg

38 Synchronization2010722

Introduction (13)Introduction (13)( )( )

loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods

loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead

loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the

cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of

computational complexity therefore it may be not be available in short-burst wireless communication

39 Synchronization2010722

Introduction (23)Introduction (23)( )( )

loz Transcevier

40 Synchronization2010722

Introduction (33)Introduction (33)( )( )

loz Synchronization erroryloz Symbol timing offset (TO)

loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)

loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect

41 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg

42 Synchronization2010722

Signal model (14)Signal model (14)g ( )g ( )

loz Time domain sample p1

2

0

1 Nj nk N

n kk

x X eN

πminus

minus= sum

Xk the frequency domain data at the k-th subcarrier0kN =

loz The multipath fading channel

( ) ( ) ( )1L

h t h t δminus

sum

loz Received time domain signal

( ) ( ) ( )0

l ll

h t h tτ τ δ τ τ=

= minussum

loz Received time-domain signal

( ) ( )1

L

n n ly h n l x w nminus

minus= +sum43

( ) ( )0l=sum

Synchronization2010722

Signal model (24)Signal model (24)g ( )g ( )

loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting

12

0

Nj kn N

k nn

Y y e πminus

minus

=

= sum

( )1 1

2

0

N Lj kn N

n l kn l

h n l x e Wπminus minus

minusminus

=

= +sumsum

loz The equation above could be expressed in matrix form

0n l

loz The equation above could be expressed in matrix form

= +Y HX W

44 Synchronization2010722

Signal model (34)Signal model (34)g ( )g ( )

H= = = = +Y Fy FCx FCF X HX W

loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal

y

H=H FCFcirculant matrix leading to be a diagonal matrix

=H FCF

( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0

h h L L hh h

⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥

( )( ) ( )

( ) ( )

1 10 1 0 0i

h Lh L h N L

⎢ ⎥⎢ ⎥minus minus⎢ ⎥

equiv= minus minus⎢ ⎥⎢ ⎥

C( ) ( )

( ) ( )

0 1 1 100

0 0 1 1 0

h L h N L

h N L L h N

⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

45

( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦

Synchronization2010722

Signal model (44)Signal model (44)g ( )g ( )

loz In a time-invariant channel the channel matrix H would be a diagonal matrix

k k k kY H X W= +

( )1 1

2

0 0

N Lj kl N

kn l

H h n l e πminus minus

minus

= =

= sumsum

loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI

k k k k kY H X I Wα= + +k k k k k

( ) ( )1 1 1

2 2 N N L

j m k Nj ml Nk mI X h n l e e ππ

minus minus minusminusminus⎛ ⎞

= ⎜ ⎟⎝ ⎠

sum sum sumICI

46

0 0 0m n lm k= = =ne

⎝ ⎠

Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

47 Synchronization2010722

Impact of timing offset Impact of timing offset p gp g

loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively

( )I f s

Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ

( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr

2 fj k Nk k k kY X H e Wπδminus= +

loz For integer part of timing offset k k k kY X H e W+

I sTδ ( ) 0g IN L δminus minus le leFFT

2 Ij k NY X H e Wπδminus= +

( ) ( ) 2 I sFFT

j f TI sh t T H f e π δδ minusminus rarr

48

k k k kY X H e W= +

Synchronization2010722

Impact of timing offset derivationImpact of timing offset derivationp gp g

49 Synchronization2010722

Discussion of symbol boundaryDiscussion of symbol boundaryy yy y

50 Synchronization2010722

Impact of timing offsetImpact of timing offsetp gp g

loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI

2 Ij k NIk k k k

NY X H e ISI W

Nπδδ minusminus

= + +

loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk

e e

p kloz Phase Shift can be resolved by Differential Encoding

Decoding or carrier recovery with pilots

51 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

52 Synchronization2010722

Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )

53 Synchronization2010722

Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )

( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s

i n t i N N T N T nTy y t e

= + + +=

( ) ( )( ) ( ) ( ) 12

sin

sin

g gI f f

I I

i N N N Nj jf N Ni k i k k

f

Y X H e eN

π ε ε π ε

ε ε

πεπε

+ + minus+

minus minus=⎛ ⎞⎜ ⎟

Faded signal attenuated and rotated by CFO

( )( ) ( ) ( ) ( )11 2

sin

sin g gI f I f

i N N N NN j j l kI f

NN

l k π ε ε π ε επ ε ε + + minusminus + + + minus

⎜ ⎟⎝ ⎠

+ + minussum

( )( )( )

( ) ( )2

0

sin

I f I f

I

j j l kI f N Ni l l

l l k I f

X H e el k

NN

π ε ε π ε ε

ε π ε ε

+ + +

= ne minus

+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠

sum

i k

N

W

⎜ ⎟⎝ ⎠

+ Inter-Carrier Interference (ICI)

54 Synchronization2010722

Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )

loz An easier representation of received signal with only p g yfractional CFO is

1N

( ) ( )1

0

0N

k k k l l kl l k

Y H X C H X C l k Wminus

= ne

= + minus +⎡ ⎤⎣ ⎦sum

where the ICI coefficient is given by

( )( )( ) ( )

sin 1exp 1fl k

C l k j l kπ ε

π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )

( )exp 1sin

f

f

C l k j l kNN l k

N

π επ ε

minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠

55 Synchronization2010722

Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )

loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality

loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift

56 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

57 Synchronization2010722

TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )

loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq

58 Synchronization2010722

TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )

loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below

loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and

rN2 can be expressed by

0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+

( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +

59

( )

Synchronization2010722

TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )

loz Without CFO the first half is identical to the second half (in time order)

loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below

( ) 12 in discretesft T f NT

NTφ π π π ε πε

⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟

⎝ ⎠

helliphellip

sNT⎝ ⎠

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)

60

Length=N2

Synchronization2010722

TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )

loz Received signal without CFOg0

0 0

jr a e θ=jθ 2 0

2 2 0

Nj j

N Nr a e a e

θ θ= =

loz Received signal with CFO0 0

0 0

j jr a e eθ φ= Extra phase rotation due to CFO0 0

2 2 20

2 2 0

N N Nj j jj

N Nr a e e a e e

θ φ φθ= =

due to CFO

Phase difference 2 2 0N N

( )2 02 20 2 0 0

Nj j T fN

r r a e a eφ φ πminus Δ= =

which contains the information about CFO

61

0 2 0 0Nr r a e a e

Synchronization2010722

TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )

loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be

loz Note that d is a time index corresponding to the first( ) ( ) 2 1

20

Nd m d m Nm

P d r rminus

+ + +== sum

loz Note that d is a time index corresponding to the first sample in a window of N samples

loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by

( )22 1

20

Nd m NR d rminus

+ += sumloz A timing metric can be defined as

))(()(

)( 2

2

dRdP

dM =

( ) 20 d m Nm + +=sum

62

))(( dR

Synchronization2010722

TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )

63 Synchronization2010722

TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )

loz If drsquo is the correct symbol timing offset

( )2

2 2

0 1 12

j T f j T f j T fN

P d a e a e a eπ π πΔ Δ Δ

minus= + + +

22

2 2

0 1 12

j T fN

e a a aπ Δ

minus

⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭

loz If drsquo falls behind the correct symbol timing offset 1 sample

2⎪ ⎪⎩ ⎭

64 Synchronization2010722

TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )

65 Synchronization2010722

TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )

loz Drawback Plateau effect

Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)

((N2) ((N2) ((N2)

helliphellip

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)

Sliding window (length = N)

66 Synchronization2010722

TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )

loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula

( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus

扣掉相加

( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +

loz The method also called delay correlator

67 Synchronization2010722

IEEE 80211aIEEE 80211a

68 Synchronization2010722

Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )

loz Minnrsquos Method [2][ ]B B －B －B

( )( )

( )

22

2Minn

P dM d =

1 4 1N minus

( )( )( )2

2

MinnR d

( ) ( ) ( ) ( )

1 4 1

2 2 2 40 0

N

d N m k d N m k Nm k

P d r r+ + + + += =

= sum sum

( ) ( ) ( )2

21 4 1

2 40 0

N

d N m k Nk

R d rminus

+ + += sum sum69

0 0m k= =

Synchronization2010722

Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )

loz Parkrsquos Method [3][ ]

where D is the symmetric version of CC D C D

where D is the symmetric version of C

( )( ) 2

3k

P dM d =( )

( )( )23

ParkM dR d

2N

( ) 2

30

N

d k d kk

P d r rminus +=

= sdotsum

( )3

22

0

N

d kk

R d r += sum70

0k=

Synchronization2010722

Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )

loz Renrsquos Method [4][ ]

AS AS

where S is the PN sequence weighted of the original bl 2preamble

( )( )( )( )

24

Re 24

n

P dM d

R d=

( )( )4R d

( ) 2 1

4 2 2

N

k k N d k d k NP d s s r rminus

+ + + += sum0k=sum

( )21

41 N

d kR d rminus

= sum71

( )402 d k

kR d r +

=sum

Synchronization2010722

Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )

72 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

73 Synchronization2010722

Fractional CFO estimation Fractional CFO estimation

loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected

11 Rminus⎛ ⎞sum

h L i th di t b t t id ti l bl k R i

0

12 d m d m L

ms

f r rLTπ + + +

=

⎛ ⎞= ang⎜ ⎟⎝ ⎠sum

where L is the distance between two identical block R is the block size

loz For exampleNN cp

L = N R = NL = N R = Ncp74 Synchronization2010722

Estimation rangeEstimation rangegg

loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval

f ( ) ( )s s⎣ ⎦

loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)

loz DVB-T (L=N) 802 11 (L N4)

[ ]05 05s sf fminus

[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)

(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus

loz Note that is the subcarrier spacing1sf NT=

75

sNT

Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz

plusmng q y

loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn

76 Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz

plusmng q y

loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for

subcarrier spacing is required

plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn

p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs

77 Synchronization2010722

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

loz If eq(13) is satisfied

by using IDFTby using IDFT

q ( ) loz is the frequency domain signal loz is the time domain signal

( )ntkfjneA φπ +Δ02

)( tkSs Δ gloz is the sub-channel spacingloz is the symbol duration in each sub-channel

)(s

fΔtNΔloz s e sy bo du o e c sub c e

loz This outcome is the same as the result obtained in the

N

loz This outcome is the same as the result obtained in the system of Figure 1 Therefore IDFT can be used to generate an OFDM transmission signalgenerate an OFDM transmission signal

20

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

( )tf02cos π

( )0a

by using IDFTby using IDFT

( )nd ( )tDI( )tf02sin π

( )0a

( )0b

bull

tf s Δ=

1 ( ) ( ) njbna +

( )nd

tNΔffnffn Δ

=Δ+=1 0

( )tDChannelInput

( )tf N 12cos minusπ

( )1minusNa

bullbull

tNΔ

( )tf N 12sin minusπ

( )1minusNb

( )0d

( )nd ( )tDChannelInput

bull bull

( )( )1d

( )2d

tf s Δ=

1 ( ) ( ) njbna +

fNΔffnffn Δ

=Δ+=1 0

ChannelInput

bullbullbullbull

bullbullbullbull

( )1minusNd

21

Frequency Error Results in ICIFrequency Error Results in ICIq yq y

22

Synchronization Error Results in ICISynchronization Error Results in ICIyy

Not Orthogonal Any More

23

Cyclic PrefixCyclic Prefixyy

loz In multipath channel delayed replicas of previous OFDM signal lead to ISI between successive OFDM signals

loz Solution Insert a guard interval between successive OFDMloz Solution Insert a guard interval between successive OFDM signals

24

Cyclic PrefixCyclic Prefix

loz Guard interval leads to intercarrier interference (ICI) in OFDM d d l i

yy

demodulation

loz In DFT interval difference between two subcarriers does not maintain integer number of cycles loss of orthogonality

loz Delayed version of subcarrier 2 causes ICI in the process of d d l ti b i 1demodulating subcarrier 1

25

Cyclic PrefixCyclic Prefixloz Cyclic prefix (CP) A copy of the last part of OFDM signal is

attached to the front of itself

yy

attached to the front of itself

[ ]0d

[ ]1d

[ ]0D[ ]1D

[ ]nd[ ]kD~

symbols dataInput

bullbullbullbull

[ ]1d

[ ]2d

[ ][ ]2D

bullbullbullbullbull

y

bullbullbullbullbull

[ ]1minusNd [ ]1minusND

[ ]gNND minus

bullbullbull

26

Cyclic PrefixCyclic Prefix

loz All delayed replicas of subcarriers always have an

yy

y p yinteger number of cycles within DFT interval no ICI

27

Cyclic PrefixCyclic Prefix

loz Linear convolution vs circular convolution

yy

N

28

Cyclic PrefixCyclic Prefix

loz Channel effect with cyclic prefix

yy

Signal after removed CP

29

Cyclic PrefixCyclic Prefix

loz Time-Domain Explanation

yy

p

30

Cyclic PrefixCyclic Prefixyy

31

Cyclic PrefixCyclic Prefix

loz Spectrum of channel response with length (smaller [ ]nh hL

yy

than )h

gN[ ] nhFFTHk =

loz Received complete OFDM signal[ ] [ ] [ ] 20 ~~ minus++leleotimes= hg LNNnnhnDnr

loz Received useful part [ ]nr[ ] [ ] [ ]nhnDnr Notimes=

where is N-point circular convolution (due to CP)loz Received symbol at k-th subcarrier

[ ] [ ] [ ]N

Notimes

[ ] [ ] [ ] kkNk HXnhnDFFTnrFFTY =otimes== kYX =rArr ldquoUseful property for OFDM system to reduce

kk H

X =rArrcomplexity of channel equalizationrdquo

32

Cyclic Cyclic Prefix (OFDM Prefix (OFDM Receiver)Receiver)yy (( ))

bull bull( )( )nr~Symbols

dataOutput

bullbullbull

bullbullbull( )nr

bullbull

bullbull

33

Cyclic PrefixCyclic Prefix

loz One of the most important reasons to do OFDM is the

yy

pefficient way it deals with multipath delay spread

loz To eliminate inter-symbol interference (ISI) almostloz To eliminate inter symbol interference (ISI) almost completely a guard time is introduced for each OFDM symbolsymbol(The guard time is chosen larger than the delay spread)

34

Bandwidth EfficiencyBandwidth Efficiencyyy

loz In a classical parallel system the channel is divided into p y N non-overlapping sub-channels to avoid inter-carrier interference (ICI)( )

loz The diagram for bandwidth efficiency of OFDM system is shown below

35

SummarySummaryyy

loz The advantage of the FFT-based OFDM system g yloz The use of IFFTFFT can reduce the computation complexityloz The orthogonality between the adjacent sub-carriers will make g y j

the use of transmission bandwidth more efficientloz The guard interval is used to resist the inter-symbol interference

(ISI)loz The main advantage of the OFDM transmission technique is its

high performance even in frequency selective channels

loz The drawbacks of the OFDM system loz It is highly vulnerable to synchronization errorsloz Peak to Average Power Ratio (PAPR) problems

36

SynchronizationSynchronization

Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offset (TO)loz Impact of timing offset (TO)loz Impact of carrier frequency offset (CFO)loz TO estimation algorithmloz CFO estimation algorithmg

38 Synchronization2010722

Introduction (13)Introduction (13)( )( )

loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods

loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead

loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the

cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of

computational complexity therefore it may be not be available in short-burst wireless communication

39 Synchronization2010722

Introduction (23)Introduction (23)( )( )

loz Transcevier

40 Synchronization2010722

Introduction (33)Introduction (33)( )( )

loz Synchronization erroryloz Symbol timing offset (TO)

loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)

loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect

41 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg

42 Synchronization2010722

Signal model (14)Signal model (14)g ( )g ( )

loz Time domain sample p1

2

0

1 Nj nk N

n kk

x X eN

πminus

minus= sum

Xk the frequency domain data at the k-th subcarrier0kN =

loz The multipath fading channel

( ) ( ) ( )1L

h t h t δminus

sum

loz Received time domain signal

( ) ( ) ( )0

l ll

h t h tτ τ δ τ τ=

= minussum

loz Received time-domain signal

( ) ( )1

L

n n ly h n l x w nminus

minus= +sum43

( ) ( )0l=sum

Synchronization2010722

Signal model (24)Signal model (24)g ( )g ( )

loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting

12

0

Nj kn N

k nn

Y y e πminus

minus

=

= sum

( )1 1

2

0

N Lj kn N

n l kn l

h n l x e Wπminus minus

minusminus

=

= +sumsum

loz The equation above could be expressed in matrix form

0n l

loz The equation above could be expressed in matrix form

= +Y HX W

44 Synchronization2010722

Signal model (34)Signal model (34)g ( )g ( )

H= = = = +Y Fy FCx FCF X HX W

loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal

y

H=H FCFcirculant matrix leading to be a diagonal matrix

=H FCF

( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0

h h L L hh h

⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥

( )( ) ( )

( ) ( )

1 10 1 0 0i

h Lh L h N L

⎢ ⎥⎢ ⎥minus minus⎢ ⎥

equiv= minus minus⎢ ⎥⎢ ⎥

C( ) ( )

( ) ( )

0 1 1 100

0 0 1 1 0

h L h N L

h N L L h N

⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

45

( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦

Synchronization2010722

Signal model (44)Signal model (44)g ( )g ( )

loz In a time-invariant channel the channel matrix H would be a diagonal matrix

k k k kY H X W= +

( )1 1

2

0 0

N Lj kl N

kn l

H h n l e πminus minus

minus

= =

= sumsum

loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI

k k k k kY H X I Wα= + +k k k k k

( ) ( )1 1 1

2 2 N N L

j m k Nj ml Nk mI X h n l e e ππ

minus minus minusminusminus⎛ ⎞

= ⎜ ⎟⎝ ⎠

sum sum sumICI

46

0 0 0m n lm k= = =ne

⎝ ⎠

Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

47 Synchronization2010722

Impact of timing offset Impact of timing offset p gp g

loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively

( )I f s

Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ

( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr

2 fj k Nk k k kY X H e Wπδminus= +

loz For integer part of timing offset k k k kY X H e W+

I sTδ ( ) 0g IN L δminus minus le leFFT

2 Ij k NY X H e Wπδminus= +

( ) ( ) 2 I sFFT

j f TI sh t T H f e π δδ minusminus rarr

48

k k k kY X H e W= +

Synchronization2010722

Impact of timing offset derivationImpact of timing offset derivationp gp g

49 Synchronization2010722

Discussion of symbol boundaryDiscussion of symbol boundaryy yy y

50 Synchronization2010722

Impact of timing offsetImpact of timing offsetp gp g

loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI

2 Ij k NIk k k k

NY X H e ISI W

Nπδδ minusminus

= + +

loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk

e e

p kloz Phase Shift can be resolved by Differential Encoding

Decoding or carrier recovery with pilots

51 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

52 Synchronization2010722

Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )

53 Synchronization2010722

Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )

( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s

i n t i N N T N T nTy y t e

= + + +=

( ) ( )( ) ( ) ( ) 12

sin

sin

g gI f f

I I

i N N N Nj jf N Ni k i k k

f

Y X H e eN

π ε ε π ε

ε ε

πεπε

+ + minus+

minus minus=⎛ ⎞⎜ ⎟

Faded signal attenuated and rotated by CFO

( )( ) ( ) ( ) ( )11 2

sin

sin g gI f I f

i N N N NN j j l kI f

NN

l k π ε ε π ε επ ε ε + + minusminus + + + minus

⎜ ⎟⎝ ⎠

+ + minussum

( )( )( )

( ) ( )2

0

sin

I f I f

I

j j l kI f N Ni l l

l l k I f

X H e el k

NN

π ε ε π ε ε

ε π ε ε

+ + +

= ne minus

+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠

sum

i k

N

W

⎜ ⎟⎝ ⎠

+ Inter-Carrier Interference (ICI)

54 Synchronization2010722

Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )

loz An easier representation of received signal with only p g yfractional CFO is

1N

( ) ( )1

0

0N

k k k l l kl l k

Y H X C H X C l k Wminus

= ne

= + minus +⎡ ⎤⎣ ⎦sum

where the ICI coefficient is given by

( )( )( ) ( )

sin 1exp 1fl k

C l k j l kπ ε

π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )

( )exp 1sin

f

f

C l k j l kNN l k

N

π επ ε

minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠

55 Synchronization2010722

Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )

loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality

loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift

56 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

57 Synchronization2010722

TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )

loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq

58 Synchronization2010722

TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )

loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below

loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and

rN2 can be expressed by

0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+

( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +

59

( )

Synchronization2010722

TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )

loz Without CFO the first half is identical to the second half (in time order)

loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below

( ) 12 in discretesft T f NT

NTφ π π π ε πε

⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟

⎝ ⎠

helliphellip

sNT⎝ ⎠

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)

60

Length=N2

Synchronization2010722

TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )

loz Received signal without CFOg0

0 0

jr a e θ=jθ 2 0

2 2 0

Nj j

N Nr a e a e

θ θ= =

loz Received signal with CFO0 0

0 0

j jr a e eθ φ= Extra phase rotation due to CFO0 0

2 2 20

2 2 0

N N Nj j jj

N Nr a e e a e e

θ φ φθ= =

due to CFO

Phase difference 2 2 0N N

( )2 02 20 2 0 0

Nj j T fN

r r a e a eφ φ πminus Δ= =

which contains the information about CFO

61

0 2 0 0Nr r a e a e

Synchronization2010722

TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )

loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be

loz Note that d is a time index corresponding to the first( ) ( ) 2 1

20

Nd m d m Nm

P d r rminus

+ + +== sum

loz Note that d is a time index corresponding to the first sample in a window of N samples

loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by

( )22 1

20

Nd m NR d rminus

+ += sumloz A timing metric can be defined as

))(()(

)( 2

2

dRdP

dM =

( ) 20 d m Nm + +=sum

62

))(( dR

Synchronization2010722

TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )

63 Synchronization2010722

TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )

loz If drsquo is the correct symbol timing offset

( )2

2 2

0 1 12

j T f j T f j T fN

P d a e a e a eπ π πΔ Δ Δ

minus= + + +

22

2 2

0 1 12

j T fN

e a a aπ Δ

minus

⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭

loz If drsquo falls behind the correct symbol timing offset 1 sample

2⎪ ⎪⎩ ⎭

64 Synchronization2010722

TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )

65 Synchronization2010722

TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )

loz Drawback Plateau effect

Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)

((N2) ((N2) ((N2)

helliphellip

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)

Sliding window (length = N)

66 Synchronization2010722

TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )

loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula

( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus

扣掉相加

( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +

loz The method also called delay correlator

67 Synchronization2010722

IEEE 80211aIEEE 80211a

68 Synchronization2010722

Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )

loz Minnrsquos Method [2][ ]B B －B －B

( )( )

( )

22

2Minn

P dM d =

1 4 1N minus

( )( )( )2

2

MinnR d

( ) ( ) ( ) ( )

1 4 1

2 2 2 40 0

N

d N m k d N m k Nm k

P d r r+ + + + += =

= sum sum

( ) ( ) ( )2

21 4 1

2 40 0

N

d N m k Nk

R d rminus

+ + += sum sum69

0 0m k= =

Synchronization2010722

Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )

loz Parkrsquos Method [3][ ]

where D is the symmetric version of CC D C D

where D is the symmetric version of C

( )( ) 2

3k

P dM d =( )

( )( )23

ParkM dR d

2N

( ) 2

30

N

d k d kk

P d r rminus +=

= sdotsum

( )3

22

0

N

d kk

R d r += sum70

0k=

Synchronization2010722

Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )

loz Renrsquos Method [4][ ]

AS AS

where S is the PN sequence weighted of the original bl 2preamble

( )( )( )( )

24

Re 24

n

P dM d

R d=

( )( )4R d

( ) 2 1

4 2 2

N

k k N d k d k NP d s s r rminus

+ + + += sum0k=sum

( )21

41 N

d kR d rminus

= sum71

( )402 d k

kR d r +

=sum

Synchronization2010722

Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )

72 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

73 Synchronization2010722

Fractional CFO estimation Fractional CFO estimation

loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected

11 Rminus⎛ ⎞sum

h L i th di t b t t id ti l bl k R i

0

12 d m d m L

ms

f r rLTπ + + +

=

⎛ ⎞= ang⎜ ⎟⎝ ⎠sum

where L is the distance between two identical block R is the block size

loz For exampleNN cp

L = N R = NL = N R = Ncp74 Synchronization2010722

Estimation rangeEstimation rangegg

loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval

f ( ) ( )s s⎣ ⎦

loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)

loz DVB-T (L=N) 802 11 (L N4)

[ ]05 05s sf fminus

[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)

(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus

loz Note that is the subcarrier spacing1sf NT=

75

sNT

Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz

plusmng q y

loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn

76 Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz

plusmng q y

loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for

subcarrier spacing is required

plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn

p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs

77 Synchronization2010722

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT

( )tf02cos π

( )0a

by using IDFTby using IDFT

( )nd ( )tDI( )tf02sin π

( )0a

( )0b

bull

tf s Δ=

1 ( ) ( ) njbna +

( )nd

tNΔffnffn Δ

=Δ+=1 0

( )tDChannelInput

( )tf N 12cos minusπ

( )1minusNa

bullbull

tNΔ

( )tf N 12sin minusπ

( )1minusNb

( )0d

( )nd ( )tDChannelInput

bull bull

( )( )1d

( )2d

tf s Δ=

1 ( ) ( ) njbna +

fNΔffnffn Δ

=Δ+=1 0

ChannelInput

bullbullbullbull

bullbullbullbull

( )1minusNd

21

Frequency Error Results in ICIFrequency Error Results in ICIq yq y

22

Synchronization Error Results in ICISynchronization Error Results in ICIyy

Not Orthogonal Any More

23

Cyclic PrefixCyclic Prefixyy

loz In multipath channel delayed replicas of previous OFDM signal lead to ISI between successive OFDM signals

loz Solution Insert a guard interval between successive OFDMloz Solution Insert a guard interval between successive OFDM signals

24

Cyclic PrefixCyclic Prefix

loz Guard interval leads to intercarrier interference (ICI) in OFDM d d l i

yy

demodulation

loz In DFT interval difference between two subcarriers does not maintain integer number of cycles loss of orthogonality

loz Delayed version of subcarrier 2 causes ICI in the process of d d l ti b i 1demodulating subcarrier 1

25

Cyclic PrefixCyclic Prefixloz Cyclic prefix (CP) A copy of the last part of OFDM signal is

attached to the front of itself

yy

attached to the front of itself

[ ]0d

[ ]1d

[ ]0D[ ]1D

[ ]nd[ ]kD~

symbols dataInput

bullbullbullbull

[ ]1d

[ ]2d

[ ][ ]2D

bullbullbullbullbull

y

bullbullbullbullbull

[ ]1minusNd [ ]1minusND

[ ]gNND minus

bullbullbull

26

Cyclic PrefixCyclic Prefix

loz All delayed replicas of subcarriers always have an

yy

y p yinteger number of cycles within DFT interval no ICI

27

Cyclic PrefixCyclic Prefix

loz Linear convolution vs circular convolution

yy

N

28

Cyclic PrefixCyclic Prefix

loz Channel effect with cyclic prefix

yy

Signal after removed CP

29

Cyclic PrefixCyclic Prefix

loz Time-Domain Explanation

yy

p

30

Cyclic PrefixCyclic Prefixyy

31

Cyclic PrefixCyclic Prefix

loz Spectrum of channel response with length (smaller [ ]nh hL

yy

than )h

gN[ ] nhFFTHk =

loz Received complete OFDM signal[ ] [ ] [ ] 20 ~~ minus++leleotimes= hg LNNnnhnDnr

loz Received useful part [ ]nr[ ] [ ] [ ]nhnDnr Notimes=

where is N-point circular convolution (due to CP)loz Received symbol at k-th subcarrier

[ ] [ ] [ ]N

Notimes

[ ] [ ] [ ] kkNk HXnhnDFFTnrFFTY =otimes== kYX =rArr ldquoUseful property for OFDM system to reduce

kk H

X =rArrcomplexity of channel equalizationrdquo

32

Cyclic Cyclic Prefix (OFDM Prefix (OFDM Receiver)Receiver)yy (( ))

bull bull( )( )nr~Symbols

dataOutput

bullbullbull

bullbullbull( )nr

bullbull

bullbull

33

Cyclic PrefixCyclic Prefix

loz One of the most important reasons to do OFDM is the

yy

pefficient way it deals with multipath delay spread

loz To eliminate inter-symbol interference (ISI) almostloz To eliminate inter symbol interference (ISI) almost completely a guard time is introduced for each OFDM symbolsymbol(The guard time is chosen larger than the delay spread)

34

Bandwidth EfficiencyBandwidth Efficiencyyy

loz In a classical parallel system the channel is divided into p y N non-overlapping sub-channels to avoid inter-carrier interference (ICI)( )

loz The diagram for bandwidth efficiency of OFDM system is shown below

35

SummarySummaryyy

loz The advantage of the FFT-based OFDM system g yloz The use of IFFTFFT can reduce the computation complexityloz The orthogonality between the adjacent sub-carriers will make g y j

the use of transmission bandwidth more efficientloz The guard interval is used to resist the inter-symbol interference

(ISI)loz The main advantage of the OFDM transmission technique is its

high performance even in frequency selective channels

loz The drawbacks of the OFDM system loz It is highly vulnerable to synchronization errorsloz Peak to Average Power Ratio (PAPR) problems

36

SynchronizationSynchronization

Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offset (TO)loz Impact of timing offset (TO)loz Impact of carrier frequency offset (CFO)loz TO estimation algorithmloz CFO estimation algorithmg

38 Synchronization2010722

Introduction (13)Introduction (13)( )( )

loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods

loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead

loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the

cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of

computational complexity therefore it may be not be available in short-burst wireless communication

39 Synchronization2010722

Introduction (23)Introduction (23)( )( )

loz Transcevier

40 Synchronization2010722

Introduction (33)Introduction (33)( )( )

loz Synchronization erroryloz Symbol timing offset (TO)

loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)

loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect

41 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg

42 Synchronization2010722

Signal model (14)Signal model (14)g ( )g ( )

loz Time domain sample p1

2

0

1 Nj nk N

n kk

x X eN

πminus

minus= sum

Xk the frequency domain data at the k-th subcarrier0kN =

loz The multipath fading channel

( ) ( ) ( )1L

h t h t δminus

sum

loz Received time domain signal

( ) ( ) ( )0

l ll

h t h tτ τ δ τ τ=

= minussum

loz Received time-domain signal

( ) ( )1

L

n n ly h n l x w nminus

minus= +sum43

( ) ( )0l=sum

Synchronization2010722

Signal model (24)Signal model (24)g ( )g ( )

loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting

12

0

Nj kn N

k nn

Y y e πminus

minus

=

= sum

( )1 1

2

0

N Lj kn N

n l kn l

h n l x e Wπminus minus

minusminus

=

= +sumsum

loz The equation above could be expressed in matrix form

0n l

loz The equation above could be expressed in matrix form

= +Y HX W

44 Synchronization2010722

Signal model (34)Signal model (34)g ( )g ( )

H= = = = +Y Fy FCx FCF X HX W

loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal

y

H=H FCFcirculant matrix leading to be a diagonal matrix

=H FCF

( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0

h h L L hh h

⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥

( )( ) ( )

( ) ( )

1 10 1 0 0i

h Lh L h N L

⎢ ⎥⎢ ⎥minus minus⎢ ⎥

equiv= minus minus⎢ ⎥⎢ ⎥

C( ) ( )

( ) ( )

0 1 1 100

0 0 1 1 0

h L h N L

h N L L h N

⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

45

( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦

Synchronization2010722

Signal model (44)Signal model (44)g ( )g ( )

loz In a time-invariant channel the channel matrix H would be a diagonal matrix

k k k kY H X W= +

( )1 1

2

0 0

N Lj kl N

kn l

H h n l e πminus minus

minus

= =

= sumsum

loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI

k k k k kY H X I Wα= + +k k k k k

( ) ( )1 1 1

2 2 N N L

j m k Nj ml Nk mI X h n l e e ππ

minus minus minusminusminus⎛ ⎞

= ⎜ ⎟⎝ ⎠

sum sum sumICI

46

0 0 0m n lm k= = =ne

⎝ ⎠

Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

47 Synchronization2010722

Impact of timing offset Impact of timing offset p gp g

loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively

( )I f s

Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ

( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr

2 fj k Nk k k kY X H e Wπδminus= +

loz For integer part of timing offset k k k kY X H e W+

I sTδ ( ) 0g IN L δminus minus le leFFT

2 Ij k NY X H e Wπδminus= +

( ) ( ) 2 I sFFT

j f TI sh t T H f e π δδ minusminus rarr

48

k k k kY X H e W= +

Synchronization2010722

Impact of timing offset derivationImpact of timing offset derivationp gp g

49 Synchronization2010722

Discussion of symbol boundaryDiscussion of symbol boundaryy yy y

50 Synchronization2010722

Impact of timing offsetImpact of timing offsetp gp g

loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI

2 Ij k NIk k k k

NY X H e ISI W

Nπδδ minusminus

= + +

loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk

e e

p kloz Phase Shift can be resolved by Differential Encoding

Decoding or carrier recovery with pilots

51 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

52 Synchronization2010722

Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )

53 Synchronization2010722

Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )

( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s

i n t i N N T N T nTy y t e

= + + +=

( ) ( )( ) ( ) ( ) 12

sin

sin

g gI f f

I I

i N N N Nj jf N Ni k i k k

f

Y X H e eN

π ε ε π ε

ε ε

πεπε

+ + minus+

minus minus=⎛ ⎞⎜ ⎟

Faded signal attenuated and rotated by CFO

( )( ) ( ) ( ) ( )11 2

sin

sin g gI f I f

i N N N NN j j l kI f

NN

l k π ε ε π ε επ ε ε + + minusminus + + + minus

⎜ ⎟⎝ ⎠

+ + minussum

( )( )( )

( ) ( )2

0

sin

I f I f

I

j j l kI f N Ni l l

l l k I f

X H e el k

NN

π ε ε π ε ε

ε π ε ε

+ + +

= ne minus

+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠

sum

i k

N

W

⎜ ⎟⎝ ⎠

+ Inter-Carrier Interference (ICI)

54 Synchronization2010722

Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )

loz An easier representation of received signal with only p g yfractional CFO is

1N

( ) ( )1

0

0N

k k k l l kl l k

Y H X C H X C l k Wminus

= ne

= + minus +⎡ ⎤⎣ ⎦sum

where the ICI coefficient is given by

( )( )( ) ( )

sin 1exp 1fl k

C l k j l kπ ε

π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )

( )exp 1sin

f

f

C l k j l kNN l k

N

π επ ε

minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠

55 Synchronization2010722

Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )

loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality

loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift

56 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

57 Synchronization2010722

TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )

loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq

58 Synchronization2010722

TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )

loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below

loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and

rN2 can be expressed by

0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+

( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +

59

( )

Synchronization2010722

TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )

loz Without CFO the first half is identical to the second half (in time order)

loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below

( ) 12 in discretesft T f NT

NTφ π π π ε πε

⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟

⎝ ⎠

helliphellip

sNT⎝ ⎠

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)

60

Length=N2

Synchronization2010722

TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )

loz Received signal without CFOg0

0 0

jr a e θ=jθ 2 0

2 2 0

Nj j

N Nr a e a e

θ θ= =

loz Received signal with CFO0 0

0 0

j jr a e eθ φ= Extra phase rotation due to CFO0 0

2 2 20

2 2 0

N N Nj j jj

N Nr a e e a e e

θ φ φθ= =

due to CFO

Phase difference 2 2 0N N

( )2 02 20 2 0 0

Nj j T fN

r r a e a eφ φ πminus Δ= =

which contains the information about CFO

61

0 2 0 0Nr r a e a e

Synchronization2010722

TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )

loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be

loz Note that d is a time index corresponding to the first( ) ( ) 2 1

20

Nd m d m Nm

P d r rminus

+ + +== sum

loz Note that d is a time index corresponding to the first sample in a window of N samples

loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by

( )22 1

20

Nd m NR d rminus

+ += sumloz A timing metric can be defined as

))(()(

)( 2

2

dRdP

dM =

( ) 20 d m Nm + +=sum

62

))(( dR

Synchronization2010722

TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )

63 Synchronization2010722

TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )

loz If drsquo is the correct symbol timing offset

( )2

2 2

0 1 12

j T f j T f j T fN

P d a e a e a eπ π πΔ Δ Δ

minus= + + +

22

2 2

0 1 12

j T fN

e a a aπ Δ

minus

⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭

loz If drsquo falls behind the correct symbol timing offset 1 sample

2⎪ ⎪⎩ ⎭

64 Synchronization2010722

TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )

65 Synchronization2010722

TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )

loz Drawback Plateau effect

Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)

((N2) ((N2) ((N2)

helliphellip

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)

Sliding window (length = N)

66 Synchronization2010722

TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )

loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula

( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus

扣掉相加

( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +

loz The method also called delay correlator

67 Synchronization2010722

IEEE 80211aIEEE 80211a

68 Synchronization2010722

Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )

loz Minnrsquos Method [2][ ]B B －B －B

( )( )

( )

22

2Minn

P dM d =

1 4 1N minus

( )( )( )2

2

MinnR d

( ) ( ) ( ) ( )

1 4 1

2 2 2 40 0

N

d N m k d N m k Nm k

P d r r+ + + + += =

= sum sum

( ) ( ) ( )2

21 4 1

2 40 0

N

d N m k Nk

R d rminus

+ + += sum sum69

0 0m k= =

Synchronization2010722

Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )

loz Parkrsquos Method [3][ ]

where D is the symmetric version of CC D C D

where D is the symmetric version of C

( )( ) 2

3k

P dM d =( )

( )( )23

ParkM dR d

2N

( ) 2

30

N

d k d kk

P d r rminus +=

= sdotsum

( )3

22

0

N

d kk

R d r += sum70

0k=

Synchronization2010722

Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )

loz Renrsquos Method [4][ ]

AS AS

where S is the PN sequence weighted of the original bl 2preamble

( )( )( )( )

24

Re 24

n

P dM d

R d=

( )( )4R d

( ) 2 1

4 2 2

N

k k N d k d k NP d s s r rminus

+ + + += sum0k=sum

( )21

41 N

d kR d rminus

= sum71

( )402 d k

kR d r +

=sum

Synchronization2010722

Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )

72 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

73 Synchronization2010722

Fractional CFO estimation Fractional CFO estimation

loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected

11 Rminus⎛ ⎞sum

h L i th di t b t t id ti l bl k R i

0

12 d m d m L

ms

f r rLTπ + + +

=

⎛ ⎞= ang⎜ ⎟⎝ ⎠sum

where L is the distance between two identical block R is the block size

loz For exampleNN cp

L = N R = NL = N R = Ncp74 Synchronization2010722

Estimation rangeEstimation rangegg

loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval

f ( ) ( )s s⎣ ⎦

loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)

loz DVB-T (L=N) 802 11 (L N4)

[ ]05 05s sf fminus

[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)

(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus

loz Note that is the subcarrier spacing1sf NT=

75

sNT

Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz

plusmng q y

loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn

76 Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz

plusmng q y

loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for

subcarrier spacing is required

plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn

p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs

77 Synchronization2010722

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

Frequency Error Results in ICIFrequency Error Results in ICIq yq y

22

Synchronization Error Results in ICISynchronization Error Results in ICIyy

Not Orthogonal Any More

23

Cyclic PrefixCyclic Prefixyy

loz In multipath channel delayed replicas of previous OFDM signal lead to ISI between successive OFDM signals

loz Solution Insert a guard interval between successive OFDMloz Solution Insert a guard interval between successive OFDM signals

24

Cyclic PrefixCyclic Prefix

loz Guard interval leads to intercarrier interference (ICI) in OFDM d d l i

yy

demodulation

loz In DFT interval difference between two subcarriers does not maintain integer number of cycles loss of orthogonality

loz Delayed version of subcarrier 2 causes ICI in the process of d d l ti b i 1demodulating subcarrier 1

25

Cyclic PrefixCyclic Prefixloz Cyclic prefix (CP) A copy of the last part of OFDM signal is

attached to the front of itself

yy

attached to the front of itself

[ ]0d

[ ]1d

[ ]0D[ ]1D

[ ]nd[ ]kD~

symbols dataInput

bullbullbullbull

[ ]1d

[ ]2d

[ ][ ]2D

bullbullbullbullbull

y

bullbullbullbullbull

[ ]1minusNd [ ]1minusND

[ ]gNND minus

bullbullbull

26

Cyclic PrefixCyclic Prefix

loz All delayed replicas of subcarriers always have an

yy

y p yinteger number of cycles within DFT interval no ICI

27

Cyclic PrefixCyclic Prefix

loz Linear convolution vs circular convolution

yy

N

28

Cyclic PrefixCyclic Prefix

loz Channel effect with cyclic prefix

yy

Signal after removed CP

29

Cyclic PrefixCyclic Prefix

loz Time-Domain Explanation

yy

p

30

Cyclic PrefixCyclic Prefixyy

31

Cyclic PrefixCyclic Prefix

loz Spectrum of channel response with length (smaller [ ]nh hL

yy

than )h

gN[ ] nhFFTHk =

loz Received complete OFDM signal[ ] [ ] [ ] 20 ~~ minus++leleotimes= hg LNNnnhnDnr

loz Received useful part [ ]nr[ ] [ ] [ ]nhnDnr Notimes=

where is N-point circular convolution (due to CP)loz Received symbol at k-th subcarrier

[ ] [ ] [ ]N

Notimes

[ ] [ ] [ ] kkNk HXnhnDFFTnrFFTY =otimes== kYX =rArr ldquoUseful property for OFDM system to reduce

kk H

X =rArrcomplexity of channel equalizationrdquo

32

Cyclic Cyclic Prefix (OFDM Prefix (OFDM Receiver)Receiver)yy (( ))

bull bull( )( )nr~Symbols

dataOutput

bullbullbull

bullbullbull( )nr

bullbull

bullbull

33

Cyclic PrefixCyclic Prefix

loz One of the most important reasons to do OFDM is the

yy

pefficient way it deals with multipath delay spread

loz To eliminate inter-symbol interference (ISI) almostloz To eliminate inter symbol interference (ISI) almost completely a guard time is introduced for each OFDM symbolsymbol(The guard time is chosen larger than the delay spread)

34

Bandwidth EfficiencyBandwidth Efficiencyyy

loz In a classical parallel system the channel is divided into p y N non-overlapping sub-channels to avoid inter-carrier interference (ICI)( )

loz The diagram for bandwidth efficiency of OFDM system is shown below

35

SummarySummaryyy

loz The advantage of the FFT-based OFDM system g yloz The use of IFFTFFT can reduce the computation complexityloz The orthogonality between the adjacent sub-carriers will make g y j

the use of transmission bandwidth more efficientloz The guard interval is used to resist the inter-symbol interference

(ISI)loz The main advantage of the OFDM transmission technique is its

high performance even in frequency selective channels

loz The drawbacks of the OFDM system loz It is highly vulnerable to synchronization errorsloz Peak to Average Power Ratio (PAPR) problems

36

SynchronizationSynchronization

Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offset (TO)loz Impact of timing offset (TO)loz Impact of carrier frequency offset (CFO)loz TO estimation algorithmloz CFO estimation algorithmg

38 Synchronization2010722

Introduction (13)Introduction (13)( )( )

loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods

loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead

loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the

cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of

computational complexity therefore it may be not be available in short-burst wireless communication

39 Synchronization2010722

Introduction (23)Introduction (23)( )( )

loz Transcevier

40 Synchronization2010722

Introduction (33)Introduction (33)( )( )

loz Synchronization erroryloz Symbol timing offset (TO)

loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)

loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect

41 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg

42 Synchronization2010722

Signal model (14)Signal model (14)g ( )g ( )

loz Time domain sample p1

2

0

1 Nj nk N

n kk

x X eN

πminus

minus= sum

Xk the frequency domain data at the k-th subcarrier0kN =

loz The multipath fading channel

( ) ( ) ( )1L

h t h t δminus

sum

loz Received time domain signal

( ) ( ) ( )0

l ll

h t h tτ τ δ τ τ=

= minussum

loz Received time-domain signal

( ) ( )1

L

n n ly h n l x w nminus

minus= +sum43

( ) ( )0l=sum

Synchronization2010722

Signal model (24)Signal model (24)g ( )g ( )

loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting

12

0

Nj kn N

k nn

Y y e πminus

minus

=

= sum

( )1 1

2

0

N Lj kn N

n l kn l

h n l x e Wπminus minus

minusminus

=

= +sumsum

loz The equation above could be expressed in matrix form

0n l

loz The equation above could be expressed in matrix form

= +Y HX W

44 Synchronization2010722

Signal model (34)Signal model (34)g ( )g ( )

H= = = = +Y Fy FCx FCF X HX W

loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal

y

H=H FCFcirculant matrix leading to be a diagonal matrix

=H FCF

( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0

h h L L hh h

⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥

( )( ) ( )

( ) ( )

1 10 1 0 0i

h Lh L h N L

⎢ ⎥⎢ ⎥minus minus⎢ ⎥

equiv= minus minus⎢ ⎥⎢ ⎥

C( ) ( )

( ) ( )

0 1 1 100

0 0 1 1 0

h L h N L

h N L L h N

⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

45

( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦

Synchronization2010722

Signal model (44)Signal model (44)g ( )g ( )

loz In a time-invariant channel the channel matrix H would be a diagonal matrix

k k k kY H X W= +

( )1 1

2

0 0

N Lj kl N

kn l

H h n l e πminus minus

minus

= =

= sumsum

loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI

k k k k kY H X I Wα= + +k k k k k

( ) ( )1 1 1

2 2 N N L

j m k Nj ml Nk mI X h n l e e ππ

minus minus minusminusminus⎛ ⎞

= ⎜ ⎟⎝ ⎠

sum sum sumICI

46

0 0 0m n lm k= = =ne

⎝ ⎠

Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

47 Synchronization2010722

Impact of timing offset Impact of timing offset p gp g

loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively

( )I f s

Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ

( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr

2 fj k Nk k k kY X H e Wπδminus= +

loz For integer part of timing offset k k k kY X H e W+

I sTδ ( ) 0g IN L δminus minus le leFFT

2 Ij k NY X H e Wπδminus= +

( ) ( ) 2 I sFFT

j f TI sh t T H f e π δδ minusminus rarr

48

k k k kY X H e W= +

Synchronization2010722

Impact of timing offset derivationImpact of timing offset derivationp gp g

49 Synchronization2010722

Discussion of symbol boundaryDiscussion of symbol boundaryy yy y

50 Synchronization2010722

Impact of timing offsetImpact of timing offsetp gp g

loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI

2 Ij k NIk k k k

NY X H e ISI W

Nπδδ minusminus

= + +

loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk

e e

p kloz Phase Shift can be resolved by Differential Encoding

Decoding or carrier recovery with pilots

51 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

52 Synchronization2010722

Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )

53 Synchronization2010722

Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )

( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s

i n t i N N T N T nTy y t e

= + + +=

( ) ( )( ) ( ) ( ) 12

sin

sin

g gI f f

I I

i N N N Nj jf N Ni k i k k

f

Y X H e eN

π ε ε π ε

ε ε

πεπε

+ + minus+

minus minus=⎛ ⎞⎜ ⎟

Faded signal attenuated and rotated by CFO

( )( ) ( ) ( ) ( )11 2

sin

sin g gI f I f

i N N N NN j j l kI f

NN

l k π ε ε π ε επ ε ε + + minusminus + + + minus

⎜ ⎟⎝ ⎠

+ + minussum

( )( )( )

( ) ( )2

0

sin

I f I f

I

j j l kI f N Ni l l

l l k I f

X H e el k

NN

π ε ε π ε ε

ε π ε ε

+ + +

= ne minus

+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠

sum

i k

N

W

⎜ ⎟⎝ ⎠

+ Inter-Carrier Interference (ICI)

54 Synchronization2010722

Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )

loz An easier representation of received signal with only p g yfractional CFO is

1N

( ) ( )1

0

0N

k k k l l kl l k

Y H X C H X C l k Wminus

= ne

= + minus +⎡ ⎤⎣ ⎦sum

where the ICI coefficient is given by

( )( )( ) ( )

sin 1exp 1fl k

C l k j l kπ ε

π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )

( )exp 1sin

f

f

C l k j l kNN l k

N

π επ ε

minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠

55 Synchronization2010722

Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )

loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality

loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift

56 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

57 Synchronization2010722

TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )

loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq

58 Synchronization2010722

TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )

loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below

loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and

rN2 can be expressed by

0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+

( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +

59

( )

Synchronization2010722

TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )

loz Without CFO the first half is identical to the second half (in time order)

loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below

( ) 12 in discretesft T f NT

NTφ π π π ε πε

⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟

⎝ ⎠

helliphellip

sNT⎝ ⎠

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)

60

Length=N2

Synchronization2010722

TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )

loz Received signal without CFOg0

0 0

jr a e θ=jθ 2 0

2 2 0

Nj j

N Nr a e a e

θ θ= =

loz Received signal with CFO0 0

0 0

j jr a e eθ φ= Extra phase rotation due to CFO0 0

2 2 20

2 2 0

N N Nj j jj

N Nr a e e a e e

θ φ φθ= =

due to CFO

Phase difference 2 2 0N N

( )2 02 20 2 0 0

Nj j T fN

r r a e a eφ φ πminus Δ= =

which contains the information about CFO

61

0 2 0 0Nr r a e a e

Synchronization2010722

TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )

loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be

loz Note that d is a time index corresponding to the first( ) ( ) 2 1

20

Nd m d m Nm

P d r rminus

+ + +== sum

loz Note that d is a time index corresponding to the first sample in a window of N samples

loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by

( )22 1

20

Nd m NR d rminus

+ += sumloz A timing metric can be defined as

))(()(

)( 2

2

dRdP

dM =

( ) 20 d m Nm + +=sum

62

))(( dR

Synchronization2010722

TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )

63 Synchronization2010722

TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )

loz If drsquo is the correct symbol timing offset

( )2

2 2

0 1 12

j T f j T f j T fN

P d a e a e a eπ π πΔ Δ Δ

minus= + + +

22

2 2

0 1 12

j T fN

e a a aπ Δ

minus

⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭

loz If drsquo falls behind the correct symbol timing offset 1 sample

2⎪ ⎪⎩ ⎭

64 Synchronization2010722

TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )

65 Synchronization2010722

TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )

loz Drawback Plateau effect

Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)

((N2) ((N2) ((N2)

helliphellip

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)

Sliding window (length = N)

66 Synchronization2010722

TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )

loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula

( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus

扣掉相加

( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +

loz The method also called delay correlator

67 Synchronization2010722

IEEE 80211aIEEE 80211a

68 Synchronization2010722

Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )

loz Minnrsquos Method [2][ ]B B －B －B

( )( )

( )

22

2Minn

P dM d =

1 4 1N minus

( )( )( )2

2

MinnR d

( ) ( ) ( ) ( )

1 4 1

2 2 2 40 0

N

d N m k d N m k Nm k

P d r r+ + + + += =

= sum sum

( ) ( ) ( )2

21 4 1

2 40 0

N

d N m k Nk

R d rminus

+ + += sum sum69

0 0m k= =

Synchronization2010722

Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )

loz Parkrsquos Method [3][ ]

where D is the symmetric version of CC D C D

where D is the symmetric version of C

( )( ) 2

3k

P dM d =( )

( )( )23

ParkM dR d

2N

( ) 2

30

N

d k d kk

P d r rminus +=

= sdotsum

( )3

22

0

N

d kk

R d r += sum70

0k=

Synchronization2010722

Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )

loz Renrsquos Method [4][ ]

AS AS

where S is the PN sequence weighted of the original bl 2preamble

( )( )( )( )

24

Re 24

n

P dM d

R d=

( )( )4R d

( ) 2 1

4 2 2

N

k k N d k d k NP d s s r rminus

+ + + += sum0k=sum

( )21

41 N

d kR d rminus

= sum71

( )402 d k

kR d r +

=sum

Synchronization2010722

Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )

72 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

73 Synchronization2010722

Fractional CFO estimation Fractional CFO estimation

loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected

11 Rminus⎛ ⎞sum

h L i th di t b t t id ti l bl k R i

0

12 d m d m L

ms

f r rLTπ + + +

=

⎛ ⎞= ang⎜ ⎟⎝ ⎠sum

where L is the distance between two identical block R is the block size

loz For exampleNN cp

L = N R = NL = N R = Ncp74 Synchronization2010722

Estimation rangeEstimation rangegg

loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval

f ( ) ( )s s⎣ ⎦

loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)

loz DVB-T (L=N) 802 11 (L N4)

[ ]05 05s sf fminus

[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)

(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus

loz Note that is the subcarrier spacing1sf NT=

75

sNT

Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz

plusmng q y

loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn

76 Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz

plusmng q y

loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for

subcarrier spacing is required

plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn

p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs

77 Synchronization2010722

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

Synchronization Error Results in ICISynchronization Error Results in ICIyy

Not Orthogonal Any More

23

Cyclic PrefixCyclic Prefixyy

loz In multipath channel delayed replicas of previous OFDM signal lead to ISI between successive OFDM signals

loz Solution Insert a guard interval between successive OFDMloz Solution Insert a guard interval between successive OFDM signals

24

Cyclic PrefixCyclic Prefix

loz Guard interval leads to intercarrier interference (ICI) in OFDM d d l i

yy

demodulation

loz In DFT interval difference between two subcarriers does not maintain integer number of cycles loss of orthogonality

loz Delayed version of subcarrier 2 causes ICI in the process of d d l ti b i 1demodulating subcarrier 1

25

Cyclic PrefixCyclic Prefixloz Cyclic prefix (CP) A copy of the last part of OFDM signal is

attached to the front of itself

yy

attached to the front of itself

[ ]0d

[ ]1d

[ ]0D[ ]1D

[ ]nd[ ]kD~

symbols dataInput

bullbullbullbull

[ ]1d

[ ]2d

[ ][ ]2D

bullbullbullbullbull

y

bullbullbullbullbull

[ ]1minusNd [ ]1minusND

[ ]gNND minus

bullbullbull

26

Cyclic PrefixCyclic Prefix

loz All delayed replicas of subcarriers always have an

yy

y p yinteger number of cycles within DFT interval no ICI

27

Cyclic PrefixCyclic Prefix

loz Linear convolution vs circular convolution

yy

N

28

Cyclic PrefixCyclic Prefix

loz Channel effect with cyclic prefix

yy

Signal after removed CP

29

Cyclic PrefixCyclic Prefix

loz Time-Domain Explanation

yy

p

30

Cyclic PrefixCyclic Prefixyy

31

Cyclic PrefixCyclic Prefix

loz Spectrum of channel response with length (smaller [ ]nh hL

yy

than )h

gN[ ] nhFFTHk =

loz Received complete OFDM signal[ ] [ ] [ ] 20 ~~ minus++leleotimes= hg LNNnnhnDnr

loz Received useful part [ ]nr[ ] [ ] [ ]nhnDnr Notimes=

where is N-point circular convolution (due to CP)loz Received symbol at k-th subcarrier

[ ] [ ] [ ]N

Notimes

[ ] [ ] [ ] kkNk HXnhnDFFTnrFFTY =otimes== kYX =rArr ldquoUseful property for OFDM system to reduce

kk H

X =rArrcomplexity of channel equalizationrdquo

32

Cyclic Cyclic Prefix (OFDM Prefix (OFDM Receiver)Receiver)yy (( ))

bull bull( )( )nr~Symbols

dataOutput

bullbullbull

bullbullbull( )nr

bullbull

bullbull

33

Cyclic PrefixCyclic Prefix

loz One of the most important reasons to do OFDM is the

yy

pefficient way it deals with multipath delay spread

loz To eliminate inter-symbol interference (ISI) almostloz To eliminate inter symbol interference (ISI) almost completely a guard time is introduced for each OFDM symbolsymbol(The guard time is chosen larger than the delay spread)

34

Bandwidth EfficiencyBandwidth Efficiencyyy

loz In a classical parallel system the channel is divided into p y N non-overlapping sub-channels to avoid inter-carrier interference (ICI)( )

loz The diagram for bandwidth efficiency of OFDM system is shown below

35

SummarySummaryyy

loz The advantage of the FFT-based OFDM system g yloz The use of IFFTFFT can reduce the computation complexityloz The orthogonality between the adjacent sub-carriers will make g y j

the use of transmission bandwidth more efficientloz The guard interval is used to resist the inter-symbol interference

(ISI)loz The main advantage of the OFDM transmission technique is its

high performance even in frequency selective channels

loz The drawbacks of the OFDM system loz It is highly vulnerable to synchronization errorsloz Peak to Average Power Ratio (PAPR) problems

36

SynchronizationSynchronization

Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offset (TO)loz Impact of timing offset (TO)loz Impact of carrier frequency offset (CFO)loz TO estimation algorithmloz CFO estimation algorithmg

38 Synchronization2010722

Introduction (13)Introduction (13)( )( )

loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods

loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead

loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the

cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of

computational complexity therefore it may be not be available in short-burst wireless communication

39 Synchronization2010722

Introduction (23)Introduction (23)( )( )

loz Transcevier

40 Synchronization2010722

Introduction (33)Introduction (33)( )( )

loz Synchronization erroryloz Symbol timing offset (TO)

loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)

loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect

41 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg

42 Synchronization2010722

Signal model (14)Signal model (14)g ( )g ( )

loz Time domain sample p1

2

0

1 Nj nk N

n kk

x X eN

πminus

minus= sum

Xk the frequency domain data at the k-th subcarrier0kN =

loz The multipath fading channel

( ) ( ) ( )1L

h t h t δminus

sum

loz Received time domain signal

( ) ( ) ( )0

l ll

h t h tτ τ δ τ τ=

= minussum

loz Received time-domain signal

( ) ( )1

L

n n ly h n l x w nminus

minus= +sum43

( ) ( )0l=sum

Synchronization2010722

Signal model (24)Signal model (24)g ( )g ( )

loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting

12

0

Nj kn N

k nn

Y y e πminus

minus

=

= sum

( )1 1

2

0

N Lj kn N

n l kn l

h n l x e Wπminus minus

minusminus

=

= +sumsum

loz The equation above could be expressed in matrix form

0n l

loz The equation above could be expressed in matrix form

= +Y HX W

44 Synchronization2010722

Signal model (34)Signal model (34)g ( )g ( )

H= = = = +Y Fy FCx FCF X HX W

loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal

y

H=H FCFcirculant matrix leading to be a diagonal matrix

=H FCF

( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0

h h L L hh h

⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥

( )( ) ( )

( ) ( )

1 10 1 0 0i

h Lh L h N L

⎢ ⎥⎢ ⎥minus minus⎢ ⎥

equiv= minus minus⎢ ⎥⎢ ⎥

C( ) ( )

( ) ( )

0 1 1 100

0 0 1 1 0

h L h N L

h N L L h N

⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

45

( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦

Synchronization2010722

Signal model (44)Signal model (44)g ( )g ( )

loz In a time-invariant channel the channel matrix H would be a diagonal matrix

k k k kY H X W= +

( )1 1

2

0 0

N Lj kl N

kn l

H h n l e πminus minus

minus

= =

= sumsum

loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI

k k k k kY H X I Wα= + +k k k k k

( ) ( )1 1 1

2 2 N N L

j m k Nj ml Nk mI X h n l e e ππ

minus minus minusminusminus⎛ ⎞

= ⎜ ⎟⎝ ⎠

sum sum sumICI

46

0 0 0m n lm k= = =ne

⎝ ⎠

Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

47 Synchronization2010722

Impact of timing offset Impact of timing offset p gp g

loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively

( )I f s

Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ

( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr

2 fj k Nk k k kY X H e Wπδminus= +

loz For integer part of timing offset k k k kY X H e W+

I sTδ ( ) 0g IN L δminus minus le leFFT

2 Ij k NY X H e Wπδminus= +

( ) ( ) 2 I sFFT

j f TI sh t T H f e π δδ minusminus rarr

48

k k k kY X H e W= +

Synchronization2010722

Impact of timing offset derivationImpact of timing offset derivationp gp g

49 Synchronization2010722

Discussion of symbol boundaryDiscussion of symbol boundaryy yy y

50 Synchronization2010722

Impact of timing offsetImpact of timing offsetp gp g

loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI

2 Ij k NIk k k k

NY X H e ISI W

Nπδδ minusminus

= + +

loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk

e e

p kloz Phase Shift can be resolved by Differential Encoding

Decoding or carrier recovery with pilots

51 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

52 Synchronization2010722

Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )

53 Synchronization2010722

Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )

( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s

i n t i N N T N T nTy y t e

= + + +=

( ) ( )( ) ( ) ( ) 12

sin

sin

g gI f f

I I

i N N N Nj jf N Ni k i k k

f

Y X H e eN

π ε ε π ε

ε ε

πεπε

+ + minus+

minus minus=⎛ ⎞⎜ ⎟

Faded signal attenuated and rotated by CFO

( )( ) ( ) ( ) ( )11 2

sin

sin g gI f I f

i N N N NN j j l kI f

NN

l k π ε ε π ε επ ε ε + + minusminus + + + minus

⎜ ⎟⎝ ⎠

+ + minussum

( )( )( )

( ) ( )2

0

sin

I f I f

I

j j l kI f N Ni l l

l l k I f

X H e el k

NN

π ε ε π ε ε

ε π ε ε

+ + +

= ne minus

+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠

sum

i k

N

W

⎜ ⎟⎝ ⎠

+ Inter-Carrier Interference (ICI)

54 Synchronization2010722

Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )

loz An easier representation of received signal with only p g yfractional CFO is

1N

( ) ( )1

0

0N

k k k l l kl l k

Y H X C H X C l k Wminus

= ne

= + minus +⎡ ⎤⎣ ⎦sum

where the ICI coefficient is given by

( )( )( ) ( )

sin 1exp 1fl k

C l k j l kπ ε

π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )

( )exp 1sin

f

f

C l k j l kNN l k

N

π επ ε

minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠

55 Synchronization2010722

Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )

loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality

loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift

56 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

57 Synchronization2010722

TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )

loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq

58 Synchronization2010722

TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )

loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below

loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and

rN2 can be expressed by

0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+

( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +

59

( )

Synchronization2010722

TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )

loz Without CFO the first half is identical to the second half (in time order)

loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below

( ) 12 in discretesft T f NT

NTφ π π π ε πε

⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟

⎝ ⎠

helliphellip

sNT⎝ ⎠

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)

60

Length=N2

Synchronization2010722

TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )

loz Received signal without CFOg0

0 0

jr a e θ=jθ 2 0

2 2 0

Nj j

N Nr a e a e

θ θ= =

loz Received signal with CFO0 0

0 0

j jr a e eθ φ= Extra phase rotation due to CFO0 0

2 2 20

2 2 0

N N Nj j jj

N Nr a e e a e e

θ φ φθ= =

due to CFO

Phase difference 2 2 0N N

( )2 02 20 2 0 0

Nj j T fN

r r a e a eφ φ πminus Δ= =

which contains the information about CFO

61

0 2 0 0Nr r a e a e

Synchronization2010722

TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )

loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be

loz Note that d is a time index corresponding to the first( ) ( ) 2 1

20

Nd m d m Nm

P d r rminus

+ + +== sum

loz Note that d is a time index corresponding to the first sample in a window of N samples

loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by

( )22 1

20

Nd m NR d rminus

+ += sumloz A timing metric can be defined as

))(()(

)( 2

2

dRdP

dM =

( ) 20 d m Nm + +=sum

62

))(( dR

Synchronization2010722

TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )

63 Synchronization2010722

TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )

loz If drsquo is the correct symbol timing offset

( )2

2 2

0 1 12

j T f j T f j T fN

P d a e a e a eπ π πΔ Δ Δ

minus= + + +

22

2 2

0 1 12

j T fN

e a a aπ Δ

minus

⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭

loz If drsquo falls behind the correct symbol timing offset 1 sample

2⎪ ⎪⎩ ⎭

64 Synchronization2010722

TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )

65 Synchronization2010722

TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )

loz Drawback Plateau effect

Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)

((N2) ((N2) ((N2)

helliphellip

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)

Sliding window (length = N)

66 Synchronization2010722

TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )

loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula

( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus

扣掉相加

( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +

loz The method also called delay correlator

67 Synchronization2010722

IEEE 80211aIEEE 80211a

68 Synchronization2010722

Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )

loz Minnrsquos Method [2][ ]B B －B －B

( )( )

( )

22

2Minn

P dM d =

1 4 1N minus

( )( )( )2

2

MinnR d

( ) ( ) ( ) ( )

1 4 1

2 2 2 40 0

N

d N m k d N m k Nm k

P d r r+ + + + += =

= sum sum

( ) ( ) ( )2

21 4 1

2 40 0

N

d N m k Nk

R d rminus

+ + += sum sum69

0 0m k= =

Synchronization2010722

Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )

loz Parkrsquos Method [3][ ]

where D is the symmetric version of CC D C D

where D is the symmetric version of C

( )( ) 2

3k

P dM d =( )

( )( )23

ParkM dR d

2N

( ) 2

30

N

d k d kk

P d r rminus +=

= sdotsum

( )3

22

0

N

d kk

R d r += sum70

0k=

Synchronization2010722

Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )

loz Renrsquos Method [4][ ]

AS AS

where S is the PN sequence weighted of the original bl 2preamble

( )( )( )( )

24

Re 24

n

P dM d

R d=

( )( )4R d

( ) 2 1

4 2 2

N

k k N d k d k NP d s s r rminus

+ + + += sum0k=sum

( )21

41 N

d kR d rminus

= sum71

( )402 d k

kR d r +

=sum

Synchronization2010722

Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )

72 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

73 Synchronization2010722

Fractional CFO estimation Fractional CFO estimation

loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected

11 Rminus⎛ ⎞sum

h L i th di t b t t id ti l bl k R i

0

12 d m d m L

ms

f r rLTπ + + +

=

⎛ ⎞= ang⎜ ⎟⎝ ⎠sum

where L is the distance between two identical block R is the block size

loz For exampleNN cp

L = N R = NL = N R = Ncp74 Synchronization2010722

Estimation rangeEstimation rangegg

loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval

f ( ) ( )s s⎣ ⎦

loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)

loz DVB-T (L=N) 802 11 (L N4)

[ ]05 05s sf fminus

[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)

(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus

loz Note that is the subcarrier spacing1sf NT=

75

sNT

Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz

plusmng q y

loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn

76 Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz

plusmng q y

loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for

subcarrier spacing is required

plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn

p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs

77 Synchronization2010722

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

Cyclic PrefixCyclic Prefixyy

loz In multipath channel delayed replicas of previous OFDM signal lead to ISI between successive OFDM signals

loz Solution Insert a guard interval between successive OFDMloz Solution Insert a guard interval between successive OFDM signals

24

Cyclic PrefixCyclic Prefix

loz Guard interval leads to intercarrier interference (ICI) in OFDM d d l i

yy

demodulation

loz In DFT interval difference between two subcarriers does not maintain integer number of cycles loss of orthogonality

loz Delayed version of subcarrier 2 causes ICI in the process of d d l ti b i 1demodulating subcarrier 1

25

Cyclic PrefixCyclic Prefixloz Cyclic prefix (CP) A copy of the last part of OFDM signal is

attached to the front of itself

yy

attached to the front of itself

[ ]0d

[ ]1d

[ ]0D[ ]1D

[ ]nd[ ]kD~

symbols dataInput

bullbullbullbull

[ ]1d

[ ]2d

[ ][ ]2D

bullbullbullbullbull

y

bullbullbullbullbull

[ ]1minusNd [ ]1minusND

[ ]gNND minus

bullbullbull

26

Cyclic PrefixCyclic Prefix

loz All delayed replicas of subcarriers always have an

yy

y p yinteger number of cycles within DFT interval no ICI

27

Cyclic PrefixCyclic Prefix

loz Linear convolution vs circular convolution

yy

N

28

Cyclic PrefixCyclic Prefix

loz Channel effect with cyclic prefix

yy

Signal after removed CP

29

Cyclic PrefixCyclic Prefix

loz Time-Domain Explanation

yy

p

30

Cyclic PrefixCyclic Prefixyy

31

Cyclic PrefixCyclic Prefix

loz Spectrum of channel response with length (smaller [ ]nh hL

yy

than )h

gN[ ] nhFFTHk =

loz Received complete OFDM signal[ ] [ ] [ ] 20 ~~ minus++leleotimes= hg LNNnnhnDnr

loz Received useful part [ ]nr[ ] [ ] [ ]nhnDnr Notimes=

where is N-point circular convolution (due to CP)loz Received symbol at k-th subcarrier

[ ] [ ] [ ]N

Notimes

[ ] [ ] [ ] kkNk HXnhnDFFTnrFFTY =otimes== kYX =rArr ldquoUseful property for OFDM system to reduce

kk H

X =rArrcomplexity of channel equalizationrdquo

32

Cyclic Cyclic Prefix (OFDM Prefix (OFDM Receiver)Receiver)yy (( ))

bull bull( )( )nr~Symbols

dataOutput

bullbullbull

bullbullbull( )nr

bullbull

bullbull

33

Cyclic PrefixCyclic Prefix

loz One of the most important reasons to do OFDM is the

yy

pefficient way it deals with multipath delay spread

loz To eliminate inter-symbol interference (ISI) almostloz To eliminate inter symbol interference (ISI) almost completely a guard time is introduced for each OFDM symbolsymbol(The guard time is chosen larger than the delay spread)

34

Bandwidth EfficiencyBandwidth Efficiencyyy

loz In a classical parallel system the channel is divided into p y N non-overlapping sub-channels to avoid inter-carrier interference (ICI)( )

loz The diagram for bandwidth efficiency of OFDM system is shown below

35

SummarySummaryyy

loz The advantage of the FFT-based OFDM system g yloz The use of IFFTFFT can reduce the computation complexityloz The orthogonality between the adjacent sub-carriers will make g y j

the use of transmission bandwidth more efficientloz The guard interval is used to resist the inter-symbol interference

(ISI)loz The main advantage of the OFDM transmission technique is its

high performance even in frequency selective channels

loz The drawbacks of the OFDM system loz It is highly vulnerable to synchronization errorsloz Peak to Average Power Ratio (PAPR) problems

36

SynchronizationSynchronization

Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offset (TO)loz Impact of timing offset (TO)loz Impact of carrier frequency offset (CFO)loz TO estimation algorithmloz CFO estimation algorithmg

38 Synchronization2010722

Introduction (13)Introduction (13)( )( )

loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods

loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead

loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the

cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of

computational complexity therefore it may be not be available in short-burst wireless communication

39 Synchronization2010722

Introduction (23)Introduction (23)( )( )

loz Transcevier

40 Synchronization2010722

Introduction (33)Introduction (33)( )( )

loz Synchronization erroryloz Symbol timing offset (TO)

loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)

loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect

41 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg

42 Synchronization2010722

Signal model (14)Signal model (14)g ( )g ( )

loz Time domain sample p1

2

0

1 Nj nk N

n kk

x X eN

πminus

minus= sum

Xk the frequency domain data at the k-th subcarrier0kN =

loz The multipath fading channel

( ) ( ) ( )1L

h t h t δminus

sum

loz Received time domain signal

( ) ( ) ( )0

l ll

h t h tτ τ δ τ τ=

= minussum

loz Received time-domain signal

( ) ( )1

L

n n ly h n l x w nminus

minus= +sum43

( ) ( )0l=sum

Synchronization2010722

Signal model (24)Signal model (24)g ( )g ( )

loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting

12

0

Nj kn N

k nn

Y y e πminus

minus

=

= sum

( )1 1

2

0

N Lj kn N

n l kn l

h n l x e Wπminus minus

minusminus

=

= +sumsum

loz The equation above could be expressed in matrix form

0n l

loz The equation above could be expressed in matrix form

= +Y HX W

44 Synchronization2010722

Signal model (34)Signal model (34)g ( )g ( )

H= = = = +Y Fy FCx FCF X HX W

loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal

y

H=H FCFcirculant matrix leading to be a diagonal matrix

=H FCF

( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0

h h L L hh h

⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥

( )( ) ( )

( ) ( )

1 10 1 0 0i

h Lh L h N L

⎢ ⎥⎢ ⎥minus minus⎢ ⎥

equiv= minus minus⎢ ⎥⎢ ⎥

C( ) ( )

( ) ( )

0 1 1 100

0 0 1 1 0

h L h N L

h N L L h N

⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

45

( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦

Synchronization2010722

Signal model (44)Signal model (44)g ( )g ( )

loz In a time-invariant channel the channel matrix H would be a diagonal matrix

k k k kY H X W= +

( )1 1

2

0 0

N Lj kl N

kn l

H h n l e πminus minus

minus

= =

= sumsum

loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI

k k k k kY H X I Wα= + +k k k k k

( ) ( )1 1 1

2 2 N N L

j m k Nj ml Nk mI X h n l e e ππ

minus minus minusminusminus⎛ ⎞

= ⎜ ⎟⎝ ⎠

sum sum sumICI

46

0 0 0m n lm k= = =ne

⎝ ⎠

Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

47 Synchronization2010722

Impact of timing offset Impact of timing offset p gp g

loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively

( )I f s

Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ

( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr

2 fj k Nk k k kY X H e Wπδminus= +

loz For integer part of timing offset k k k kY X H e W+

I sTδ ( ) 0g IN L δminus minus le leFFT

2 Ij k NY X H e Wπδminus= +

( ) ( ) 2 I sFFT

j f TI sh t T H f e π δδ minusminus rarr

48

k k k kY X H e W= +

Synchronization2010722

Impact of timing offset derivationImpact of timing offset derivationp gp g

49 Synchronization2010722

Discussion of symbol boundaryDiscussion of symbol boundaryy yy y

50 Synchronization2010722

Impact of timing offsetImpact of timing offsetp gp g

loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI

2 Ij k NIk k k k

NY X H e ISI W

Nπδδ minusminus

= + +

loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk

e e

p kloz Phase Shift can be resolved by Differential Encoding

Decoding or carrier recovery with pilots

51 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

52 Synchronization2010722

Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )

53 Synchronization2010722

Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )

( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s

i n t i N N T N T nTy y t e

= + + +=

( ) ( )( ) ( ) ( ) 12

sin

sin

g gI f f

I I

i N N N Nj jf N Ni k i k k

f

Y X H e eN

π ε ε π ε

ε ε

πεπε

+ + minus+

minus minus=⎛ ⎞⎜ ⎟

Faded signal attenuated and rotated by CFO

( )( ) ( ) ( ) ( )11 2

sin

sin g gI f I f

i N N N NN j j l kI f

NN

l k π ε ε π ε επ ε ε + + minusminus + + + minus

⎜ ⎟⎝ ⎠

+ + minussum

( )( )( )

( ) ( )2

0

sin

I f I f

I

j j l kI f N Ni l l

l l k I f

X H e el k

NN

π ε ε π ε ε

ε π ε ε

+ + +

= ne minus

+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠

sum

i k

N

W

⎜ ⎟⎝ ⎠

+ Inter-Carrier Interference (ICI)

54 Synchronization2010722

Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )

loz An easier representation of received signal with only p g yfractional CFO is

1N

( ) ( )1

0

0N

k k k l l kl l k

Y H X C H X C l k Wminus

= ne

= + minus +⎡ ⎤⎣ ⎦sum

where the ICI coefficient is given by

( )( )( ) ( )

sin 1exp 1fl k

C l k j l kπ ε

π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )

( )exp 1sin

f

f

C l k j l kNN l k

N

π επ ε

minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠

55 Synchronization2010722

Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )

loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality

loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift

56 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

57 Synchronization2010722

TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )

loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq

58 Synchronization2010722

TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )

loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below

loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and

rN2 can be expressed by

0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+

( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +

59

( )

Synchronization2010722

TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )

loz Without CFO the first half is identical to the second half (in time order)

loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below

( ) 12 in discretesft T f NT

NTφ π π π ε πε

⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟

⎝ ⎠

helliphellip

sNT⎝ ⎠

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)

60

Length=N2

Synchronization2010722

TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )

loz Received signal without CFOg0

0 0

jr a e θ=jθ 2 0

2 2 0

Nj j

N Nr a e a e

θ θ= =

loz Received signal with CFO0 0

0 0

j jr a e eθ φ= Extra phase rotation due to CFO0 0

2 2 20

2 2 0

N N Nj j jj

N Nr a e e a e e

θ φ φθ= =

due to CFO

Phase difference 2 2 0N N

( )2 02 20 2 0 0

Nj j T fN

r r a e a eφ φ πminus Δ= =

which contains the information about CFO

61

0 2 0 0Nr r a e a e

Synchronization2010722

TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )

loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be

loz Note that d is a time index corresponding to the first( ) ( ) 2 1

20

Nd m d m Nm

P d r rminus

+ + +== sum

loz Note that d is a time index corresponding to the first sample in a window of N samples

loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by

( )22 1

20

Nd m NR d rminus

+ += sumloz A timing metric can be defined as

))(()(

)( 2

2

dRdP

dM =

( ) 20 d m Nm + +=sum

62

))(( dR

Synchronization2010722

TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )

63 Synchronization2010722

TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )

loz If drsquo is the correct symbol timing offset

( )2

2 2

0 1 12

j T f j T f j T fN

P d a e a e a eπ π πΔ Δ Δ

minus= + + +

22

2 2

0 1 12

j T fN

e a a aπ Δ

minus

⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭

loz If drsquo falls behind the correct symbol timing offset 1 sample

2⎪ ⎪⎩ ⎭

64 Synchronization2010722

TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )

65 Synchronization2010722

TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )

loz Drawback Plateau effect

Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)

((N2) ((N2) ((N2)

helliphellip

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)

Sliding window (length = N)

66 Synchronization2010722

TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )

loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula

( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus

扣掉相加

( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +

loz The method also called delay correlator

67 Synchronization2010722

IEEE 80211aIEEE 80211a

68 Synchronization2010722

Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )

loz Minnrsquos Method [2][ ]B B －B －B

( )( )

( )

22

2Minn

P dM d =

1 4 1N minus

( )( )( )2

2

MinnR d

( ) ( ) ( ) ( )

1 4 1

2 2 2 40 0

N

d N m k d N m k Nm k

P d r r+ + + + += =

= sum sum

( ) ( ) ( )2

21 4 1

2 40 0

N

d N m k Nk

R d rminus

+ + += sum sum69

0 0m k= =

Synchronization2010722

Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )

loz Parkrsquos Method [3][ ]

where D is the symmetric version of CC D C D

where D is the symmetric version of C

( )( ) 2

3k

P dM d =( )

( )( )23

ParkM dR d

2N

( ) 2

30

N

d k d kk

P d r rminus +=

= sdotsum

( )3

22

0

N

d kk

R d r += sum70

0k=

Synchronization2010722

Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )

loz Renrsquos Method [4][ ]

AS AS

where S is the PN sequence weighted of the original bl 2preamble

( )( )( )( )

24

Re 24

n

P dM d

R d=

( )( )4R d

( ) 2 1

4 2 2

N

k k N d k d k NP d s s r rminus

+ + + += sum0k=sum

( )21

41 N

d kR d rminus

= sum71

( )402 d k

kR d r +

=sum

Synchronization2010722

Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )

72 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

73 Synchronization2010722

Fractional CFO estimation Fractional CFO estimation

loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected

11 Rminus⎛ ⎞sum

h L i th di t b t t id ti l bl k R i

0

12 d m d m L

ms

f r rLTπ + + +

=

⎛ ⎞= ang⎜ ⎟⎝ ⎠sum

where L is the distance between two identical block R is the block size

loz For exampleNN cp

L = N R = NL = N R = Ncp74 Synchronization2010722

Estimation rangeEstimation rangegg

loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval

f ( ) ( )s s⎣ ⎦

loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)

loz DVB-T (L=N) 802 11 (L N4)

[ ]05 05s sf fminus

[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)

(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus

loz Note that is the subcarrier spacing1sf NT=

75

sNT

Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz

plusmng q y

loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn

76 Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz

plusmng q y

loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for

subcarrier spacing is required

plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn

p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs

77 Synchronization2010722

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

Cyclic PrefixCyclic Prefix

loz Guard interval leads to intercarrier interference (ICI) in OFDM d d l i

yy

demodulation

loz In DFT interval difference between two subcarriers does not maintain integer number of cycles loss of orthogonality

loz Delayed version of subcarrier 2 causes ICI in the process of d d l ti b i 1demodulating subcarrier 1

25

Cyclic PrefixCyclic Prefixloz Cyclic prefix (CP) A copy of the last part of OFDM signal is

attached to the front of itself

yy

attached to the front of itself

[ ]0d

[ ]1d

[ ]0D[ ]1D

[ ]nd[ ]kD~

symbols dataInput

bullbullbullbull

[ ]1d

[ ]2d

[ ][ ]2D

bullbullbullbullbull

y

bullbullbullbullbull

[ ]1minusNd [ ]1minusND

[ ]gNND minus

bullbullbull

26

Cyclic PrefixCyclic Prefix

loz All delayed replicas of subcarriers always have an

yy

y p yinteger number of cycles within DFT interval no ICI

27

Cyclic PrefixCyclic Prefix

loz Linear convolution vs circular convolution

yy

N

28

Cyclic PrefixCyclic Prefix

loz Channel effect with cyclic prefix

yy

Signal after removed CP

29

Cyclic PrefixCyclic Prefix

loz Time-Domain Explanation

yy

p

30

Cyclic PrefixCyclic Prefixyy

31

Cyclic PrefixCyclic Prefix

loz Spectrum of channel response with length (smaller [ ]nh hL

yy

than )h

gN[ ] nhFFTHk =

loz Received complete OFDM signal[ ] [ ] [ ] 20 ~~ minus++leleotimes= hg LNNnnhnDnr

loz Received useful part [ ]nr[ ] [ ] [ ]nhnDnr Notimes=

where is N-point circular convolution (due to CP)loz Received symbol at k-th subcarrier

[ ] [ ] [ ]N

Notimes

[ ] [ ] [ ] kkNk HXnhnDFFTnrFFTY =otimes== kYX =rArr ldquoUseful property for OFDM system to reduce

kk H

X =rArrcomplexity of channel equalizationrdquo

32

Cyclic Cyclic Prefix (OFDM Prefix (OFDM Receiver)Receiver)yy (( ))

bull bull( )( )nr~Symbols

dataOutput

bullbullbull

bullbullbull( )nr

bullbull

bullbull

33

Cyclic PrefixCyclic Prefix

loz One of the most important reasons to do OFDM is the

yy

pefficient way it deals with multipath delay spread

loz To eliminate inter-symbol interference (ISI) almostloz To eliminate inter symbol interference (ISI) almost completely a guard time is introduced for each OFDM symbolsymbol(The guard time is chosen larger than the delay spread)

34

Bandwidth EfficiencyBandwidth Efficiencyyy

loz In a classical parallel system the channel is divided into p y N non-overlapping sub-channels to avoid inter-carrier interference (ICI)( )

loz The diagram for bandwidth efficiency of OFDM system is shown below

35

SummarySummaryyy

loz The advantage of the FFT-based OFDM system g yloz The use of IFFTFFT can reduce the computation complexityloz The orthogonality between the adjacent sub-carriers will make g y j

the use of transmission bandwidth more efficientloz The guard interval is used to resist the inter-symbol interference

(ISI)loz The main advantage of the OFDM transmission technique is its

high performance even in frequency selective channels

loz The drawbacks of the OFDM system loz It is highly vulnerable to synchronization errorsloz Peak to Average Power Ratio (PAPR) problems

36

SynchronizationSynchronization

Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offset (TO)loz Impact of timing offset (TO)loz Impact of carrier frequency offset (CFO)loz TO estimation algorithmloz CFO estimation algorithmg

38 Synchronization2010722

Introduction (13)Introduction (13)( )( )

loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods

loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead

loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the

cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of

computational complexity therefore it may be not be available in short-burst wireless communication

39 Synchronization2010722

Introduction (23)Introduction (23)( )( )

loz Transcevier

40 Synchronization2010722

Introduction (33)Introduction (33)( )( )

loz Synchronization erroryloz Symbol timing offset (TO)

loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)

loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect

41 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg

42 Synchronization2010722

Signal model (14)Signal model (14)g ( )g ( )

loz Time domain sample p1

2

0

1 Nj nk N

n kk

x X eN

πminus

minus= sum

Xk the frequency domain data at the k-th subcarrier0kN =

loz The multipath fading channel

( ) ( ) ( )1L

h t h t δminus

sum

loz Received time domain signal

( ) ( ) ( )0

l ll

h t h tτ τ δ τ τ=

= minussum

loz Received time-domain signal

( ) ( )1

L

n n ly h n l x w nminus

minus= +sum43

( ) ( )0l=sum

Synchronization2010722

Signal model (24)Signal model (24)g ( )g ( )

loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting

12

0

Nj kn N

k nn

Y y e πminus

minus

=

= sum

( )1 1

2

0

N Lj kn N

n l kn l

h n l x e Wπminus minus

minusminus

=

= +sumsum

loz The equation above could be expressed in matrix form

0n l

loz The equation above could be expressed in matrix form

= +Y HX W

44 Synchronization2010722

Signal model (34)Signal model (34)g ( )g ( )

H= = = = +Y Fy FCx FCF X HX W

loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal

y

H=H FCFcirculant matrix leading to be a diagonal matrix

=H FCF

( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0

h h L L hh h

⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥

( )( ) ( )

( ) ( )

1 10 1 0 0i

h Lh L h N L

⎢ ⎥⎢ ⎥minus minus⎢ ⎥

equiv= minus minus⎢ ⎥⎢ ⎥

C( ) ( )

( ) ( )

0 1 1 100

0 0 1 1 0

h L h N L

h N L L h N

⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

45

( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦

Synchronization2010722

Signal model (44)Signal model (44)g ( )g ( )

loz In a time-invariant channel the channel matrix H would be a diagonal matrix

k k k kY H X W= +

( )1 1

2

0 0

N Lj kl N

kn l

H h n l e πminus minus

minus

= =

= sumsum

loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI

k k k k kY H X I Wα= + +k k k k k

( ) ( )1 1 1

2 2 N N L

j m k Nj ml Nk mI X h n l e e ππ

minus minus minusminusminus⎛ ⎞

= ⎜ ⎟⎝ ⎠

sum sum sumICI

46

0 0 0m n lm k= = =ne

⎝ ⎠

Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

47 Synchronization2010722

Impact of timing offset Impact of timing offset p gp g

loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively

( )I f s

Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ

( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr

2 fj k Nk k k kY X H e Wπδminus= +

loz For integer part of timing offset k k k kY X H e W+

I sTδ ( ) 0g IN L δminus minus le leFFT

2 Ij k NY X H e Wπδminus= +

( ) ( ) 2 I sFFT

j f TI sh t T H f e π δδ minusminus rarr

48

k k k kY X H e W= +

Synchronization2010722

Impact of timing offset derivationImpact of timing offset derivationp gp g

49 Synchronization2010722

Discussion of symbol boundaryDiscussion of symbol boundaryy yy y

50 Synchronization2010722

Impact of timing offsetImpact of timing offsetp gp g

loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI

2 Ij k NIk k k k

NY X H e ISI W

Nπδδ minusminus

= + +

loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk

e e

p kloz Phase Shift can be resolved by Differential Encoding

Decoding or carrier recovery with pilots

51 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

52 Synchronization2010722

Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )

53 Synchronization2010722

Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )

( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s

i n t i N N T N T nTy y t e

= + + +=

( ) ( )( ) ( ) ( ) 12

sin

sin

g gI f f

I I

i N N N Nj jf N Ni k i k k

f

Y X H e eN

π ε ε π ε

ε ε

πεπε

+ + minus+

minus minus=⎛ ⎞⎜ ⎟

Faded signal attenuated and rotated by CFO

( )( ) ( ) ( ) ( )11 2

sin

sin g gI f I f

i N N N NN j j l kI f

NN

l k π ε ε π ε επ ε ε + + minusminus + + + minus

⎜ ⎟⎝ ⎠

+ + minussum

( )( )( )

( ) ( )2

0

sin

I f I f

I

j j l kI f N Ni l l

l l k I f

X H e el k

NN

π ε ε π ε ε

ε π ε ε

+ + +

= ne minus

+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠

sum

i k

N

W

⎜ ⎟⎝ ⎠

+ Inter-Carrier Interference (ICI)

54 Synchronization2010722

Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )

loz An easier representation of received signal with only p g yfractional CFO is

1N

( ) ( )1

0

0N

k k k l l kl l k

Y H X C H X C l k Wminus

= ne

= + minus +⎡ ⎤⎣ ⎦sum

where the ICI coefficient is given by

( )( )( ) ( )

sin 1exp 1fl k

C l k j l kπ ε

π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )

( )exp 1sin

f

f

C l k j l kNN l k

N

π επ ε

minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠

55 Synchronization2010722

Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )

loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality

loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift

56 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

57 Synchronization2010722

TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )

loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq

58 Synchronization2010722

TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )

loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below

loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and

rN2 can be expressed by

0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+

( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +

59

( )

Synchronization2010722

TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )

loz Without CFO the first half is identical to the second half (in time order)

loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below

( ) 12 in discretesft T f NT

NTφ π π π ε πε

⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟

⎝ ⎠

helliphellip

sNT⎝ ⎠

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)

60

Length=N2

Synchronization2010722

TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )

loz Received signal without CFOg0

0 0

jr a e θ=jθ 2 0

2 2 0

Nj j

N Nr a e a e

θ θ= =

loz Received signal with CFO0 0

0 0

j jr a e eθ φ= Extra phase rotation due to CFO0 0

2 2 20

2 2 0

N N Nj j jj

N Nr a e e a e e

θ φ φθ= =

due to CFO

Phase difference 2 2 0N N

( )2 02 20 2 0 0

Nj j T fN

r r a e a eφ φ πminus Δ= =

which contains the information about CFO

61

0 2 0 0Nr r a e a e

Synchronization2010722

TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )

loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be

loz Note that d is a time index corresponding to the first( ) ( ) 2 1

20

Nd m d m Nm

P d r rminus

+ + +== sum

loz Note that d is a time index corresponding to the first sample in a window of N samples

loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by

( )22 1

20

Nd m NR d rminus

+ += sumloz A timing metric can be defined as

))(()(

)( 2

2

dRdP

dM =

( ) 20 d m Nm + +=sum

62

))(( dR

Synchronization2010722

TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )

63 Synchronization2010722

TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )

loz If drsquo is the correct symbol timing offset

( )2

2 2

0 1 12

j T f j T f j T fN

P d a e a e a eπ π πΔ Δ Δ

minus= + + +

22

2 2

0 1 12

j T fN

e a a aπ Δ

minus

⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭

loz If drsquo falls behind the correct symbol timing offset 1 sample

2⎪ ⎪⎩ ⎭

64 Synchronization2010722

TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )

65 Synchronization2010722

TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )

loz Drawback Plateau effect

Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)

((N2) ((N2) ((N2)

helliphellip

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)

Sliding window (length = N)

66 Synchronization2010722

TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )

loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula

( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus

扣掉相加

( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +

loz The method also called delay correlator

67 Synchronization2010722

IEEE 80211aIEEE 80211a

68 Synchronization2010722

Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )

loz Minnrsquos Method [2][ ]B B －B －B

( )( )

( )

22

2Minn

P dM d =

1 4 1N minus

( )( )( )2

2

MinnR d

( ) ( ) ( ) ( )

1 4 1

2 2 2 40 0

N

d N m k d N m k Nm k

P d r r+ + + + += =

= sum sum

( ) ( ) ( )2

21 4 1

2 40 0

N

d N m k Nk

R d rminus

+ + += sum sum69

0 0m k= =

Synchronization2010722

Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )

loz Parkrsquos Method [3][ ]

where D is the symmetric version of CC D C D

where D is the symmetric version of C

( )( ) 2

3k

P dM d =( )

( )( )23

ParkM dR d

2N

( ) 2

30

N

d k d kk

P d r rminus +=

= sdotsum

( )3

22

0

N

d kk

R d r += sum70

0k=

Synchronization2010722

Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )

loz Renrsquos Method [4][ ]

AS AS

where S is the PN sequence weighted of the original bl 2preamble

( )( )( )( )

24

Re 24

n

P dM d

R d=

( )( )4R d

( ) 2 1

4 2 2

N

k k N d k d k NP d s s r rminus

+ + + += sum0k=sum

( )21

41 N

d kR d rminus

= sum71

( )402 d k

kR d r +

=sum

Synchronization2010722

Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )

72 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

73 Synchronization2010722

Fractional CFO estimation Fractional CFO estimation

loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected

11 Rminus⎛ ⎞sum

h L i th di t b t t id ti l bl k R i

0

12 d m d m L

ms

f r rLTπ + + +

=

⎛ ⎞= ang⎜ ⎟⎝ ⎠sum

where L is the distance between two identical block R is the block size

loz For exampleNN cp

L = N R = NL = N R = Ncp74 Synchronization2010722

Estimation rangeEstimation rangegg

loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval

f ( ) ( )s s⎣ ⎦

loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)

loz DVB-T (L=N) 802 11 (L N4)

[ ]05 05s sf fminus

[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)

(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus

loz Note that is the subcarrier spacing1sf NT=

75

sNT

Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz

plusmng q y

loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn

76 Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz

plusmng q y

loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for

subcarrier spacing is required

plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn

p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs

77 Synchronization2010722

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

Cyclic PrefixCyclic Prefixloz Cyclic prefix (CP) A copy of the last part of OFDM signal is

attached to the front of itself

yy

attached to the front of itself

[ ]0d

[ ]1d

[ ]0D[ ]1D

[ ]nd[ ]kD~

symbols dataInput

bullbullbullbull

[ ]1d

[ ]2d

[ ][ ]2D

bullbullbullbullbull

y

bullbullbullbullbull

[ ]1minusNd [ ]1minusND

[ ]gNND minus

bullbullbull

26

Cyclic PrefixCyclic Prefix

loz All delayed replicas of subcarriers always have an

yy

y p yinteger number of cycles within DFT interval no ICI

27

Cyclic PrefixCyclic Prefix

loz Linear convolution vs circular convolution

yy

N

28

Cyclic PrefixCyclic Prefix

loz Channel effect with cyclic prefix

yy

Signal after removed CP

29

Cyclic PrefixCyclic Prefix

loz Time-Domain Explanation

yy

p

30

Cyclic PrefixCyclic Prefixyy

31

Cyclic PrefixCyclic Prefix

loz Spectrum of channel response with length (smaller [ ]nh hL

yy

than )h

gN[ ] nhFFTHk =

loz Received complete OFDM signal[ ] [ ] [ ] 20 ~~ minus++leleotimes= hg LNNnnhnDnr

loz Received useful part [ ]nr[ ] [ ] [ ]nhnDnr Notimes=

where is N-point circular convolution (due to CP)loz Received symbol at k-th subcarrier

[ ] [ ] [ ]N

Notimes

[ ] [ ] [ ] kkNk HXnhnDFFTnrFFTY =otimes== kYX =rArr ldquoUseful property for OFDM system to reduce

kk H

X =rArrcomplexity of channel equalizationrdquo

32

Cyclic Cyclic Prefix (OFDM Prefix (OFDM Receiver)Receiver)yy (( ))

bull bull( )( )nr~Symbols

dataOutput

bullbullbull

bullbullbull( )nr

bullbull

bullbull

33

Cyclic PrefixCyclic Prefix

loz One of the most important reasons to do OFDM is the

yy

pefficient way it deals with multipath delay spread

loz To eliminate inter-symbol interference (ISI) almostloz To eliminate inter symbol interference (ISI) almost completely a guard time is introduced for each OFDM symbolsymbol(The guard time is chosen larger than the delay spread)

34

Bandwidth EfficiencyBandwidth Efficiencyyy

loz In a classical parallel system the channel is divided into p y N non-overlapping sub-channels to avoid inter-carrier interference (ICI)( )

loz The diagram for bandwidth efficiency of OFDM system is shown below

35

SummarySummaryyy

loz The advantage of the FFT-based OFDM system g yloz The use of IFFTFFT can reduce the computation complexityloz The orthogonality between the adjacent sub-carriers will make g y j

the use of transmission bandwidth more efficientloz The guard interval is used to resist the inter-symbol interference

(ISI)loz The main advantage of the OFDM transmission technique is its

high performance even in frequency selective channels

loz The drawbacks of the OFDM system loz It is highly vulnerable to synchronization errorsloz Peak to Average Power Ratio (PAPR) problems

36

SynchronizationSynchronization

Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offset (TO)loz Impact of timing offset (TO)loz Impact of carrier frequency offset (CFO)loz TO estimation algorithmloz CFO estimation algorithmg

38 Synchronization2010722

Introduction (13)Introduction (13)( )( )

loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods

loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead

loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the

cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of

computational complexity therefore it may be not be available in short-burst wireless communication

39 Synchronization2010722

Introduction (23)Introduction (23)( )( )

loz Transcevier

40 Synchronization2010722

Introduction (33)Introduction (33)( )( )

loz Synchronization erroryloz Symbol timing offset (TO)

loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)

loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect

41 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg

42 Synchronization2010722

Signal model (14)Signal model (14)g ( )g ( )

loz Time domain sample p1

2

0

1 Nj nk N

n kk

x X eN

πminus

minus= sum

Xk the frequency domain data at the k-th subcarrier0kN =

loz The multipath fading channel

( ) ( ) ( )1L

h t h t δminus

sum

loz Received time domain signal

( ) ( ) ( )0

l ll

h t h tτ τ δ τ τ=

= minussum

loz Received time-domain signal

( ) ( )1

L

n n ly h n l x w nminus

minus= +sum43

( ) ( )0l=sum

Synchronization2010722

Signal model (24)Signal model (24)g ( )g ( )

loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting

12

0

Nj kn N

k nn

Y y e πminus

minus

=

= sum

( )1 1

2

0

N Lj kn N

n l kn l

h n l x e Wπminus minus

minusminus

=

= +sumsum

loz The equation above could be expressed in matrix form

0n l

loz The equation above could be expressed in matrix form

= +Y HX W

44 Synchronization2010722

Signal model (34)Signal model (34)g ( )g ( )

H= = = = +Y Fy FCx FCF X HX W

loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal

y

H=H FCFcirculant matrix leading to be a diagonal matrix

=H FCF

( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0

h h L L hh h

⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥

( )( ) ( )

( ) ( )

1 10 1 0 0i

h Lh L h N L

⎢ ⎥⎢ ⎥minus minus⎢ ⎥

equiv= minus minus⎢ ⎥⎢ ⎥

C( ) ( )

( ) ( )

0 1 1 100

0 0 1 1 0

h L h N L

h N L L h N

⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

45

( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦

Synchronization2010722

Signal model (44)Signal model (44)g ( )g ( )

loz In a time-invariant channel the channel matrix H would be a diagonal matrix

k k k kY H X W= +

( )1 1

2

0 0

N Lj kl N

kn l

H h n l e πminus minus

minus

= =

= sumsum

loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI

k k k k kY H X I Wα= + +k k k k k

( ) ( )1 1 1

2 2 N N L

j m k Nj ml Nk mI X h n l e e ππ

minus minus minusminusminus⎛ ⎞

= ⎜ ⎟⎝ ⎠

sum sum sumICI

46

0 0 0m n lm k= = =ne

⎝ ⎠

Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

47 Synchronization2010722

Impact of timing offset Impact of timing offset p gp g

loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively

( )I f s

Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ

( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr

2 fj k Nk k k kY X H e Wπδminus= +

loz For integer part of timing offset k k k kY X H e W+

I sTδ ( ) 0g IN L δminus minus le leFFT

2 Ij k NY X H e Wπδminus= +

( ) ( ) 2 I sFFT

j f TI sh t T H f e π δδ minusminus rarr

48

k k k kY X H e W= +

Synchronization2010722

Impact of timing offset derivationImpact of timing offset derivationp gp g

49 Synchronization2010722

Discussion of symbol boundaryDiscussion of symbol boundaryy yy y

50 Synchronization2010722

Impact of timing offsetImpact of timing offsetp gp g

loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI

2 Ij k NIk k k k

NY X H e ISI W

Nπδδ minusminus

= + +

loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk

e e

p kloz Phase Shift can be resolved by Differential Encoding

Decoding or carrier recovery with pilots

51 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

52 Synchronization2010722

Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )

53 Synchronization2010722

Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )

( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s

i n t i N N T N T nTy y t e

= + + +=

( ) ( )( ) ( ) ( ) 12

sin

sin

g gI f f

I I

i N N N Nj jf N Ni k i k k

f

Y X H e eN

π ε ε π ε

ε ε

πεπε

+ + minus+

minus minus=⎛ ⎞⎜ ⎟

Faded signal attenuated and rotated by CFO

( )( ) ( ) ( ) ( )11 2

sin

sin g gI f I f

i N N N NN j j l kI f

NN

l k π ε ε π ε επ ε ε + + minusminus + + + minus

⎜ ⎟⎝ ⎠

+ + minussum

( )( )( )

( ) ( )2

0

sin

I f I f

I

j j l kI f N Ni l l

l l k I f

X H e el k

NN

π ε ε π ε ε

ε π ε ε

+ + +

= ne minus

+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠

sum

i k

N

W

⎜ ⎟⎝ ⎠

+ Inter-Carrier Interference (ICI)

54 Synchronization2010722

Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )

loz An easier representation of received signal with only p g yfractional CFO is

1N

( ) ( )1

0

0N

k k k l l kl l k

Y H X C H X C l k Wminus

= ne

= + minus +⎡ ⎤⎣ ⎦sum

where the ICI coefficient is given by

( )( )( ) ( )

sin 1exp 1fl k

C l k j l kπ ε

π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )

( )exp 1sin

f

f

C l k j l kNN l k

N

π επ ε

minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠

55 Synchronization2010722

Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )

loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality

loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift

56 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

57 Synchronization2010722

TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )

loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq

58 Synchronization2010722

TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )

loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below

loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and

rN2 can be expressed by

0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+

( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +

59

( )

Synchronization2010722

TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )

loz Without CFO the first half is identical to the second half (in time order)

loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below

( ) 12 in discretesft T f NT

NTφ π π π ε πε

⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟

⎝ ⎠

helliphellip

sNT⎝ ⎠

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)

60

Length=N2

Synchronization2010722

TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )

loz Received signal without CFOg0

0 0

jr a e θ=jθ 2 0

2 2 0

Nj j

N Nr a e a e

θ θ= =

loz Received signal with CFO0 0

0 0

j jr a e eθ φ= Extra phase rotation due to CFO0 0

2 2 20

2 2 0

N N Nj j jj

N Nr a e e a e e

θ φ φθ= =

due to CFO

Phase difference 2 2 0N N

( )2 02 20 2 0 0

Nj j T fN

r r a e a eφ φ πminus Δ= =

which contains the information about CFO

61

0 2 0 0Nr r a e a e

Synchronization2010722

TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )

loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be

loz Note that d is a time index corresponding to the first( ) ( ) 2 1

20

Nd m d m Nm

P d r rminus

+ + +== sum

loz Note that d is a time index corresponding to the first sample in a window of N samples

loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by

( )22 1

20

Nd m NR d rminus

+ += sumloz A timing metric can be defined as

))(()(

)( 2

2

dRdP

dM =

( ) 20 d m Nm + +=sum

62

))(( dR

Synchronization2010722

TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )

63 Synchronization2010722

TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )

loz If drsquo is the correct symbol timing offset

( )2

2 2

0 1 12

j T f j T f j T fN

P d a e a e a eπ π πΔ Δ Δ

minus= + + +

22

2 2

0 1 12

j T fN

e a a aπ Δ

minus

⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭

loz If drsquo falls behind the correct symbol timing offset 1 sample

2⎪ ⎪⎩ ⎭

64 Synchronization2010722

TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )

65 Synchronization2010722

TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )

loz Drawback Plateau effect

Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)

((N2) ((N2) ((N2)

helliphellip

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)

Sliding window (length = N)

66 Synchronization2010722

TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )

loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula

( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus

扣掉相加

( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +

loz The method also called delay correlator

67 Synchronization2010722

IEEE 80211aIEEE 80211a

68 Synchronization2010722

Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )

loz Minnrsquos Method [2][ ]B B －B －B

( )( )

( )

22

2Minn

P dM d =

1 4 1N minus

( )( )( )2

2

MinnR d

( ) ( ) ( ) ( )

1 4 1

2 2 2 40 0

N

d N m k d N m k Nm k

P d r r+ + + + += =

= sum sum

( ) ( ) ( )2

21 4 1

2 40 0

N

d N m k Nk

R d rminus

+ + += sum sum69

0 0m k= =

Synchronization2010722

Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )

loz Parkrsquos Method [3][ ]

where D is the symmetric version of CC D C D

where D is the symmetric version of C

( )( ) 2

3k

P dM d =( )

( )( )23

ParkM dR d

2N

( ) 2

30

N

d k d kk

P d r rminus +=

= sdotsum

( )3

22

0

N

d kk

R d r += sum70

0k=

Synchronization2010722

Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )

loz Renrsquos Method [4][ ]

AS AS

where S is the PN sequence weighted of the original bl 2preamble

( )( )( )( )

24

Re 24

n

P dM d

R d=

( )( )4R d

( ) 2 1

4 2 2

N

k k N d k d k NP d s s r rminus

+ + + += sum0k=sum

( )21

41 N

d kR d rminus

= sum71

( )402 d k

kR d r +

=sum

Synchronization2010722

Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )

72 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

73 Synchronization2010722

Fractional CFO estimation Fractional CFO estimation

loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected

11 Rminus⎛ ⎞sum

h L i th di t b t t id ti l bl k R i

0

12 d m d m L

ms

f r rLTπ + + +

=

⎛ ⎞= ang⎜ ⎟⎝ ⎠sum

where L is the distance between two identical block R is the block size

loz For exampleNN cp

L = N R = NL = N R = Ncp74 Synchronization2010722

Estimation rangeEstimation rangegg

loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval

f ( ) ( )s s⎣ ⎦

loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)

loz DVB-T (L=N) 802 11 (L N4)

[ ]05 05s sf fminus

[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)

(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus

loz Note that is the subcarrier spacing1sf NT=

75

sNT

Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz

plusmng q y

loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn

76 Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz

plusmng q y

loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for

subcarrier spacing is required

plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn

p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs

77 Synchronization2010722

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

Cyclic PrefixCyclic Prefix

loz All delayed replicas of subcarriers always have an

yy

y p yinteger number of cycles within DFT interval no ICI

27

Cyclic PrefixCyclic Prefix

loz Linear convolution vs circular convolution

yy

N

28

Cyclic PrefixCyclic Prefix

loz Channel effect with cyclic prefix

yy

Signal after removed CP

29

Cyclic PrefixCyclic Prefix

loz Time-Domain Explanation

yy

p

30

Cyclic PrefixCyclic Prefixyy

31

Cyclic PrefixCyclic Prefix

loz Spectrum of channel response with length (smaller [ ]nh hL

yy

than )h

gN[ ] nhFFTHk =

loz Received complete OFDM signal[ ] [ ] [ ] 20 ~~ minus++leleotimes= hg LNNnnhnDnr

loz Received useful part [ ]nr[ ] [ ] [ ]nhnDnr Notimes=

where is N-point circular convolution (due to CP)loz Received symbol at k-th subcarrier

[ ] [ ] [ ]N

Notimes

[ ] [ ] [ ] kkNk HXnhnDFFTnrFFTY =otimes== kYX =rArr ldquoUseful property for OFDM system to reduce

kk H

X =rArrcomplexity of channel equalizationrdquo

32

Cyclic Cyclic Prefix (OFDM Prefix (OFDM Receiver)Receiver)yy (( ))

bull bull( )( )nr~Symbols

dataOutput

bullbullbull

bullbullbull( )nr

bullbull

bullbull

33

Cyclic PrefixCyclic Prefix

loz One of the most important reasons to do OFDM is the

yy

pefficient way it deals with multipath delay spread

loz To eliminate inter-symbol interference (ISI) almostloz To eliminate inter symbol interference (ISI) almost completely a guard time is introduced for each OFDM symbolsymbol(The guard time is chosen larger than the delay spread)

34

Bandwidth EfficiencyBandwidth Efficiencyyy

loz In a classical parallel system the channel is divided into p y N non-overlapping sub-channels to avoid inter-carrier interference (ICI)( )

loz The diagram for bandwidth efficiency of OFDM system is shown below

35

SummarySummaryyy

loz The advantage of the FFT-based OFDM system g yloz The use of IFFTFFT can reduce the computation complexityloz The orthogonality between the adjacent sub-carriers will make g y j

the use of transmission bandwidth more efficientloz The guard interval is used to resist the inter-symbol interference

(ISI)loz The main advantage of the OFDM transmission technique is its

high performance even in frequency selective channels

loz The drawbacks of the OFDM system loz It is highly vulnerable to synchronization errorsloz Peak to Average Power Ratio (PAPR) problems

36

SynchronizationSynchronization

Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offset (TO)loz Impact of timing offset (TO)loz Impact of carrier frequency offset (CFO)loz TO estimation algorithmloz CFO estimation algorithmg

38 Synchronization2010722

Introduction (13)Introduction (13)( )( )

loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods

loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead

loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the

cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of

computational complexity therefore it may be not be available in short-burst wireless communication

39 Synchronization2010722

Introduction (23)Introduction (23)( )( )

loz Transcevier

40 Synchronization2010722

Introduction (33)Introduction (33)( )( )

loz Synchronization erroryloz Symbol timing offset (TO)

loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)

loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect

41 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg

42 Synchronization2010722

Signal model (14)Signal model (14)g ( )g ( )

loz Time domain sample p1

2

0

1 Nj nk N

n kk

x X eN

πminus

minus= sum

Xk the frequency domain data at the k-th subcarrier0kN =

loz The multipath fading channel

( ) ( ) ( )1L

h t h t δminus

sum

loz Received time domain signal

( ) ( ) ( )0

l ll

h t h tτ τ δ τ τ=

= minussum

loz Received time-domain signal

( ) ( )1

L

n n ly h n l x w nminus

minus= +sum43

( ) ( )0l=sum

Synchronization2010722

Signal model (24)Signal model (24)g ( )g ( )

loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting

12

0

Nj kn N

k nn

Y y e πminus

minus

=

= sum

( )1 1

2

0

N Lj kn N

n l kn l

h n l x e Wπminus minus

minusminus

=

= +sumsum

loz The equation above could be expressed in matrix form

0n l

loz The equation above could be expressed in matrix form

= +Y HX W

44 Synchronization2010722

Signal model (34)Signal model (34)g ( )g ( )

H= = = = +Y Fy FCx FCF X HX W

loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal

y

H=H FCFcirculant matrix leading to be a diagonal matrix

=H FCF

( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0

h h L L hh h

⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥

( )( ) ( )

( ) ( )

1 10 1 0 0i

h Lh L h N L

⎢ ⎥⎢ ⎥minus minus⎢ ⎥

equiv= minus minus⎢ ⎥⎢ ⎥

C( ) ( )

( ) ( )

0 1 1 100

0 0 1 1 0

h L h N L

h N L L h N

⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

45

( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦

Synchronization2010722

Signal model (44)Signal model (44)g ( )g ( )

loz In a time-invariant channel the channel matrix H would be a diagonal matrix

k k k kY H X W= +

( )1 1

2

0 0

N Lj kl N

kn l

H h n l e πminus minus

minus

= =

= sumsum

loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI

k k k k kY H X I Wα= + +k k k k k

( ) ( )1 1 1

2 2 N N L

j m k Nj ml Nk mI X h n l e e ππ

minus minus minusminusminus⎛ ⎞

= ⎜ ⎟⎝ ⎠

sum sum sumICI

46

0 0 0m n lm k= = =ne

⎝ ⎠

Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

47 Synchronization2010722

Impact of timing offset Impact of timing offset p gp g

loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively

( )I f s

Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ

( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr

2 fj k Nk k k kY X H e Wπδminus= +

loz For integer part of timing offset k k k kY X H e W+

I sTδ ( ) 0g IN L δminus minus le leFFT

2 Ij k NY X H e Wπδminus= +

( ) ( ) 2 I sFFT

j f TI sh t T H f e π δδ minusminus rarr

48

k k k kY X H e W= +

Synchronization2010722

Impact of timing offset derivationImpact of timing offset derivationp gp g

49 Synchronization2010722

Discussion of symbol boundaryDiscussion of symbol boundaryy yy y

50 Synchronization2010722

Impact of timing offsetImpact of timing offsetp gp g

loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI

2 Ij k NIk k k k

NY X H e ISI W

Nπδδ minusminus

= + +

loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk

e e

p kloz Phase Shift can be resolved by Differential Encoding

Decoding or carrier recovery with pilots

51 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

52 Synchronization2010722

Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )

53 Synchronization2010722

Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )

( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s

i n t i N N T N T nTy y t e

= + + +=

( ) ( )( ) ( ) ( ) 12

sin

sin

g gI f f

I I

i N N N Nj jf N Ni k i k k

f

Y X H e eN

π ε ε π ε

ε ε

πεπε

+ + minus+

minus minus=⎛ ⎞⎜ ⎟

Faded signal attenuated and rotated by CFO

( )( ) ( ) ( ) ( )11 2

sin

sin g gI f I f

i N N N NN j j l kI f

NN

l k π ε ε π ε επ ε ε + + minusminus + + + minus

⎜ ⎟⎝ ⎠

+ + minussum

( )( )( )

( ) ( )2

0

sin

I f I f

I

j j l kI f N Ni l l

l l k I f

X H e el k

NN

π ε ε π ε ε

ε π ε ε

+ + +

= ne minus

+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠

sum

i k

N

W

⎜ ⎟⎝ ⎠

+ Inter-Carrier Interference (ICI)

54 Synchronization2010722

Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )

loz An easier representation of received signal with only p g yfractional CFO is

1N

( ) ( )1

0

0N

k k k l l kl l k

Y H X C H X C l k Wminus

= ne

= + minus +⎡ ⎤⎣ ⎦sum

where the ICI coefficient is given by

( )( )( ) ( )

sin 1exp 1fl k

C l k j l kπ ε

π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )

( )exp 1sin

f

f

C l k j l kNN l k

N

π επ ε

minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠

55 Synchronization2010722

Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )

loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality

loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift

56 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

57 Synchronization2010722

TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )

loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq

58 Synchronization2010722

TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )

loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below

loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and

rN2 can be expressed by

0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+

( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +

59

( )

Synchronization2010722

TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )

loz Without CFO the first half is identical to the second half (in time order)

loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below

( ) 12 in discretesft T f NT

NTφ π π π ε πε

⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟

⎝ ⎠

helliphellip

sNT⎝ ⎠

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)

60

Length=N2

Synchronization2010722

TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )

loz Received signal without CFOg0

0 0

jr a e θ=jθ 2 0

2 2 0

Nj j

N Nr a e a e

θ θ= =

loz Received signal with CFO0 0

0 0

j jr a e eθ φ= Extra phase rotation due to CFO0 0

2 2 20

2 2 0

N N Nj j jj

N Nr a e e a e e

θ φ φθ= =

due to CFO

Phase difference 2 2 0N N

( )2 02 20 2 0 0

Nj j T fN

r r a e a eφ φ πminus Δ= =

which contains the information about CFO

61

0 2 0 0Nr r a e a e

Synchronization2010722

TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )

loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be

loz Note that d is a time index corresponding to the first( ) ( ) 2 1

20

Nd m d m Nm

P d r rminus

+ + +== sum

loz Note that d is a time index corresponding to the first sample in a window of N samples

loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by

( )22 1

20

Nd m NR d rminus

+ += sumloz A timing metric can be defined as

))(()(

)( 2

2

dRdP

dM =

( ) 20 d m Nm + +=sum

62

))(( dR

Synchronization2010722

TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )

63 Synchronization2010722

TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )

loz If drsquo is the correct symbol timing offset

( )2

2 2

0 1 12

j T f j T f j T fN

P d a e a e a eπ π πΔ Δ Δ

minus= + + +

22

2 2

0 1 12

j T fN

e a a aπ Δ

minus

⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭

loz If drsquo falls behind the correct symbol timing offset 1 sample

2⎪ ⎪⎩ ⎭

64 Synchronization2010722

TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )

65 Synchronization2010722

TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )

loz Drawback Plateau effect

Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)

((N2) ((N2) ((N2)

helliphellip

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)

Sliding window (length = N)

66 Synchronization2010722

TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )

loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula

( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus

扣掉相加

( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +

loz The method also called delay correlator

67 Synchronization2010722

IEEE 80211aIEEE 80211a

68 Synchronization2010722

Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )

loz Minnrsquos Method [2][ ]B B －B －B

( )( )

( )

22

2Minn

P dM d =

1 4 1N minus

( )( )( )2

2

MinnR d

( ) ( ) ( ) ( )

1 4 1

2 2 2 40 0

N

d N m k d N m k Nm k

P d r r+ + + + += =

= sum sum

( ) ( ) ( )2

21 4 1

2 40 0

N

d N m k Nk

R d rminus

+ + += sum sum69

0 0m k= =

Synchronization2010722

Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )

loz Parkrsquos Method [3][ ]

where D is the symmetric version of CC D C D

where D is the symmetric version of C

( )( ) 2

3k

P dM d =( )

( )( )23

ParkM dR d

2N

( ) 2

30

N

d k d kk

P d r rminus +=

= sdotsum

( )3

22

0

N

d kk

R d r += sum70

0k=

Synchronization2010722

Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )

loz Renrsquos Method [4][ ]

AS AS

where S is the PN sequence weighted of the original bl 2preamble

( )( )( )( )

24

Re 24

n

P dM d

R d=

( )( )4R d

( ) 2 1

4 2 2

N

k k N d k d k NP d s s r rminus

+ + + += sum0k=sum

( )21

41 N

d kR d rminus

= sum71

( )402 d k

kR d r +

=sum

Synchronization2010722

Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )

72 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

73 Synchronization2010722

Fractional CFO estimation Fractional CFO estimation

loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected

11 Rminus⎛ ⎞sum

h L i th di t b t t id ti l bl k R i

0

12 d m d m L

ms

f r rLTπ + + +

=

⎛ ⎞= ang⎜ ⎟⎝ ⎠sum

where L is the distance between two identical block R is the block size

loz For exampleNN cp

L = N R = NL = N R = Ncp74 Synchronization2010722

Estimation rangeEstimation rangegg

loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval

f ( ) ( )s s⎣ ⎦

loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)

loz DVB-T (L=N) 802 11 (L N4)

[ ]05 05s sf fminus

[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)

(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus

loz Note that is the subcarrier spacing1sf NT=

75

sNT

Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz

plusmng q y

loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn

76 Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz

plusmng q y

loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for

subcarrier spacing is required

plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn

p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs

77 Synchronization2010722

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

Cyclic PrefixCyclic Prefix

loz Linear convolution vs circular convolution

yy

N

28

Cyclic PrefixCyclic Prefix

loz Channel effect with cyclic prefix

yy

Signal after removed CP

29

Cyclic PrefixCyclic Prefix

loz Time-Domain Explanation

yy

p

30

Cyclic PrefixCyclic Prefixyy

31

Cyclic PrefixCyclic Prefix

loz Spectrum of channel response with length (smaller [ ]nh hL

yy

than )h

gN[ ] nhFFTHk =

loz Received complete OFDM signal[ ] [ ] [ ] 20 ~~ minus++leleotimes= hg LNNnnhnDnr

loz Received useful part [ ]nr[ ] [ ] [ ]nhnDnr Notimes=

where is N-point circular convolution (due to CP)loz Received symbol at k-th subcarrier

[ ] [ ] [ ]N

Notimes

[ ] [ ] [ ] kkNk HXnhnDFFTnrFFTY =otimes== kYX =rArr ldquoUseful property for OFDM system to reduce

kk H

X =rArrcomplexity of channel equalizationrdquo

32

Cyclic Cyclic Prefix (OFDM Prefix (OFDM Receiver)Receiver)yy (( ))

bull bull( )( )nr~Symbols

dataOutput

bullbullbull

bullbullbull( )nr

bullbull

bullbull

33

Cyclic PrefixCyclic Prefix

loz One of the most important reasons to do OFDM is the

yy

pefficient way it deals with multipath delay spread

loz To eliminate inter-symbol interference (ISI) almostloz To eliminate inter symbol interference (ISI) almost completely a guard time is introduced for each OFDM symbolsymbol(The guard time is chosen larger than the delay spread)

34

Bandwidth EfficiencyBandwidth Efficiencyyy

loz In a classical parallel system the channel is divided into p y N non-overlapping sub-channels to avoid inter-carrier interference (ICI)( )

loz The diagram for bandwidth efficiency of OFDM system is shown below

35

SummarySummaryyy

loz The advantage of the FFT-based OFDM system g yloz The use of IFFTFFT can reduce the computation complexityloz The orthogonality between the adjacent sub-carriers will make g y j

the use of transmission bandwidth more efficientloz The guard interval is used to resist the inter-symbol interference

(ISI)loz The main advantage of the OFDM transmission technique is its

high performance even in frequency selective channels

loz The drawbacks of the OFDM system loz It is highly vulnerable to synchronization errorsloz Peak to Average Power Ratio (PAPR) problems

36

SynchronizationSynchronization

Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offset (TO)loz Impact of timing offset (TO)loz Impact of carrier frequency offset (CFO)loz TO estimation algorithmloz CFO estimation algorithmg

38 Synchronization2010722

Introduction (13)Introduction (13)( )( )

loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods

loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead

loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the

cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of

computational complexity therefore it may be not be available in short-burst wireless communication

39 Synchronization2010722

Introduction (23)Introduction (23)( )( )

loz Transcevier

40 Synchronization2010722

Introduction (33)Introduction (33)( )( )

loz Synchronization erroryloz Symbol timing offset (TO)

loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)

loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect

41 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg

42 Synchronization2010722

Signal model (14)Signal model (14)g ( )g ( )

loz Time domain sample p1

2

0

1 Nj nk N

n kk

x X eN

πminus

minus= sum

Xk the frequency domain data at the k-th subcarrier0kN =

loz The multipath fading channel

( ) ( ) ( )1L

h t h t δminus

sum

loz Received time domain signal

( ) ( ) ( )0

l ll

h t h tτ τ δ τ τ=

= minussum

loz Received time-domain signal

( ) ( )1

L

n n ly h n l x w nminus

minus= +sum43

( ) ( )0l=sum

Synchronization2010722

Signal model (24)Signal model (24)g ( )g ( )

loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting

12

0

Nj kn N

k nn

Y y e πminus

minus

=

= sum

( )1 1

2

0

N Lj kn N

n l kn l

h n l x e Wπminus minus

minusminus

=

= +sumsum

loz The equation above could be expressed in matrix form

0n l

loz The equation above could be expressed in matrix form

= +Y HX W

44 Synchronization2010722

Signal model (34)Signal model (34)g ( )g ( )

H= = = = +Y Fy FCx FCF X HX W

loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal

y

H=H FCFcirculant matrix leading to be a diagonal matrix

=H FCF

( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0

h h L L hh h

⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥

( )( ) ( )

( ) ( )

1 10 1 0 0i

h Lh L h N L

⎢ ⎥⎢ ⎥minus minus⎢ ⎥

equiv= minus minus⎢ ⎥⎢ ⎥

C( ) ( )

( ) ( )

0 1 1 100

0 0 1 1 0

h L h N L

h N L L h N

⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

45

( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦

Synchronization2010722

Signal model (44)Signal model (44)g ( )g ( )

loz In a time-invariant channel the channel matrix H would be a diagonal matrix

k k k kY H X W= +

( )1 1

2

0 0

N Lj kl N

kn l

H h n l e πminus minus

minus

= =

= sumsum

loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI

k k k k kY H X I Wα= + +k k k k k

( ) ( )1 1 1

2 2 N N L

j m k Nj ml Nk mI X h n l e e ππ

minus minus minusminusminus⎛ ⎞

= ⎜ ⎟⎝ ⎠

sum sum sumICI

46

0 0 0m n lm k= = =ne

⎝ ⎠

Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

47 Synchronization2010722

Impact of timing offset Impact of timing offset p gp g

loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively

( )I f s

Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ

( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr

2 fj k Nk k k kY X H e Wπδminus= +

loz For integer part of timing offset k k k kY X H e W+

I sTδ ( ) 0g IN L δminus minus le leFFT

2 Ij k NY X H e Wπδminus= +

( ) ( ) 2 I sFFT

j f TI sh t T H f e π δδ minusminus rarr

48

k k k kY X H e W= +

Synchronization2010722

Impact of timing offset derivationImpact of timing offset derivationp gp g

49 Synchronization2010722

Discussion of symbol boundaryDiscussion of symbol boundaryy yy y

50 Synchronization2010722

Impact of timing offsetImpact of timing offsetp gp g

loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI

2 Ij k NIk k k k

NY X H e ISI W

Nπδδ minusminus

= + +

loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk

e e

p kloz Phase Shift can be resolved by Differential Encoding

Decoding or carrier recovery with pilots

51 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

52 Synchronization2010722

Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )

53 Synchronization2010722

Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )

( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s

i n t i N N T N T nTy y t e

= + + +=

( ) ( )( ) ( ) ( ) 12

sin

sin

g gI f f

I I

i N N N Nj jf N Ni k i k k

f

Y X H e eN

π ε ε π ε

ε ε

πεπε

+ + minus+

minus minus=⎛ ⎞⎜ ⎟

Faded signal attenuated and rotated by CFO

( )( ) ( ) ( ) ( )11 2

sin

sin g gI f I f

i N N N NN j j l kI f

NN

l k π ε ε π ε επ ε ε + + minusminus + + + minus

⎜ ⎟⎝ ⎠

+ + minussum

( )( )( )

( ) ( )2

0

sin

I f I f

I

j j l kI f N Ni l l

l l k I f

X H e el k

NN

π ε ε π ε ε

ε π ε ε

+ + +

= ne minus

+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠

sum

i k

N

W

⎜ ⎟⎝ ⎠

+ Inter-Carrier Interference (ICI)

54 Synchronization2010722

Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )

loz An easier representation of received signal with only p g yfractional CFO is

1N

( ) ( )1

0

0N

k k k l l kl l k

Y H X C H X C l k Wminus

= ne

= + minus +⎡ ⎤⎣ ⎦sum

where the ICI coefficient is given by

( )( )( ) ( )

sin 1exp 1fl k

C l k j l kπ ε

π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )

( )exp 1sin

f

f

C l k j l kNN l k

N

π επ ε

minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠

55 Synchronization2010722

Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )

loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality

loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift

56 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

57 Synchronization2010722

TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )

loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq

58 Synchronization2010722

TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )

loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below

loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and

rN2 can be expressed by

0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+

( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +

59

( )

Synchronization2010722

TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )

loz Without CFO the first half is identical to the second half (in time order)

loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below

( ) 12 in discretesft T f NT

NTφ π π π ε πε

⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟

⎝ ⎠

helliphellip

sNT⎝ ⎠

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)

60

Length=N2

Synchronization2010722

TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )

loz Received signal without CFOg0

0 0

jr a e θ=jθ 2 0

2 2 0

Nj j

N Nr a e a e

θ θ= =

loz Received signal with CFO0 0

0 0

j jr a e eθ φ= Extra phase rotation due to CFO0 0

2 2 20

2 2 0

N N Nj j jj

N Nr a e e a e e

θ φ φθ= =

due to CFO

Phase difference 2 2 0N N

( )2 02 20 2 0 0

Nj j T fN

r r a e a eφ φ πminus Δ= =

which contains the information about CFO

61

0 2 0 0Nr r a e a e

Synchronization2010722

TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )

loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be

loz Note that d is a time index corresponding to the first( ) ( ) 2 1

20

Nd m d m Nm

P d r rminus

+ + +== sum

loz Note that d is a time index corresponding to the first sample in a window of N samples

loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by

( )22 1

20

Nd m NR d rminus

+ += sumloz A timing metric can be defined as

))(()(

)( 2

2

dRdP

dM =

( ) 20 d m Nm + +=sum

62

))(( dR

Synchronization2010722

TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )

63 Synchronization2010722

TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )

loz If drsquo is the correct symbol timing offset

( )2

2 2

0 1 12

j T f j T f j T fN

P d a e a e a eπ π πΔ Δ Δ

minus= + + +

22

2 2

0 1 12

j T fN

e a a aπ Δ

minus

⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭

loz If drsquo falls behind the correct symbol timing offset 1 sample

2⎪ ⎪⎩ ⎭

64 Synchronization2010722

TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )

65 Synchronization2010722

TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )

loz Drawback Plateau effect

Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)

((N2) ((N2) ((N2)

helliphellip

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)

Sliding window (length = N)

66 Synchronization2010722

TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )

loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula

( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus

扣掉相加

( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +

loz The method also called delay correlator

67 Synchronization2010722

IEEE 80211aIEEE 80211a

68 Synchronization2010722

Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )

loz Minnrsquos Method [2][ ]B B －B －B

( )( )

( )

22

2Minn

P dM d =

1 4 1N minus

( )( )( )2

2

MinnR d

( ) ( ) ( ) ( )

1 4 1

2 2 2 40 0

N

d N m k d N m k Nm k

P d r r+ + + + += =

= sum sum

( ) ( ) ( )2

21 4 1

2 40 0

N

d N m k Nk

R d rminus

+ + += sum sum69

0 0m k= =

Synchronization2010722

Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )

loz Parkrsquos Method [3][ ]

where D is the symmetric version of CC D C D

where D is the symmetric version of C

( )( ) 2

3k

P dM d =( )

( )( )23

ParkM dR d

2N

( ) 2

30

N

d k d kk

P d r rminus +=

= sdotsum

( )3

22

0

N

d kk

R d r += sum70

0k=

Synchronization2010722

Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )

loz Renrsquos Method [4][ ]

AS AS

where S is the PN sequence weighted of the original bl 2preamble

( )( )( )( )

24

Re 24

n

P dM d

R d=

( )( )4R d

( ) 2 1

4 2 2

N

k k N d k d k NP d s s r rminus

+ + + += sum0k=sum

( )21

41 N

d kR d rminus

= sum71

( )402 d k

kR d r +

=sum

Synchronization2010722

Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )

72 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

73 Synchronization2010722

Fractional CFO estimation Fractional CFO estimation

loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected

11 Rminus⎛ ⎞sum

h L i th di t b t t id ti l bl k R i

0

12 d m d m L

ms

f r rLTπ + + +

=

⎛ ⎞= ang⎜ ⎟⎝ ⎠sum

where L is the distance between two identical block R is the block size

loz For exampleNN cp

L = N R = NL = N R = Ncp74 Synchronization2010722

Estimation rangeEstimation rangegg

loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval

f ( ) ( )s s⎣ ⎦

loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)

loz DVB-T (L=N) 802 11 (L N4)

[ ]05 05s sf fminus

[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)

(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus

loz Note that is the subcarrier spacing1sf NT=

75

sNT

Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz

plusmng q y

loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn

76 Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz

plusmng q y

loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for

subcarrier spacing is required

plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn

p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs

77 Synchronization2010722

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

Cyclic PrefixCyclic Prefix

loz Channel effect with cyclic prefix

yy

Signal after removed CP

29

Cyclic PrefixCyclic Prefix

loz Time-Domain Explanation

yy

p

30

Cyclic PrefixCyclic Prefixyy

31

Cyclic PrefixCyclic Prefix

loz Spectrum of channel response with length (smaller [ ]nh hL

yy

than )h

gN[ ] nhFFTHk =

loz Received complete OFDM signal[ ] [ ] [ ] 20 ~~ minus++leleotimes= hg LNNnnhnDnr

loz Received useful part [ ]nr[ ] [ ] [ ]nhnDnr Notimes=

where is N-point circular convolution (due to CP)loz Received symbol at k-th subcarrier

[ ] [ ] [ ]N

Notimes

[ ] [ ] [ ] kkNk HXnhnDFFTnrFFTY =otimes== kYX =rArr ldquoUseful property for OFDM system to reduce

kk H

X =rArrcomplexity of channel equalizationrdquo

32

Cyclic Cyclic Prefix (OFDM Prefix (OFDM Receiver)Receiver)yy (( ))

bull bull( )( )nr~Symbols

dataOutput

bullbullbull

bullbullbull( )nr

bullbull

bullbull

33

Cyclic PrefixCyclic Prefix

loz One of the most important reasons to do OFDM is the

yy

pefficient way it deals with multipath delay spread

loz To eliminate inter-symbol interference (ISI) almostloz To eliminate inter symbol interference (ISI) almost completely a guard time is introduced for each OFDM symbolsymbol(The guard time is chosen larger than the delay spread)

34

Bandwidth EfficiencyBandwidth Efficiencyyy

loz In a classical parallel system the channel is divided into p y N non-overlapping sub-channels to avoid inter-carrier interference (ICI)( )

loz The diagram for bandwidth efficiency of OFDM system is shown below

35

SummarySummaryyy

loz The advantage of the FFT-based OFDM system g yloz The use of IFFTFFT can reduce the computation complexityloz The orthogonality between the adjacent sub-carriers will make g y j

the use of transmission bandwidth more efficientloz The guard interval is used to resist the inter-symbol interference

(ISI)loz The main advantage of the OFDM transmission technique is its

high performance even in frequency selective channels

loz The drawbacks of the OFDM system loz It is highly vulnerable to synchronization errorsloz Peak to Average Power Ratio (PAPR) problems

36

SynchronizationSynchronization

Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offset (TO)loz Impact of timing offset (TO)loz Impact of carrier frequency offset (CFO)loz TO estimation algorithmloz CFO estimation algorithmg

38 Synchronization2010722

Introduction (13)Introduction (13)( )( )

loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods

loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead

loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the

cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of

computational complexity therefore it may be not be available in short-burst wireless communication

39 Synchronization2010722

Introduction (23)Introduction (23)( )( )

loz Transcevier

40 Synchronization2010722

Introduction (33)Introduction (33)( )( )

loz Synchronization erroryloz Symbol timing offset (TO)

loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)

loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect

41 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg

42 Synchronization2010722

Signal model (14)Signal model (14)g ( )g ( )

loz Time domain sample p1

2

0

1 Nj nk N

n kk

x X eN

πminus

minus= sum

Xk the frequency domain data at the k-th subcarrier0kN =

loz The multipath fading channel

( ) ( ) ( )1L

h t h t δminus

sum

loz Received time domain signal

( ) ( ) ( )0

l ll

h t h tτ τ δ τ τ=

= minussum

loz Received time-domain signal

( ) ( )1

L

n n ly h n l x w nminus

minus= +sum43

( ) ( )0l=sum

Synchronization2010722

Signal model (24)Signal model (24)g ( )g ( )

loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting

12

0

Nj kn N

k nn

Y y e πminus

minus

=

= sum

( )1 1

2

0

N Lj kn N

n l kn l

h n l x e Wπminus minus

minusminus

=

= +sumsum

loz The equation above could be expressed in matrix form

0n l

loz The equation above could be expressed in matrix form

= +Y HX W

44 Synchronization2010722

Signal model (34)Signal model (34)g ( )g ( )

H= = = = +Y Fy FCx FCF X HX W

loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal

y

H=H FCFcirculant matrix leading to be a diagonal matrix

=H FCF

( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0

h h L L hh h

⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥

( )( ) ( )

( ) ( )

1 10 1 0 0i

h Lh L h N L

⎢ ⎥⎢ ⎥minus minus⎢ ⎥

equiv= minus minus⎢ ⎥⎢ ⎥

C( ) ( )

( ) ( )

0 1 1 100

0 0 1 1 0

h L h N L

h N L L h N

⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

45

( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦

Synchronization2010722

Signal model (44)Signal model (44)g ( )g ( )

loz In a time-invariant channel the channel matrix H would be a diagonal matrix

k k k kY H X W= +

( )1 1

2

0 0

N Lj kl N

kn l

H h n l e πminus minus

minus

= =

= sumsum

loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI

k k k k kY H X I Wα= + +k k k k k

( ) ( )1 1 1

2 2 N N L

j m k Nj ml Nk mI X h n l e e ππ

minus minus minusminusminus⎛ ⎞

= ⎜ ⎟⎝ ⎠

sum sum sumICI

46

0 0 0m n lm k= = =ne

⎝ ⎠

Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

47 Synchronization2010722

Impact of timing offset Impact of timing offset p gp g

loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively

( )I f s

Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ

( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr

2 fj k Nk k k kY X H e Wπδminus= +

loz For integer part of timing offset k k k kY X H e W+

I sTδ ( ) 0g IN L δminus minus le leFFT

2 Ij k NY X H e Wπδminus= +

( ) ( ) 2 I sFFT

j f TI sh t T H f e π δδ minusminus rarr

48

k k k kY X H e W= +

Synchronization2010722

Impact of timing offset derivationImpact of timing offset derivationp gp g

49 Synchronization2010722

Discussion of symbol boundaryDiscussion of symbol boundaryy yy y

50 Synchronization2010722

Impact of timing offsetImpact of timing offsetp gp g

loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI

2 Ij k NIk k k k

NY X H e ISI W

Nπδδ minusminus

= + +

loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk

e e

p kloz Phase Shift can be resolved by Differential Encoding

Decoding or carrier recovery with pilots

51 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

52 Synchronization2010722

Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )

53 Synchronization2010722

Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )

( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s

i n t i N N T N T nTy y t e

= + + +=

( ) ( )( ) ( ) ( ) 12

sin

sin

g gI f f

I I

i N N N Nj jf N Ni k i k k

f

Y X H e eN

π ε ε π ε

ε ε

πεπε

+ + minus+

minus minus=⎛ ⎞⎜ ⎟

Faded signal attenuated and rotated by CFO

( )( ) ( ) ( ) ( )11 2

sin

sin g gI f I f

i N N N NN j j l kI f

NN

l k π ε ε π ε επ ε ε + + minusminus + + + minus

⎜ ⎟⎝ ⎠

+ + minussum

( )( )( )

( ) ( )2

0

sin

I f I f

I

j j l kI f N Ni l l

l l k I f

X H e el k

NN

π ε ε π ε ε

ε π ε ε

+ + +

= ne minus

+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠

sum

i k

N

W

⎜ ⎟⎝ ⎠

+ Inter-Carrier Interference (ICI)

54 Synchronization2010722

Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )

loz An easier representation of received signal with only p g yfractional CFO is

1N

( ) ( )1

0

0N

k k k l l kl l k

Y H X C H X C l k Wminus

= ne

= + minus +⎡ ⎤⎣ ⎦sum

where the ICI coefficient is given by

( )( )( ) ( )

sin 1exp 1fl k

C l k j l kπ ε

π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )

( )exp 1sin

f

f

C l k j l kNN l k

N

π επ ε

minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠

55 Synchronization2010722

Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )

loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality

loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift

56 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

57 Synchronization2010722

TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )

loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq

58 Synchronization2010722

TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )

loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below

loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and

rN2 can be expressed by

0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+

( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +

59

( )

Synchronization2010722

TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )

loz Without CFO the first half is identical to the second half (in time order)

loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below

( ) 12 in discretesft T f NT

NTφ π π π ε πε

⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟

⎝ ⎠

helliphellip

sNT⎝ ⎠

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)

60

Length=N2

Synchronization2010722

TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )

loz Received signal without CFOg0

0 0

jr a e θ=jθ 2 0

2 2 0

Nj j

N Nr a e a e

θ θ= =

loz Received signal with CFO0 0

0 0

j jr a e eθ φ= Extra phase rotation due to CFO0 0

2 2 20

2 2 0

N N Nj j jj

N Nr a e e a e e

θ φ φθ= =

due to CFO

Phase difference 2 2 0N N

( )2 02 20 2 0 0

Nj j T fN

r r a e a eφ φ πminus Δ= =

which contains the information about CFO

61

0 2 0 0Nr r a e a e

Synchronization2010722

TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )

loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be

loz Note that d is a time index corresponding to the first( ) ( ) 2 1

20

Nd m d m Nm

P d r rminus

+ + +== sum

loz Note that d is a time index corresponding to the first sample in a window of N samples

loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by

( )22 1

20

Nd m NR d rminus

+ += sumloz A timing metric can be defined as

))(()(

)( 2

2

dRdP

dM =

( ) 20 d m Nm + +=sum

62

))(( dR

Synchronization2010722

TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )

63 Synchronization2010722

TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )

loz If drsquo is the correct symbol timing offset

( )2

2 2

0 1 12

j T f j T f j T fN

P d a e a e a eπ π πΔ Δ Δ

minus= + + +

22

2 2

0 1 12

j T fN

e a a aπ Δ

minus

⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭

loz If drsquo falls behind the correct symbol timing offset 1 sample

2⎪ ⎪⎩ ⎭

64 Synchronization2010722

TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )

65 Synchronization2010722

TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )

loz Drawback Plateau effect

Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)

((N2) ((N2) ((N2)

helliphellip

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)

Sliding window (length = N)

66 Synchronization2010722

TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )

loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula

( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus

扣掉相加

( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +

loz The method also called delay correlator

67 Synchronization2010722

IEEE 80211aIEEE 80211a

68 Synchronization2010722

Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )

loz Minnrsquos Method [2][ ]B B －B －B

( )( )

( )

22

2Minn

P dM d =

1 4 1N minus

( )( )( )2

2

MinnR d

( ) ( ) ( ) ( )

1 4 1

2 2 2 40 0

N

d N m k d N m k Nm k

P d r r+ + + + += =

= sum sum

( ) ( ) ( )2

21 4 1

2 40 0

N

d N m k Nk

R d rminus

+ + += sum sum69

0 0m k= =

Synchronization2010722

Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )

loz Parkrsquos Method [3][ ]

where D is the symmetric version of CC D C D

where D is the symmetric version of C

( )( ) 2

3k

P dM d =( )

( )( )23

ParkM dR d

2N

( ) 2

30

N

d k d kk

P d r rminus +=

= sdotsum

( )3

22

0

N

d kk

R d r += sum70

0k=

Synchronization2010722

Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )

loz Renrsquos Method [4][ ]

AS AS

where S is the PN sequence weighted of the original bl 2preamble

( )( )( )( )

24

Re 24

n

P dM d

R d=

( )( )4R d

( ) 2 1

4 2 2

N

k k N d k d k NP d s s r rminus

+ + + += sum0k=sum

( )21

41 N

d kR d rminus

= sum71

( )402 d k

kR d r +

=sum

Synchronization2010722

Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )

72 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

73 Synchronization2010722

Fractional CFO estimation Fractional CFO estimation

loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected

11 Rminus⎛ ⎞sum

h L i th di t b t t id ti l bl k R i

0

12 d m d m L

ms

f r rLTπ + + +

=

⎛ ⎞= ang⎜ ⎟⎝ ⎠sum

where L is the distance between two identical block R is the block size

loz For exampleNN cp

L = N R = NL = N R = Ncp74 Synchronization2010722

Estimation rangeEstimation rangegg

loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval

f ( ) ( )s s⎣ ⎦

loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)

loz DVB-T (L=N) 802 11 (L N4)

[ ]05 05s sf fminus

[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)

(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus

loz Note that is the subcarrier spacing1sf NT=

75

sNT

Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz

plusmng q y

loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn

76 Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz

plusmng q y

loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for

subcarrier spacing is required

plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn

p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs

77 Synchronization2010722

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

Cyclic PrefixCyclic Prefix

loz Time-Domain Explanation

yy

p

30

Cyclic PrefixCyclic Prefixyy

31

Cyclic PrefixCyclic Prefix

loz Spectrum of channel response with length (smaller [ ]nh hL

yy

than )h

gN[ ] nhFFTHk =

loz Received complete OFDM signal[ ] [ ] [ ] 20 ~~ minus++leleotimes= hg LNNnnhnDnr

loz Received useful part [ ]nr[ ] [ ] [ ]nhnDnr Notimes=

where is N-point circular convolution (due to CP)loz Received symbol at k-th subcarrier

[ ] [ ] [ ]N

Notimes

[ ] [ ] [ ] kkNk HXnhnDFFTnrFFTY =otimes== kYX =rArr ldquoUseful property for OFDM system to reduce

kk H

X =rArrcomplexity of channel equalizationrdquo

32

Cyclic Cyclic Prefix (OFDM Prefix (OFDM Receiver)Receiver)yy (( ))

bull bull( )( )nr~Symbols

dataOutput

bullbullbull

bullbullbull( )nr

bullbull

bullbull

33

Cyclic PrefixCyclic Prefix

loz One of the most important reasons to do OFDM is the

yy

pefficient way it deals with multipath delay spread

loz To eliminate inter-symbol interference (ISI) almostloz To eliminate inter symbol interference (ISI) almost completely a guard time is introduced for each OFDM symbolsymbol(The guard time is chosen larger than the delay spread)

34

Bandwidth EfficiencyBandwidth Efficiencyyy

loz In a classical parallel system the channel is divided into p y N non-overlapping sub-channels to avoid inter-carrier interference (ICI)( )

loz The diagram for bandwidth efficiency of OFDM system is shown below

35

SummarySummaryyy

loz The advantage of the FFT-based OFDM system g yloz The use of IFFTFFT can reduce the computation complexityloz The orthogonality between the adjacent sub-carriers will make g y j

the use of transmission bandwidth more efficientloz The guard interval is used to resist the inter-symbol interference

(ISI)loz The main advantage of the OFDM transmission technique is its

high performance even in frequency selective channels

loz The drawbacks of the OFDM system loz It is highly vulnerable to synchronization errorsloz Peak to Average Power Ratio (PAPR) problems

36

SynchronizationSynchronization

Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offset (TO)loz Impact of timing offset (TO)loz Impact of carrier frequency offset (CFO)loz TO estimation algorithmloz CFO estimation algorithmg

38 Synchronization2010722

Introduction (13)Introduction (13)( )( )

loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods

loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead

loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the

cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of

computational complexity therefore it may be not be available in short-burst wireless communication

39 Synchronization2010722

Introduction (23)Introduction (23)( )( )

loz Transcevier

40 Synchronization2010722

Introduction (33)Introduction (33)( )( )

loz Synchronization erroryloz Symbol timing offset (TO)

loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)

loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect

41 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg

42 Synchronization2010722

Signal model (14)Signal model (14)g ( )g ( )

loz Time domain sample p1

2

0

1 Nj nk N

n kk

x X eN

πminus

minus= sum

Xk the frequency domain data at the k-th subcarrier0kN =

loz The multipath fading channel

( ) ( ) ( )1L

h t h t δminus

sum

loz Received time domain signal

( ) ( ) ( )0

l ll

h t h tτ τ δ τ τ=

= minussum

loz Received time-domain signal

( ) ( )1

L

n n ly h n l x w nminus

minus= +sum43

( ) ( )0l=sum

Synchronization2010722

Signal model (24)Signal model (24)g ( )g ( )

loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting

12

0

Nj kn N

k nn

Y y e πminus

minus

=

= sum

( )1 1

2

0

N Lj kn N

n l kn l

h n l x e Wπminus minus

minusminus

=

= +sumsum

loz The equation above could be expressed in matrix form

0n l

loz The equation above could be expressed in matrix form

= +Y HX W

44 Synchronization2010722

Signal model (34)Signal model (34)g ( )g ( )

H= = = = +Y Fy FCx FCF X HX W

loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal

y

H=H FCFcirculant matrix leading to be a diagonal matrix

=H FCF

( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0

h h L L hh h

⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥

( )( ) ( )

( ) ( )

1 10 1 0 0i

h Lh L h N L

⎢ ⎥⎢ ⎥minus minus⎢ ⎥

equiv= minus minus⎢ ⎥⎢ ⎥

C( ) ( )

( ) ( )

0 1 1 100

0 0 1 1 0

h L h N L

h N L L h N

⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

45

( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦

Synchronization2010722

Signal model (44)Signal model (44)g ( )g ( )

loz In a time-invariant channel the channel matrix H would be a diagonal matrix

k k k kY H X W= +

( )1 1

2

0 0

N Lj kl N

kn l

H h n l e πminus minus

minus

= =

= sumsum

loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI

k k k k kY H X I Wα= + +k k k k k

( ) ( )1 1 1

2 2 N N L

j m k Nj ml Nk mI X h n l e e ππ

minus minus minusminusminus⎛ ⎞

= ⎜ ⎟⎝ ⎠

sum sum sumICI

46

0 0 0m n lm k= = =ne

⎝ ⎠

Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

47 Synchronization2010722

Impact of timing offset Impact of timing offset p gp g

loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively

( )I f s

Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ

( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr

2 fj k Nk k k kY X H e Wπδminus= +

loz For integer part of timing offset k k k kY X H e W+

I sTδ ( ) 0g IN L δminus minus le leFFT

2 Ij k NY X H e Wπδminus= +

( ) ( ) 2 I sFFT

j f TI sh t T H f e π δδ minusminus rarr

48

k k k kY X H e W= +

Synchronization2010722

Impact of timing offset derivationImpact of timing offset derivationp gp g

49 Synchronization2010722

Discussion of symbol boundaryDiscussion of symbol boundaryy yy y

50 Synchronization2010722

Impact of timing offsetImpact of timing offsetp gp g

loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI

2 Ij k NIk k k k

NY X H e ISI W

Nπδδ minusminus

= + +

loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk

e e

p kloz Phase Shift can be resolved by Differential Encoding

Decoding or carrier recovery with pilots

51 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

52 Synchronization2010722

Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )

53 Synchronization2010722

Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )

( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s

i n t i N N T N T nTy y t e

= + + +=

( ) ( )( ) ( ) ( ) 12

sin

sin

g gI f f

I I

i N N N Nj jf N Ni k i k k

f

Y X H e eN

π ε ε π ε

ε ε

πεπε

+ + minus+

minus minus=⎛ ⎞⎜ ⎟

Faded signal attenuated and rotated by CFO

( )( ) ( ) ( ) ( )11 2

sin

sin g gI f I f

i N N N NN j j l kI f

NN

l k π ε ε π ε επ ε ε + + minusminus + + + minus

⎜ ⎟⎝ ⎠

+ + minussum

( )( )( )

( ) ( )2

0

sin

I f I f

I

j j l kI f N Ni l l

l l k I f

X H e el k

NN

π ε ε π ε ε

ε π ε ε

+ + +

= ne minus

+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠

sum

i k

N

W

⎜ ⎟⎝ ⎠

+ Inter-Carrier Interference (ICI)

54 Synchronization2010722

Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )

loz An easier representation of received signal with only p g yfractional CFO is

1N

( ) ( )1

0

0N

k k k l l kl l k

Y H X C H X C l k Wminus

= ne

= + minus +⎡ ⎤⎣ ⎦sum

where the ICI coefficient is given by

( )( )( ) ( )

sin 1exp 1fl k

C l k j l kπ ε

π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )

( )exp 1sin

f

f

C l k j l kNN l k

N

π επ ε

minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠

55 Synchronization2010722

Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )

loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality

loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift

56 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

57 Synchronization2010722

TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )

loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq

58 Synchronization2010722

TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )

loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below

loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and

rN2 can be expressed by

0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+

( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +

59

( )

Synchronization2010722

TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )

loz Without CFO the first half is identical to the second half (in time order)

loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below

( ) 12 in discretesft T f NT

NTφ π π π ε πε

⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟

⎝ ⎠

helliphellip

sNT⎝ ⎠

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)

60

Length=N2

Synchronization2010722

TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )

loz Received signal without CFOg0

0 0

jr a e θ=jθ 2 0

2 2 0

Nj j

N Nr a e a e

θ θ= =

loz Received signal with CFO0 0

0 0

j jr a e eθ φ= Extra phase rotation due to CFO0 0

2 2 20

2 2 0

N N Nj j jj

N Nr a e e a e e

θ φ φθ= =

due to CFO

Phase difference 2 2 0N N

( )2 02 20 2 0 0

Nj j T fN

r r a e a eφ φ πminus Δ= =

which contains the information about CFO

61

0 2 0 0Nr r a e a e

Synchronization2010722

TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )

loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be

loz Note that d is a time index corresponding to the first( ) ( ) 2 1

20

Nd m d m Nm

P d r rminus

+ + +== sum

loz Note that d is a time index corresponding to the first sample in a window of N samples

loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by

( )22 1

20

Nd m NR d rminus

+ += sumloz A timing metric can be defined as

))(()(

)( 2

2

dRdP

dM =

( ) 20 d m Nm + +=sum

62

))(( dR

Synchronization2010722

TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )

63 Synchronization2010722

TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )

loz If drsquo is the correct symbol timing offset

( )2

2 2

0 1 12

j T f j T f j T fN

P d a e a e a eπ π πΔ Δ Δ

minus= + + +

22

2 2

0 1 12

j T fN

e a a aπ Δ

minus

⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭

loz If drsquo falls behind the correct symbol timing offset 1 sample

2⎪ ⎪⎩ ⎭

64 Synchronization2010722

TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )

65 Synchronization2010722

TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )

loz Drawback Plateau effect

Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)

((N2) ((N2) ((N2)

helliphellip

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)

Sliding window (length = N)

66 Synchronization2010722

TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )

loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula

( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus

扣掉相加

( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +

loz The method also called delay correlator

67 Synchronization2010722

IEEE 80211aIEEE 80211a

68 Synchronization2010722

Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )

loz Minnrsquos Method [2][ ]B B －B －B

( )( )

( )

22

2Minn

P dM d =

1 4 1N minus

( )( )( )2

2

MinnR d

( ) ( ) ( ) ( )

1 4 1

2 2 2 40 0

N

d N m k d N m k Nm k

P d r r+ + + + += =

= sum sum

( ) ( ) ( )2

21 4 1

2 40 0

N

d N m k Nk

R d rminus

+ + += sum sum69

0 0m k= =

Synchronization2010722

Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )

loz Parkrsquos Method [3][ ]

where D is the symmetric version of CC D C D

where D is the symmetric version of C

( )( ) 2

3k

P dM d =( )

( )( )23

ParkM dR d

2N

( ) 2

30

N

d k d kk

P d r rminus +=

= sdotsum

( )3

22

0

N

d kk

R d r += sum70

0k=

Synchronization2010722

Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )

loz Renrsquos Method [4][ ]

AS AS

where S is the PN sequence weighted of the original bl 2preamble

( )( )( )( )

24

Re 24

n

P dM d

R d=

( )( )4R d

( ) 2 1

4 2 2

N

k k N d k d k NP d s s r rminus

+ + + += sum0k=sum

( )21

41 N

d kR d rminus

= sum71

( )402 d k

kR d r +

=sum

Synchronization2010722

Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )

72 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

73 Synchronization2010722

Fractional CFO estimation Fractional CFO estimation

loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected

11 Rminus⎛ ⎞sum

h L i th di t b t t id ti l bl k R i

0

12 d m d m L

ms

f r rLTπ + + +

=

⎛ ⎞= ang⎜ ⎟⎝ ⎠sum

where L is the distance between two identical block R is the block size

loz For exampleNN cp

L = N R = NL = N R = Ncp74 Synchronization2010722

Estimation rangeEstimation rangegg

loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval

f ( ) ( )s s⎣ ⎦

loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)

loz DVB-T (L=N) 802 11 (L N4)

[ ]05 05s sf fminus

[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)

(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus

loz Note that is the subcarrier spacing1sf NT=

75

sNT

Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz

plusmng q y

loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn

76 Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz

plusmng q y

loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for

subcarrier spacing is required

plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn

p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs

77 Synchronization2010722

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

Cyclic PrefixCyclic Prefixyy

31

Cyclic PrefixCyclic Prefix

loz Spectrum of channel response with length (smaller [ ]nh hL

yy

than )h

gN[ ] nhFFTHk =

loz Received complete OFDM signal[ ] [ ] [ ] 20 ~~ minus++leleotimes= hg LNNnnhnDnr

loz Received useful part [ ]nr[ ] [ ] [ ]nhnDnr Notimes=

where is N-point circular convolution (due to CP)loz Received symbol at k-th subcarrier

[ ] [ ] [ ]N

Notimes

[ ] [ ] [ ] kkNk HXnhnDFFTnrFFTY =otimes== kYX =rArr ldquoUseful property for OFDM system to reduce

kk H

X =rArrcomplexity of channel equalizationrdquo

32

Cyclic Cyclic Prefix (OFDM Prefix (OFDM Receiver)Receiver)yy (( ))

bull bull( )( )nr~Symbols

dataOutput

bullbullbull

bullbullbull( )nr

bullbull

bullbull

33

Cyclic PrefixCyclic Prefix

loz One of the most important reasons to do OFDM is the

yy

pefficient way it deals with multipath delay spread

loz To eliminate inter-symbol interference (ISI) almostloz To eliminate inter symbol interference (ISI) almost completely a guard time is introduced for each OFDM symbolsymbol(The guard time is chosen larger than the delay spread)

34

Bandwidth EfficiencyBandwidth Efficiencyyy

loz In a classical parallel system the channel is divided into p y N non-overlapping sub-channels to avoid inter-carrier interference (ICI)( )

loz The diagram for bandwidth efficiency of OFDM system is shown below

35

SummarySummaryyy

loz The advantage of the FFT-based OFDM system g yloz The use of IFFTFFT can reduce the computation complexityloz The orthogonality between the adjacent sub-carriers will make g y j

the use of transmission bandwidth more efficientloz The guard interval is used to resist the inter-symbol interference

(ISI)loz The main advantage of the OFDM transmission technique is its

high performance even in frequency selective channels

loz The drawbacks of the OFDM system loz It is highly vulnerable to synchronization errorsloz Peak to Average Power Ratio (PAPR) problems

36

SynchronizationSynchronization

Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offset (TO)loz Impact of timing offset (TO)loz Impact of carrier frequency offset (CFO)loz TO estimation algorithmloz CFO estimation algorithmg

38 Synchronization2010722

Introduction (13)Introduction (13)( )( )

loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods

loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead

loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the

cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of

computational complexity therefore it may be not be available in short-burst wireless communication

39 Synchronization2010722

Introduction (23)Introduction (23)( )( )

loz Transcevier

40 Synchronization2010722

Introduction (33)Introduction (33)( )( )

loz Synchronization erroryloz Symbol timing offset (TO)

loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)

loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect

41 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg

42 Synchronization2010722

Signal model (14)Signal model (14)g ( )g ( )

loz Time domain sample p1

2

0

1 Nj nk N

n kk

x X eN

πminus

minus= sum

Xk the frequency domain data at the k-th subcarrier0kN =

loz The multipath fading channel

( ) ( ) ( )1L

h t h t δminus

sum

loz Received time domain signal

( ) ( ) ( )0

l ll

h t h tτ τ δ τ τ=

= minussum

loz Received time-domain signal

( ) ( )1

L

n n ly h n l x w nminus

minus= +sum43

( ) ( )0l=sum

Synchronization2010722

Signal model (24)Signal model (24)g ( )g ( )

loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting

12

0

Nj kn N

k nn

Y y e πminus

minus

=

= sum

( )1 1

2

0

N Lj kn N

n l kn l

h n l x e Wπminus minus

minusminus

=

= +sumsum

loz The equation above could be expressed in matrix form

0n l

loz The equation above could be expressed in matrix form

= +Y HX W

44 Synchronization2010722

Signal model (34)Signal model (34)g ( )g ( )

H= = = = +Y Fy FCx FCF X HX W

loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal

y

H=H FCFcirculant matrix leading to be a diagonal matrix

=H FCF

( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0

h h L L hh h

⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥

( )( ) ( )

( ) ( )

1 10 1 0 0i

h Lh L h N L

⎢ ⎥⎢ ⎥minus minus⎢ ⎥

equiv= minus minus⎢ ⎥⎢ ⎥

C( ) ( )

( ) ( )

0 1 1 100

0 0 1 1 0

h L h N L

h N L L h N

⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

45

( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦

Synchronization2010722

Signal model (44)Signal model (44)g ( )g ( )

loz In a time-invariant channel the channel matrix H would be a diagonal matrix

k k k kY H X W= +

( )1 1

2

0 0

N Lj kl N

kn l

H h n l e πminus minus

minus

= =

= sumsum

loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI

k k k k kY H X I Wα= + +k k k k k

( ) ( )1 1 1

2 2 N N L

j m k Nj ml Nk mI X h n l e e ππ

minus minus minusminusminus⎛ ⎞

= ⎜ ⎟⎝ ⎠

sum sum sumICI

46

0 0 0m n lm k= = =ne

⎝ ⎠

Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

47 Synchronization2010722

Impact of timing offset Impact of timing offset p gp g

loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively

( )I f s

Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ

( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr

2 fj k Nk k k kY X H e Wπδminus= +

loz For integer part of timing offset k k k kY X H e W+

I sTδ ( ) 0g IN L δminus minus le leFFT

2 Ij k NY X H e Wπδminus= +

( ) ( ) 2 I sFFT

j f TI sh t T H f e π δδ minusminus rarr

48

k k k kY X H e W= +

Synchronization2010722

Impact of timing offset derivationImpact of timing offset derivationp gp g

49 Synchronization2010722

Discussion of symbol boundaryDiscussion of symbol boundaryy yy y

50 Synchronization2010722

Impact of timing offsetImpact of timing offsetp gp g

loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI

2 Ij k NIk k k k

NY X H e ISI W

Nπδδ minusminus

= + +

loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk

e e

p kloz Phase Shift can be resolved by Differential Encoding

Decoding or carrier recovery with pilots

51 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

52 Synchronization2010722

Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )

53 Synchronization2010722

Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )

( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s

i n t i N N T N T nTy y t e

= + + +=

( ) ( )( ) ( ) ( ) 12

sin

sin

g gI f f

I I

i N N N Nj jf N Ni k i k k

f

Y X H e eN

π ε ε π ε

ε ε

πεπε

+ + minus+

minus minus=⎛ ⎞⎜ ⎟

Faded signal attenuated and rotated by CFO

( )( ) ( ) ( ) ( )11 2

sin

sin g gI f I f

i N N N NN j j l kI f

NN

l k π ε ε π ε επ ε ε + + minusminus + + + minus

⎜ ⎟⎝ ⎠

+ + minussum

( )( )( )

( ) ( )2

0

sin

I f I f

I

j j l kI f N Ni l l

l l k I f

X H e el k

NN

π ε ε π ε ε

ε π ε ε

+ + +

= ne minus

+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠

sum

i k

N

W

⎜ ⎟⎝ ⎠

+ Inter-Carrier Interference (ICI)

54 Synchronization2010722

Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )

loz An easier representation of received signal with only p g yfractional CFO is

1N

( ) ( )1

0

0N

k k k l l kl l k

Y H X C H X C l k Wminus

= ne

= + minus +⎡ ⎤⎣ ⎦sum

where the ICI coefficient is given by

( )( )( ) ( )

sin 1exp 1fl k

C l k j l kπ ε

π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )

( )exp 1sin

f

f

C l k j l kNN l k

N

π επ ε

minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠

55 Synchronization2010722

Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )

loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality

loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift

56 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

57 Synchronization2010722

TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )

loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq

58 Synchronization2010722

TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )

loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below

loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and

rN2 can be expressed by

0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+

( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +

59

( )

Synchronization2010722

TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )

loz Without CFO the first half is identical to the second half (in time order)

loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below

( ) 12 in discretesft T f NT

NTφ π π π ε πε

⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟

⎝ ⎠

helliphellip

sNT⎝ ⎠

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)

60

Length=N2

Synchronization2010722

TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )

loz Received signal without CFOg0

0 0

jr a e θ=jθ 2 0

2 2 0

Nj j

N Nr a e a e

θ θ= =

loz Received signal with CFO0 0

0 0

j jr a e eθ φ= Extra phase rotation due to CFO0 0

2 2 20

2 2 0

N N Nj j jj

N Nr a e e a e e

θ φ φθ= =

due to CFO

Phase difference 2 2 0N N

( )2 02 20 2 0 0

Nj j T fN

r r a e a eφ φ πminus Δ= =

which contains the information about CFO

61

0 2 0 0Nr r a e a e

Synchronization2010722

TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )

loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be

loz Note that d is a time index corresponding to the first( ) ( ) 2 1

20

Nd m d m Nm

P d r rminus

+ + +== sum

loz Note that d is a time index corresponding to the first sample in a window of N samples

loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by

( )22 1

20

Nd m NR d rminus

+ += sumloz A timing metric can be defined as

))(()(

)( 2

2

dRdP

dM =

( ) 20 d m Nm + +=sum

62

))(( dR

Synchronization2010722

TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )

63 Synchronization2010722

TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )

loz If drsquo is the correct symbol timing offset

( )2

2 2

0 1 12

j T f j T f j T fN

P d a e a e a eπ π πΔ Δ Δ

minus= + + +

22

2 2

0 1 12

j T fN

e a a aπ Δ

minus

⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭

loz If drsquo falls behind the correct symbol timing offset 1 sample

2⎪ ⎪⎩ ⎭

64 Synchronization2010722

TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )

65 Synchronization2010722

TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )

loz Drawback Plateau effect

Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)

((N2) ((N2) ((N2)

helliphellip

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)

Sliding window (length = N)

66 Synchronization2010722

TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )

loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula

( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus

扣掉相加

( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +

loz The method also called delay correlator

67 Synchronization2010722

IEEE 80211aIEEE 80211a

68 Synchronization2010722

Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )

loz Minnrsquos Method [2][ ]B B －B －B

( )( )

( )

22

2Minn

P dM d =

1 4 1N minus

( )( )( )2

2

MinnR d

( ) ( ) ( ) ( )

1 4 1

2 2 2 40 0

N

d N m k d N m k Nm k

P d r r+ + + + += =

= sum sum

( ) ( ) ( )2

21 4 1

2 40 0

N

d N m k Nk

R d rminus

+ + += sum sum69

0 0m k= =

Synchronization2010722

Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )

loz Parkrsquos Method [3][ ]

where D is the symmetric version of CC D C D

where D is the symmetric version of C

( )( ) 2

3k

P dM d =( )

( )( )23

ParkM dR d

2N

( ) 2

30

N

d k d kk

P d r rminus +=

= sdotsum

( )3

22

0

N

d kk

R d r += sum70

0k=

Synchronization2010722

Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )

loz Renrsquos Method [4][ ]

AS AS

where S is the PN sequence weighted of the original bl 2preamble

( )( )( )( )

24

Re 24

n

P dM d

R d=

( )( )4R d

( ) 2 1

4 2 2

N

k k N d k d k NP d s s r rminus

+ + + += sum0k=sum

( )21

41 N

d kR d rminus

= sum71

( )402 d k

kR d r +

=sum

Synchronization2010722

Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )

72 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

73 Synchronization2010722

Fractional CFO estimation Fractional CFO estimation

loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected

11 Rminus⎛ ⎞sum

h L i th di t b t t id ti l bl k R i

0

12 d m d m L

ms

f r rLTπ + + +

=

⎛ ⎞= ang⎜ ⎟⎝ ⎠sum

where L is the distance between two identical block R is the block size

loz For exampleNN cp

L = N R = NL = N R = Ncp74 Synchronization2010722

Estimation rangeEstimation rangegg

loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval

f ( ) ( )s s⎣ ⎦

loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)

loz DVB-T (L=N) 802 11 (L N4)

[ ]05 05s sf fminus

[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)

(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus

loz Note that is the subcarrier spacing1sf NT=

75

sNT

Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz

plusmng q y

loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn

76 Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz

plusmng q y

loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for

subcarrier spacing is required

plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn

p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs

77 Synchronization2010722

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

Cyclic PrefixCyclic Prefix

loz Spectrum of channel response with length (smaller [ ]nh hL

yy

than )h

gN[ ] nhFFTHk =

loz Received complete OFDM signal[ ] [ ] [ ] 20 ~~ minus++leleotimes= hg LNNnnhnDnr

loz Received useful part [ ]nr[ ] [ ] [ ]nhnDnr Notimes=

where is N-point circular convolution (due to CP)loz Received symbol at k-th subcarrier

[ ] [ ] [ ]N

Notimes

[ ] [ ] [ ] kkNk HXnhnDFFTnrFFTY =otimes== kYX =rArr ldquoUseful property for OFDM system to reduce

kk H

X =rArrcomplexity of channel equalizationrdquo

32

Cyclic Cyclic Prefix (OFDM Prefix (OFDM Receiver)Receiver)yy (( ))

bull bull( )( )nr~Symbols

dataOutput

bullbullbull

bullbullbull( )nr

bullbull

bullbull

33

Cyclic PrefixCyclic Prefix

loz One of the most important reasons to do OFDM is the

yy

pefficient way it deals with multipath delay spread

loz To eliminate inter-symbol interference (ISI) almostloz To eliminate inter symbol interference (ISI) almost completely a guard time is introduced for each OFDM symbolsymbol(The guard time is chosen larger than the delay spread)

34

Bandwidth EfficiencyBandwidth Efficiencyyy

loz In a classical parallel system the channel is divided into p y N non-overlapping sub-channels to avoid inter-carrier interference (ICI)( )

loz The diagram for bandwidth efficiency of OFDM system is shown below

35

SummarySummaryyy

loz The advantage of the FFT-based OFDM system g yloz The use of IFFTFFT can reduce the computation complexityloz The orthogonality between the adjacent sub-carriers will make g y j

the use of transmission bandwidth more efficientloz The guard interval is used to resist the inter-symbol interference

(ISI)loz The main advantage of the OFDM transmission technique is its

high performance even in frequency selective channels

loz The drawbacks of the OFDM system loz It is highly vulnerable to synchronization errorsloz Peak to Average Power Ratio (PAPR) problems

36

SynchronizationSynchronization

Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offset (TO)loz Impact of timing offset (TO)loz Impact of carrier frequency offset (CFO)loz TO estimation algorithmloz CFO estimation algorithmg

38 Synchronization2010722

Introduction (13)Introduction (13)( )( )

loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods

loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead

loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the

cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of

computational complexity therefore it may be not be available in short-burst wireless communication

39 Synchronization2010722

Introduction (23)Introduction (23)( )( )

loz Transcevier

40 Synchronization2010722

Introduction (33)Introduction (33)( )( )

loz Synchronization erroryloz Symbol timing offset (TO)

loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)

loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect

41 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg

42 Synchronization2010722

Signal model (14)Signal model (14)g ( )g ( )

loz Time domain sample p1

2

0

1 Nj nk N

n kk

x X eN

πminus

minus= sum

Xk the frequency domain data at the k-th subcarrier0kN =

loz The multipath fading channel

( ) ( ) ( )1L

h t h t δminus

sum

loz Received time domain signal

( ) ( ) ( )0

l ll

h t h tτ τ δ τ τ=

= minussum

loz Received time-domain signal

( ) ( )1

L

n n ly h n l x w nminus

minus= +sum43

( ) ( )0l=sum

Synchronization2010722

Signal model (24)Signal model (24)g ( )g ( )

loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting

12

0

Nj kn N

k nn

Y y e πminus

minus

=

= sum

( )1 1

2

0

N Lj kn N

n l kn l

h n l x e Wπminus minus

minusminus

=

= +sumsum

loz The equation above could be expressed in matrix form

0n l

loz The equation above could be expressed in matrix form

= +Y HX W

44 Synchronization2010722

Signal model (34)Signal model (34)g ( )g ( )

H= = = = +Y Fy FCx FCF X HX W

loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal

y

H=H FCFcirculant matrix leading to be a diagonal matrix

=H FCF

( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0

h h L L hh h

⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥

( )( ) ( )

( ) ( )

1 10 1 0 0i

h Lh L h N L

⎢ ⎥⎢ ⎥minus minus⎢ ⎥

equiv= minus minus⎢ ⎥⎢ ⎥

C( ) ( )

( ) ( )

0 1 1 100

0 0 1 1 0

h L h N L

h N L L h N

⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

45

( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦

Synchronization2010722

Signal model (44)Signal model (44)g ( )g ( )

loz In a time-invariant channel the channel matrix H would be a diagonal matrix

k k k kY H X W= +

( )1 1

2

0 0

N Lj kl N

kn l

H h n l e πminus minus

minus

= =

= sumsum

loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI

k k k k kY H X I Wα= + +k k k k k

( ) ( )1 1 1

2 2 N N L

j m k Nj ml Nk mI X h n l e e ππ

minus minus minusminusminus⎛ ⎞

= ⎜ ⎟⎝ ⎠

sum sum sumICI

46

0 0 0m n lm k= = =ne

⎝ ⎠

Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

47 Synchronization2010722

Impact of timing offset Impact of timing offset p gp g

loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively

( )I f s

Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ

( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr

2 fj k Nk k k kY X H e Wπδminus= +

loz For integer part of timing offset k k k kY X H e W+

I sTδ ( ) 0g IN L δminus minus le leFFT

2 Ij k NY X H e Wπδminus= +

( ) ( ) 2 I sFFT

j f TI sh t T H f e π δδ minusminus rarr

48

k k k kY X H e W= +

Synchronization2010722

Impact of timing offset derivationImpact of timing offset derivationp gp g

49 Synchronization2010722

Discussion of symbol boundaryDiscussion of symbol boundaryy yy y

50 Synchronization2010722

Impact of timing offsetImpact of timing offsetp gp g

loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI

2 Ij k NIk k k k

NY X H e ISI W

Nπδδ minusminus

= + +

loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk

e e

p kloz Phase Shift can be resolved by Differential Encoding

Decoding or carrier recovery with pilots

51 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

52 Synchronization2010722

Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )

53 Synchronization2010722

Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )

( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s

i n t i N N T N T nTy y t e

= + + +=

( ) ( )( ) ( ) ( ) 12

sin

sin

g gI f f

I I

i N N N Nj jf N Ni k i k k

f

Y X H e eN

π ε ε π ε

ε ε

πεπε

+ + minus+

minus minus=⎛ ⎞⎜ ⎟

Faded signal attenuated and rotated by CFO

( )( ) ( ) ( ) ( )11 2

sin

sin g gI f I f

i N N N NN j j l kI f

NN

l k π ε ε π ε επ ε ε + + minusminus + + + minus

⎜ ⎟⎝ ⎠

+ + minussum

( )( )( )

( ) ( )2

0

sin

I f I f

I

j j l kI f N Ni l l

l l k I f

X H e el k

NN

π ε ε π ε ε

ε π ε ε

+ + +

= ne minus

+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠

sum

i k

N

W

⎜ ⎟⎝ ⎠

+ Inter-Carrier Interference (ICI)

54 Synchronization2010722

Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )

loz An easier representation of received signal with only p g yfractional CFO is

1N

( ) ( )1

0

0N

k k k l l kl l k

Y H X C H X C l k Wminus

= ne

= + minus +⎡ ⎤⎣ ⎦sum

where the ICI coefficient is given by

( )( )( ) ( )

sin 1exp 1fl k

C l k j l kπ ε

π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )

( )exp 1sin

f

f

C l k j l kNN l k

N

π επ ε

minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠

55 Synchronization2010722

Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )

loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality

loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift

56 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

57 Synchronization2010722

TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )

loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq

58 Synchronization2010722

TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )

loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below

loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and

rN2 can be expressed by

0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+

( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +

59

( )

Synchronization2010722

TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )

loz Without CFO the first half is identical to the second half (in time order)

loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below

( ) 12 in discretesft T f NT

NTφ π π π ε πε

⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟

⎝ ⎠

helliphellip

sNT⎝ ⎠

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)

60

Length=N2

Synchronization2010722

TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )

loz Received signal without CFOg0

0 0

jr a e θ=jθ 2 0

2 2 0

Nj j

N Nr a e a e

θ θ= =

loz Received signal with CFO0 0

0 0

j jr a e eθ φ= Extra phase rotation due to CFO0 0

2 2 20

2 2 0

N N Nj j jj

N Nr a e e a e e

θ φ φθ= =

due to CFO

Phase difference 2 2 0N N

( )2 02 20 2 0 0

Nj j T fN

r r a e a eφ φ πminus Δ= =

which contains the information about CFO

61

0 2 0 0Nr r a e a e

Synchronization2010722

TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )

loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be

loz Note that d is a time index corresponding to the first( ) ( ) 2 1

20

Nd m d m Nm

P d r rminus

+ + +== sum

loz Note that d is a time index corresponding to the first sample in a window of N samples

loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by

( )22 1

20

Nd m NR d rminus

+ += sumloz A timing metric can be defined as

))(()(

)( 2

2

dRdP

dM =

( ) 20 d m Nm + +=sum

62

))(( dR

Synchronization2010722

TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )

63 Synchronization2010722

TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )

loz If drsquo is the correct symbol timing offset

( )2

2 2

0 1 12

j T f j T f j T fN

P d a e a e a eπ π πΔ Δ Δ

minus= + + +

22

2 2

0 1 12

j T fN

e a a aπ Δ

minus

⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭

loz If drsquo falls behind the correct symbol timing offset 1 sample

2⎪ ⎪⎩ ⎭

64 Synchronization2010722

TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )

65 Synchronization2010722

TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )

loz Drawback Plateau effect

Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)

((N2) ((N2) ((N2)

helliphellip

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)

Sliding window (length = N)

66 Synchronization2010722

TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )

loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula

( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus

扣掉相加

( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +

loz The method also called delay correlator

67 Synchronization2010722

IEEE 80211aIEEE 80211a

68 Synchronization2010722

Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )

loz Minnrsquos Method [2][ ]B B －B －B

( )( )

( )

22

2Minn

P dM d =

1 4 1N minus

( )( )( )2

2

MinnR d

( ) ( ) ( ) ( )

1 4 1

2 2 2 40 0

N

d N m k d N m k Nm k

P d r r+ + + + += =

= sum sum

( ) ( ) ( )2

21 4 1

2 40 0

N

d N m k Nk

R d rminus

+ + += sum sum69

0 0m k= =

Synchronization2010722

Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )

loz Parkrsquos Method [3][ ]

where D is the symmetric version of CC D C D

where D is the symmetric version of C

( )( ) 2

3k

P dM d =( )

( )( )23

ParkM dR d

2N

( ) 2

30

N

d k d kk

P d r rminus +=

= sdotsum

( )3

22

0

N

d kk

R d r += sum70

0k=

Synchronization2010722

Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )

loz Renrsquos Method [4][ ]

AS AS

where S is the PN sequence weighted of the original bl 2preamble

( )( )( )( )

24

Re 24

n

P dM d

R d=

( )( )4R d

( ) 2 1

4 2 2

N

k k N d k d k NP d s s r rminus

+ + + += sum0k=sum

( )21

41 N

d kR d rminus

= sum71

( )402 d k

kR d r +

=sum

Synchronization2010722

Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )

72 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

73 Synchronization2010722

Fractional CFO estimation Fractional CFO estimation

loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected

11 Rminus⎛ ⎞sum

h L i th di t b t t id ti l bl k R i

0

12 d m d m L

ms

f r rLTπ + + +

=

⎛ ⎞= ang⎜ ⎟⎝ ⎠sum

where L is the distance between two identical block R is the block size

loz For exampleNN cp

L = N R = NL = N R = Ncp74 Synchronization2010722

Estimation rangeEstimation rangegg

loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval

f ( ) ( )s s⎣ ⎦

loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)

loz DVB-T (L=N) 802 11 (L N4)

[ ]05 05s sf fminus

[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)

(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus

loz Note that is the subcarrier spacing1sf NT=

75

sNT

Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz

plusmng q y

loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn

76 Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz

plusmng q y

loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for

subcarrier spacing is required

plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn

p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs

77 Synchronization2010722

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

Cyclic Cyclic Prefix (OFDM Prefix (OFDM Receiver)Receiver)yy (( ))

bull bull( )( )nr~Symbols

dataOutput

bullbullbull

bullbullbull( )nr

bullbull

bullbull

33

Cyclic PrefixCyclic Prefix

loz One of the most important reasons to do OFDM is the

yy

pefficient way it deals with multipath delay spread

loz To eliminate inter-symbol interference (ISI) almostloz To eliminate inter symbol interference (ISI) almost completely a guard time is introduced for each OFDM symbolsymbol(The guard time is chosen larger than the delay spread)

34

Bandwidth EfficiencyBandwidth Efficiencyyy

loz In a classical parallel system the channel is divided into p y N non-overlapping sub-channels to avoid inter-carrier interference (ICI)( )

loz The diagram for bandwidth efficiency of OFDM system is shown below

35

SummarySummaryyy

loz The advantage of the FFT-based OFDM system g yloz The use of IFFTFFT can reduce the computation complexityloz The orthogonality between the adjacent sub-carriers will make g y j

the use of transmission bandwidth more efficientloz The guard interval is used to resist the inter-symbol interference

(ISI)loz The main advantage of the OFDM transmission technique is its

high performance even in frequency selective channels

loz The drawbacks of the OFDM system loz It is highly vulnerable to synchronization errorsloz Peak to Average Power Ratio (PAPR) problems

36

SynchronizationSynchronization

Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offset (TO)loz Impact of timing offset (TO)loz Impact of carrier frequency offset (CFO)loz TO estimation algorithmloz CFO estimation algorithmg

38 Synchronization2010722

Introduction (13)Introduction (13)( )( )

loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods

loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead

loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the

cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of

computational complexity therefore it may be not be available in short-burst wireless communication

39 Synchronization2010722

Introduction (23)Introduction (23)( )( )

loz Transcevier

40 Synchronization2010722

Introduction (33)Introduction (33)( )( )

loz Synchronization erroryloz Symbol timing offset (TO)

loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)

loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect

41 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg

42 Synchronization2010722

Signal model (14)Signal model (14)g ( )g ( )

loz Time domain sample p1

2

0

1 Nj nk N

n kk

x X eN

πminus

minus= sum

Xk the frequency domain data at the k-th subcarrier0kN =

loz The multipath fading channel

( ) ( ) ( )1L

h t h t δminus

sum

loz Received time domain signal

( ) ( ) ( )0

l ll

h t h tτ τ δ τ τ=

= minussum

loz Received time-domain signal

( ) ( )1

L

n n ly h n l x w nminus

minus= +sum43

( ) ( )0l=sum

Synchronization2010722

Signal model (24)Signal model (24)g ( )g ( )

loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting

12

0

Nj kn N

k nn

Y y e πminus

minus

=

= sum

( )1 1

2

0

N Lj kn N

n l kn l

h n l x e Wπminus minus

minusminus

=

= +sumsum

loz The equation above could be expressed in matrix form

0n l

loz The equation above could be expressed in matrix form

= +Y HX W

44 Synchronization2010722

Signal model (34)Signal model (34)g ( )g ( )

H= = = = +Y Fy FCx FCF X HX W

loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal

y

H=H FCFcirculant matrix leading to be a diagonal matrix

=H FCF

( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0

h h L L hh h

⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥

( )( ) ( )

( ) ( )

1 10 1 0 0i

h Lh L h N L

⎢ ⎥⎢ ⎥minus minus⎢ ⎥

equiv= minus minus⎢ ⎥⎢ ⎥

C( ) ( )

( ) ( )

0 1 1 100

0 0 1 1 0

h L h N L

h N L L h N

⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

45

( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦

Synchronization2010722

Signal model (44)Signal model (44)g ( )g ( )

loz In a time-invariant channel the channel matrix H would be a diagonal matrix

k k k kY H X W= +

( )1 1

2

0 0

N Lj kl N

kn l

H h n l e πminus minus

minus

= =

= sumsum

loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI

k k k k kY H X I Wα= + +k k k k k

( ) ( )1 1 1

2 2 N N L

j m k Nj ml Nk mI X h n l e e ππ

minus minus minusminusminus⎛ ⎞

= ⎜ ⎟⎝ ⎠

sum sum sumICI

46

0 0 0m n lm k= = =ne

⎝ ⎠

Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

47 Synchronization2010722

Impact of timing offset Impact of timing offset p gp g

loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively

( )I f s

Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ

( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr

2 fj k Nk k k kY X H e Wπδminus= +

loz For integer part of timing offset k k k kY X H e W+

I sTδ ( ) 0g IN L δminus minus le leFFT

2 Ij k NY X H e Wπδminus= +

( ) ( ) 2 I sFFT

j f TI sh t T H f e π δδ minusminus rarr

48

k k k kY X H e W= +

Synchronization2010722

Impact of timing offset derivationImpact of timing offset derivationp gp g

49 Synchronization2010722

Discussion of symbol boundaryDiscussion of symbol boundaryy yy y

50 Synchronization2010722

Impact of timing offsetImpact of timing offsetp gp g

loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI

2 Ij k NIk k k k

NY X H e ISI W

Nπδδ minusminus

= + +

loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk

e e

p kloz Phase Shift can be resolved by Differential Encoding

Decoding or carrier recovery with pilots

51 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

52 Synchronization2010722

Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )

53 Synchronization2010722

Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )

( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s

i n t i N N T N T nTy y t e

= + + +=

( ) ( )( ) ( ) ( ) 12

sin

sin

g gI f f

I I

i N N N Nj jf N Ni k i k k

f

Y X H e eN

π ε ε π ε

ε ε

πεπε

+ + minus+

minus minus=⎛ ⎞⎜ ⎟

Faded signal attenuated and rotated by CFO

( )( ) ( ) ( ) ( )11 2

sin

sin g gI f I f

i N N N NN j j l kI f

NN

l k π ε ε π ε επ ε ε + + minusminus + + + minus

⎜ ⎟⎝ ⎠

+ + minussum

( )( )( )

( ) ( )2

0

sin

I f I f

I

j j l kI f N Ni l l

l l k I f

X H e el k

NN

π ε ε π ε ε

ε π ε ε

+ + +

= ne minus

+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠

sum

i k

N

W

⎜ ⎟⎝ ⎠

+ Inter-Carrier Interference (ICI)

54 Synchronization2010722

Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )

loz An easier representation of received signal with only p g yfractional CFO is

1N

( ) ( )1

0

0N

k k k l l kl l k

Y H X C H X C l k Wminus

= ne

= + minus +⎡ ⎤⎣ ⎦sum

where the ICI coefficient is given by

( )( )( ) ( )

sin 1exp 1fl k

C l k j l kπ ε

π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )

( )exp 1sin

f

f

C l k j l kNN l k

N

π επ ε

minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠

55 Synchronization2010722

Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )

loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality

loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift

56 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

57 Synchronization2010722

TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )

loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq

58 Synchronization2010722

TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )

loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below

loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and

rN2 can be expressed by

0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+

( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +

59

( )

Synchronization2010722

TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )

loz Without CFO the first half is identical to the second half (in time order)

loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below

( ) 12 in discretesft T f NT

NTφ π π π ε πε

⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟

⎝ ⎠

helliphellip

sNT⎝ ⎠

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)

60

Length=N2

Synchronization2010722

TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )

loz Received signal without CFOg0

0 0

jr a e θ=jθ 2 0

2 2 0

Nj j

N Nr a e a e

θ θ= =

loz Received signal with CFO0 0

0 0

j jr a e eθ φ= Extra phase rotation due to CFO0 0

2 2 20

2 2 0

N N Nj j jj

N Nr a e e a e e

θ φ φθ= =

due to CFO

Phase difference 2 2 0N N

( )2 02 20 2 0 0

Nj j T fN

r r a e a eφ φ πminus Δ= =

which contains the information about CFO

61

0 2 0 0Nr r a e a e

Synchronization2010722

TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )

loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be

loz Note that d is a time index corresponding to the first( ) ( ) 2 1

20

Nd m d m Nm

P d r rminus

+ + +== sum

loz Note that d is a time index corresponding to the first sample in a window of N samples

loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by

( )22 1

20

Nd m NR d rminus

+ += sumloz A timing metric can be defined as

))(()(

)( 2

2

dRdP

dM =

( ) 20 d m Nm + +=sum

62

))(( dR

Synchronization2010722

TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )

63 Synchronization2010722

TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )

loz If drsquo is the correct symbol timing offset

( )2

2 2

0 1 12

j T f j T f j T fN

P d a e a e a eπ π πΔ Δ Δ

minus= + + +

22

2 2

0 1 12

j T fN

e a a aπ Δ

minus

⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭

loz If drsquo falls behind the correct symbol timing offset 1 sample

2⎪ ⎪⎩ ⎭

64 Synchronization2010722

TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )

65 Synchronization2010722

TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )

loz Drawback Plateau effect

Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)

((N2) ((N2) ((N2)

helliphellip

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)

Sliding window (length = N)

66 Synchronization2010722

TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )

loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula

( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus

扣掉相加

( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +

loz The method also called delay correlator

67 Synchronization2010722

IEEE 80211aIEEE 80211a

68 Synchronization2010722

Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )

loz Minnrsquos Method [2][ ]B B －B －B

( )( )

( )

22

2Minn

P dM d =

1 4 1N minus

( )( )( )2

2

MinnR d

( ) ( ) ( ) ( )

1 4 1

2 2 2 40 0

N

d N m k d N m k Nm k

P d r r+ + + + += =

= sum sum

( ) ( ) ( )2

21 4 1

2 40 0

N

d N m k Nk

R d rminus

+ + += sum sum69

0 0m k= =

Synchronization2010722

Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )

loz Parkrsquos Method [3][ ]

where D is the symmetric version of CC D C D

where D is the symmetric version of C

( )( ) 2

3k

P dM d =( )

( )( )23

ParkM dR d

2N

( ) 2

30

N

d k d kk

P d r rminus +=

= sdotsum

( )3

22

0

N

d kk

R d r += sum70

0k=

Synchronization2010722

Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )

loz Renrsquos Method [4][ ]

AS AS

where S is the PN sequence weighted of the original bl 2preamble

( )( )( )( )

24

Re 24

n

P dM d

R d=

( )( )4R d

( ) 2 1

4 2 2

N

k k N d k d k NP d s s r rminus

+ + + += sum0k=sum

( )21

41 N

d kR d rminus

= sum71

( )402 d k

kR d r +

=sum

Synchronization2010722

Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )

72 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

73 Synchronization2010722

Fractional CFO estimation Fractional CFO estimation

loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected

11 Rminus⎛ ⎞sum

h L i th di t b t t id ti l bl k R i

0

12 d m d m L

ms

f r rLTπ + + +

=

⎛ ⎞= ang⎜ ⎟⎝ ⎠sum

where L is the distance between two identical block R is the block size

loz For exampleNN cp

L = N R = NL = N R = Ncp74 Synchronization2010722

Estimation rangeEstimation rangegg

loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval

f ( ) ( )s s⎣ ⎦

loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)

loz DVB-T (L=N) 802 11 (L N4)

[ ]05 05s sf fminus

[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)

(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus

loz Note that is the subcarrier spacing1sf NT=

75

sNT

Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz

plusmng q y

loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn

76 Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz

plusmng q y

loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for

subcarrier spacing is required

plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn

p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs

77 Synchronization2010722

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

Cyclic PrefixCyclic Prefix

loz One of the most important reasons to do OFDM is the

yy

pefficient way it deals with multipath delay spread

loz To eliminate inter-symbol interference (ISI) almostloz To eliminate inter symbol interference (ISI) almost completely a guard time is introduced for each OFDM symbolsymbol(The guard time is chosen larger than the delay spread)

34

Bandwidth EfficiencyBandwidth Efficiencyyy

loz In a classical parallel system the channel is divided into p y N non-overlapping sub-channels to avoid inter-carrier interference (ICI)( )

loz The diagram for bandwidth efficiency of OFDM system is shown below

35

SummarySummaryyy

loz The advantage of the FFT-based OFDM system g yloz The use of IFFTFFT can reduce the computation complexityloz The orthogonality between the adjacent sub-carriers will make g y j

the use of transmission bandwidth more efficientloz The guard interval is used to resist the inter-symbol interference

(ISI)loz The main advantage of the OFDM transmission technique is its

high performance even in frequency selective channels

loz The drawbacks of the OFDM system loz It is highly vulnerable to synchronization errorsloz Peak to Average Power Ratio (PAPR) problems

36

SynchronizationSynchronization

Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offset (TO)loz Impact of timing offset (TO)loz Impact of carrier frequency offset (CFO)loz TO estimation algorithmloz CFO estimation algorithmg

38 Synchronization2010722

Introduction (13)Introduction (13)( )( )

loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods

loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead

loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the

cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of

computational complexity therefore it may be not be available in short-burst wireless communication

39 Synchronization2010722

Introduction (23)Introduction (23)( )( )

loz Transcevier

40 Synchronization2010722

Introduction (33)Introduction (33)( )( )

loz Synchronization erroryloz Symbol timing offset (TO)

loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)

loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect

41 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg

42 Synchronization2010722

Signal model (14)Signal model (14)g ( )g ( )

loz Time domain sample p1

2

0

1 Nj nk N

n kk

x X eN

πminus

minus= sum

Xk the frequency domain data at the k-th subcarrier0kN =

loz The multipath fading channel

( ) ( ) ( )1L

h t h t δminus

sum

loz Received time domain signal

( ) ( ) ( )0

l ll

h t h tτ τ δ τ τ=

= minussum

loz Received time-domain signal

( ) ( )1

L

n n ly h n l x w nminus

minus= +sum43

( ) ( )0l=sum

Synchronization2010722

Signal model (24)Signal model (24)g ( )g ( )

loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting

12

0

Nj kn N

k nn

Y y e πminus

minus

=

= sum

( )1 1

2

0

N Lj kn N

n l kn l

h n l x e Wπminus minus

minusminus

=

= +sumsum

loz The equation above could be expressed in matrix form

0n l

loz The equation above could be expressed in matrix form

= +Y HX W

44 Synchronization2010722

Signal model (34)Signal model (34)g ( )g ( )

H= = = = +Y Fy FCx FCF X HX W

loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal

y

H=H FCFcirculant matrix leading to be a diagonal matrix

=H FCF

( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0

h h L L hh h

⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥

( )( ) ( )

( ) ( )

1 10 1 0 0i

h Lh L h N L

⎢ ⎥⎢ ⎥minus minus⎢ ⎥

equiv= minus minus⎢ ⎥⎢ ⎥

C( ) ( )

( ) ( )

0 1 1 100

0 0 1 1 0

h L h N L

h N L L h N

⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

45

( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦

Synchronization2010722

Signal model (44)Signal model (44)g ( )g ( )

loz In a time-invariant channel the channel matrix H would be a diagonal matrix

k k k kY H X W= +

( )1 1

2

0 0

N Lj kl N

kn l

H h n l e πminus minus

minus

= =

= sumsum

loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI

k k k k kY H X I Wα= + +k k k k k

( ) ( )1 1 1

2 2 N N L

j m k Nj ml Nk mI X h n l e e ππ

minus minus minusminusminus⎛ ⎞

= ⎜ ⎟⎝ ⎠

sum sum sumICI

46

0 0 0m n lm k= = =ne

⎝ ⎠

Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

47 Synchronization2010722

Impact of timing offset Impact of timing offset p gp g

loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively

( )I f s

Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ

( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr

2 fj k Nk k k kY X H e Wπδminus= +

loz For integer part of timing offset k k k kY X H e W+

I sTδ ( ) 0g IN L δminus minus le leFFT

2 Ij k NY X H e Wπδminus= +

( ) ( ) 2 I sFFT

j f TI sh t T H f e π δδ minusminus rarr

48

k k k kY X H e W= +

Synchronization2010722

Impact of timing offset derivationImpact of timing offset derivationp gp g

49 Synchronization2010722

Discussion of symbol boundaryDiscussion of symbol boundaryy yy y

50 Synchronization2010722

Impact of timing offsetImpact of timing offsetp gp g

loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI

2 Ij k NIk k k k

NY X H e ISI W

Nπδδ minusminus

= + +

loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk

e e

p kloz Phase Shift can be resolved by Differential Encoding

Decoding or carrier recovery with pilots

51 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

52 Synchronization2010722

Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )

53 Synchronization2010722

Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )

( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s

i n t i N N T N T nTy y t e

= + + +=

( ) ( )( ) ( ) ( ) 12

sin

sin

g gI f f

I I

i N N N Nj jf N Ni k i k k

f

Y X H e eN

π ε ε π ε

ε ε

πεπε

+ + minus+

minus minus=⎛ ⎞⎜ ⎟

Faded signal attenuated and rotated by CFO

( )( ) ( ) ( ) ( )11 2

sin

sin g gI f I f

i N N N NN j j l kI f

NN

l k π ε ε π ε επ ε ε + + minusminus + + + minus

⎜ ⎟⎝ ⎠

+ + minussum

( )( )( )

( ) ( )2

0

sin

I f I f

I

j j l kI f N Ni l l

l l k I f

X H e el k

NN

π ε ε π ε ε

ε π ε ε

+ + +

= ne minus

+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠

sum

i k

N

W

⎜ ⎟⎝ ⎠

+ Inter-Carrier Interference (ICI)

54 Synchronization2010722

Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )

loz An easier representation of received signal with only p g yfractional CFO is

1N

( ) ( )1

0

0N

k k k l l kl l k

Y H X C H X C l k Wminus

= ne

= + minus +⎡ ⎤⎣ ⎦sum

where the ICI coefficient is given by

( )( )( ) ( )

sin 1exp 1fl k

C l k j l kπ ε

π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )

( )exp 1sin

f

f

C l k j l kNN l k

N

π επ ε

minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠

55 Synchronization2010722

Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )

loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality

loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift

56 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

57 Synchronization2010722

TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )

loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq

58 Synchronization2010722

TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )

loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below

loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and

rN2 can be expressed by

0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+

( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +

59

( )

Synchronization2010722

TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )

loz Without CFO the first half is identical to the second half (in time order)

loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below

( ) 12 in discretesft T f NT

NTφ π π π ε πε

⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟

⎝ ⎠

helliphellip

sNT⎝ ⎠

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)

60

Length=N2

Synchronization2010722

TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )

loz Received signal without CFOg0

0 0

jr a e θ=jθ 2 0

2 2 0

Nj j

N Nr a e a e

θ θ= =

loz Received signal with CFO0 0

0 0

j jr a e eθ φ= Extra phase rotation due to CFO0 0

2 2 20

2 2 0

N N Nj j jj

N Nr a e e a e e

θ φ φθ= =

due to CFO

Phase difference 2 2 0N N

( )2 02 20 2 0 0

Nj j T fN

r r a e a eφ φ πminus Δ= =

which contains the information about CFO

61

0 2 0 0Nr r a e a e

Synchronization2010722

TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )

loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be

loz Note that d is a time index corresponding to the first( ) ( ) 2 1

20

Nd m d m Nm

P d r rminus

+ + +== sum

loz Note that d is a time index corresponding to the first sample in a window of N samples

loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by

( )22 1

20

Nd m NR d rminus

+ += sumloz A timing metric can be defined as

))(()(

)( 2

2

dRdP

dM =

( ) 20 d m Nm + +=sum

62

))(( dR

Synchronization2010722

TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )

63 Synchronization2010722

TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )

loz If drsquo is the correct symbol timing offset

( )2

2 2

0 1 12

j T f j T f j T fN

P d a e a e a eπ π πΔ Δ Δ

minus= + + +

22

2 2

0 1 12

j T fN

e a a aπ Δ

minus

⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭

loz If drsquo falls behind the correct symbol timing offset 1 sample

2⎪ ⎪⎩ ⎭

64 Synchronization2010722

TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )

65 Synchronization2010722

TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )

loz Drawback Plateau effect

Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)

((N2) ((N2) ((N2)

helliphellip

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)

Sliding window (length = N)

66 Synchronization2010722

TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )

loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula

( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus

扣掉相加

( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +

loz The method also called delay correlator

67 Synchronization2010722

IEEE 80211aIEEE 80211a

68 Synchronization2010722

Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )

loz Minnrsquos Method [2][ ]B B －B －B

( )( )

( )

22

2Minn

P dM d =

1 4 1N minus

( )( )( )2

2

MinnR d

( ) ( ) ( ) ( )

1 4 1

2 2 2 40 0

N

d N m k d N m k Nm k

P d r r+ + + + += =

= sum sum

( ) ( ) ( )2

21 4 1

2 40 0

N

d N m k Nk

R d rminus

+ + += sum sum69

0 0m k= =

Synchronization2010722

Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )

loz Parkrsquos Method [3][ ]

where D is the symmetric version of CC D C D

where D is the symmetric version of C

( )( ) 2

3k

P dM d =( )

( )( )23

ParkM dR d

2N

( ) 2

30

N

d k d kk

P d r rminus +=

= sdotsum

( )3

22

0

N

d kk

R d r += sum70

0k=

Synchronization2010722

Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )

loz Renrsquos Method [4][ ]

AS AS

where S is the PN sequence weighted of the original bl 2preamble

( )( )( )( )

24

Re 24

n

P dM d

R d=

( )( )4R d

( ) 2 1

4 2 2

N

k k N d k d k NP d s s r rminus

+ + + += sum0k=sum

( )21

41 N

d kR d rminus

= sum71

( )402 d k

kR d r +

=sum

Synchronization2010722

Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )

72 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

73 Synchronization2010722

Fractional CFO estimation Fractional CFO estimation

loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected

11 Rminus⎛ ⎞sum

h L i th di t b t t id ti l bl k R i

0

12 d m d m L

ms

f r rLTπ + + +

=

⎛ ⎞= ang⎜ ⎟⎝ ⎠sum

where L is the distance between two identical block R is the block size

loz For exampleNN cp

L = N R = NL = N R = Ncp74 Synchronization2010722

Estimation rangeEstimation rangegg

loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval

f ( ) ( )s s⎣ ⎦

loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)

loz DVB-T (L=N) 802 11 (L N4)

[ ]05 05s sf fminus

[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)

(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus

loz Note that is the subcarrier spacing1sf NT=

75

sNT

Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz

plusmng q y

loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn

76 Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz

plusmng q y

loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for

subcarrier spacing is required

plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn

p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs

77 Synchronization2010722

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

Bandwidth EfficiencyBandwidth Efficiencyyy

loz In a classical parallel system the channel is divided into p y N non-overlapping sub-channels to avoid inter-carrier interference (ICI)( )

loz The diagram for bandwidth efficiency of OFDM system is shown below

35

SummarySummaryyy

loz The advantage of the FFT-based OFDM system g yloz The use of IFFTFFT can reduce the computation complexityloz The orthogonality between the adjacent sub-carriers will make g y j

the use of transmission bandwidth more efficientloz The guard interval is used to resist the inter-symbol interference

(ISI)loz The main advantage of the OFDM transmission technique is its

high performance even in frequency selective channels

loz The drawbacks of the OFDM system loz It is highly vulnerable to synchronization errorsloz Peak to Average Power Ratio (PAPR) problems

36

SynchronizationSynchronization

Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offset (TO)loz Impact of timing offset (TO)loz Impact of carrier frequency offset (CFO)loz TO estimation algorithmloz CFO estimation algorithmg

38 Synchronization2010722

Introduction (13)Introduction (13)( )( )

loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods

loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead

loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the

cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of

computational complexity therefore it may be not be available in short-burst wireless communication

39 Synchronization2010722

Introduction (23)Introduction (23)( )( )

loz Transcevier

40 Synchronization2010722

Introduction (33)Introduction (33)( )( )

loz Synchronization erroryloz Symbol timing offset (TO)

loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)

loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect

41 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg

42 Synchronization2010722

Signal model (14)Signal model (14)g ( )g ( )

loz Time domain sample p1

2

0

1 Nj nk N

n kk

x X eN

πminus

minus= sum

Xk the frequency domain data at the k-th subcarrier0kN =

loz The multipath fading channel

( ) ( ) ( )1L

h t h t δminus

sum

loz Received time domain signal

( ) ( ) ( )0

l ll

h t h tτ τ δ τ τ=

= minussum

loz Received time-domain signal

( ) ( )1

L

n n ly h n l x w nminus

minus= +sum43

( ) ( )0l=sum

Synchronization2010722

Signal model (24)Signal model (24)g ( )g ( )

loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting

12

0

Nj kn N

k nn

Y y e πminus

minus

=

= sum

( )1 1

2

0

N Lj kn N

n l kn l

h n l x e Wπminus minus

minusminus

=

= +sumsum

loz The equation above could be expressed in matrix form

0n l

loz The equation above could be expressed in matrix form

= +Y HX W

44 Synchronization2010722

Signal model (34)Signal model (34)g ( )g ( )

H= = = = +Y Fy FCx FCF X HX W

loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal

y

H=H FCFcirculant matrix leading to be a diagonal matrix

=H FCF

( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0

h h L L hh h

⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥

( )( ) ( )

( ) ( )

1 10 1 0 0i

h Lh L h N L

⎢ ⎥⎢ ⎥minus minus⎢ ⎥

equiv= minus minus⎢ ⎥⎢ ⎥

C( ) ( )

( ) ( )

0 1 1 100

0 0 1 1 0

h L h N L

h N L L h N

⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

45

( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦

Synchronization2010722

Signal model (44)Signal model (44)g ( )g ( )

loz In a time-invariant channel the channel matrix H would be a diagonal matrix

k k k kY H X W= +

( )1 1

2

0 0

N Lj kl N

kn l

H h n l e πminus minus

minus

= =

= sumsum

loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI

k k k k kY H X I Wα= + +k k k k k

( ) ( )1 1 1

2 2 N N L

j m k Nj ml Nk mI X h n l e e ππ

minus minus minusminusminus⎛ ⎞

= ⎜ ⎟⎝ ⎠

sum sum sumICI

46

0 0 0m n lm k= = =ne

⎝ ⎠

Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

47 Synchronization2010722

Impact of timing offset Impact of timing offset p gp g

loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively

( )I f s

Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ

( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr

2 fj k Nk k k kY X H e Wπδminus= +

loz For integer part of timing offset k k k kY X H e W+

I sTδ ( ) 0g IN L δminus minus le leFFT

2 Ij k NY X H e Wπδminus= +

( ) ( ) 2 I sFFT

j f TI sh t T H f e π δδ minusminus rarr

48

k k k kY X H e W= +

Synchronization2010722

Impact of timing offset derivationImpact of timing offset derivationp gp g

49 Synchronization2010722

Discussion of symbol boundaryDiscussion of symbol boundaryy yy y

50 Synchronization2010722

Impact of timing offsetImpact of timing offsetp gp g

loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI

2 Ij k NIk k k k

NY X H e ISI W

Nπδδ minusminus

= + +

loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk

e e

p kloz Phase Shift can be resolved by Differential Encoding

Decoding or carrier recovery with pilots

51 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

52 Synchronization2010722

Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )

53 Synchronization2010722

Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )

( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s

i n t i N N T N T nTy y t e

= + + +=

( ) ( )( ) ( ) ( ) 12

sin

sin

g gI f f

I I

i N N N Nj jf N Ni k i k k

f

Y X H e eN

π ε ε π ε

ε ε

πεπε

+ + minus+

minus minus=⎛ ⎞⎜ ⎟

Faded signal attenuated and rotated by CFO

( )( ) ( ) ( ) ( )11 2

sin

sin g gI f I f

i N N N NN j j l kI f

NN

l k π ε ε π ε επ ε ε + + minusminus + + + minus

⎜ ⎟⎝ ⎠

+ + minussum

( )( )( )

( ) ( )2

0

sin

I f I f

I

j j l kI f N Ni l l

l l k I f

X H e el k

NN

π ε ε π ε ε

ε π ε ε

+ + +

= ne minus

+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠

sum

i k

N

W

⎜ ⎟⎝ ⎠

+ Inter-Carrier Interference (ICI)

54 Synchronization2010722

Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )

loz An easier representation of received signal with only p g yfractional CFO is

1N

( ) ( )1

0

0N

k k k l l kl l k

Y H X C H X C l k Wminus

= ne

= + minus +⎡ ⎤⎣ ⎦sum

where the ICI coefficient is given by

( )( )( ) ( )

sin 1exp 1fl k

C l k j l kπ ε

π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )

( )exp 1sin

f

f

C l k j l kNN l k

N

π επ ε

minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠

55 Synchronization2010722

Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )

loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality

loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift

56 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

57 Synchronization2010722

TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )

loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq

58 Synchronization2010722

TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )

loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below

loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and

rN2 can be expressed by

0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+

( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +

59

( )

Synchronization2010722

TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )

loz Without CFO the first half is identical to the second half (in time order)

loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below

( ) 12 in discretesft T f NT

NTφ π π π ε πε

⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟

⎝ ⎠

helliphellip

sNT⎝ ⎠

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)

60

Length=N2

Synchronization2010722

TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )

loz Received signal without CFOg0

0 0

jr a e θ=jθ 2 0

2 2 0

Nj j

N Nr a e a e

θ θ= =

loz Received signal with CFO0 0

0 0

j jr a e eθ φ= Extra phase rotation due to CFO0 0

2 2 20

2 2 0

N N Nj j jj

N Nr a e e a e e

θ φ φθ= =

due to CFO

Phase difference 2 2 0N N

( )2 02 20 2 0 0

Nj j T fN

r r a e a eφ φ πminus Δ= =

which contains the information about CFO

61

0 2 0 0Nr r a e a e

Synchronization2010722

TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )

loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be

loz Note that d is a time index corresponding to the first( ) ( ) 2 1

20

Nd m d m Nm

P d r rminus

+ + +== sum

loz Note that d is a time index corresponding to the first sample in a window of N samples

loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by

( )22 1

20

Nd m NR d rminus

+ += sumloz A timing metric can be defined as

))(()(

)( 2

2

dRdP

dM =

( ) 20 d m Nm + +=sum

62

))(( dR

Synchronization2010722

TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )

63 Synchronization2010722

TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )

loz If drsquo is the correct symbol timing offset

( )2

2 2

0 1 12

j T f j T f j T fN

P d a e a e a eπ π πΔ Δ Δ

minus= + + +

22

2 2

0 1 12

j T fN

e a a aπ Δ

minus

⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭

loz If drsquo falls behind the correct symbol timing offset 1 sample

2⎪ ⎪⎩ ⎭

64 Synchronization2010722

TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )

65 Synchronization2010722

TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )

loz Drawback Plateau effect

Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)

((N2) ((N2) ((N2)

helliphellip

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)

Sliding window (length = N)

66 Synchronization2010722

TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )

loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula

( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus

扣掉相加

( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +

loz The method also called delay correlator

67 Synchronization2010722

IEEE 80211aIEEE 80211a

68 Synchronization2010722

Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )

loz Minnrsquos Method [2][ ]B B －B －B

( )( )

( )

22

2Minn

P dM d =

1 4 1N minus

( )( )( )2

2

MinnR d

( ) ( ) ( ) ( )

1 4 1

2 2 2 40 0

N

d N m k d N m k Nm k

P d r r+ + + + += =

= sum sum

( ) ( ) ( )2

21 4 1

2 40 0

N

d N m k Nk

R d rminus

+ + += sum sum69

0 0m k= =

Synchronization2010722

Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )

loz Parkrsquos Method [3][ ]

where D is the symmetric version of CC D C D

where D is the symmetric version of C

( )( ) 2

3k

P dM d =( )

( )( )23

ParkM dR d

2N

( ) 2

30

N

d k d kk

P d r rminus +=

= sdotsum

( )3

22

0

N

d kk

R d r += sum70

0k=

Synchronization2010722

Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )

loz Renrsquos Method [4][ ]

AS AS

where S is the PN sequence weighted of the original bl 2preamble

( )( )( )( )

24

Re 24

n

P dM d

R d=

( )( )4R d

( ) 2 1

4 2 2

N

k k N d k d k NP d s s r rminus

+ + + += sum0k=sum

( )21

41 N

d kR d rminus

= sum71

( )402 d k

kR d r +

=sum

Synchronization2010722

Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )

72 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

73 Synchronization2010722

Fractional CFO estimation Fractional CFO estimation

loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected

11 Rminus⎛ ⎞sum

h L i th di t b t t id ti l bl k R i

0

12 d m d m L

ms

f r rLTπ + + +

=

⎛ ⎞= ang⎜ ⎟⎝ ⎠sum

where L is the distance between two identical block R is the block size

loz For exampleNN cp

L = N R = NL = N R = Ncp74 Synchronization2010722

Estimation rangeEstimation rangegg

loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval

f ( ) ( )s s⎣ ⎦

loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)

loz DVB-T (L=N) 802 11 (L N4)

[ ]05 05s sf fminus

[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)

(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus

loz Note that is the subcarrier spacing1sf NT=

75

sNT

Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz

plusmng q y

loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn

76 Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz

plusmng q y

loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for

subcarrier spacing is required

plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn

p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs

77 Synchronization2010722

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

SummarySummaryyy

loz The advantage of the FFT-based OFDM system g yloz The use of IFFTFFT can reduce the computation complexityloz The orthogonality between the adjacent sub-carriers will make g y j

the use of transmission bandwidth more efficientloz The guard interval is used to resist the inter-symbol interference

(ISI)loz The main advantage of the OFDM transmission technique is its

high performance even in frequency selective channels

loz The drawbacks of the OFDM system loz It is highly vulnerable to synchronization errorsloz Peak to Average Power Ratio (PAPR) problems

36

SynchronizationSynchronization

Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offset (TO)loz Impact of timing offset (TO)loz Impact of carrier frequency offset (CFO)loz TO estimation algorithmloz CFO estimation algorithmg

38 Synchronization2010722

Introduction (13)Introduction (13)( )( )

loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods

loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead

loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the

cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of

computational complexity therefore it may be not be available in short-burst wireless communication

39 Synchronization2010722

Introduction (23)Introduction (23)( )( )

loz Transcevier

40 Synchronization2010722

Introduction (33)Introduction (33)( )( )

loz Synchronization erroryloz Symbol timing offset (TO)

loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)

loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect

41 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg

42 Synchronization2010722

Signal model (14)Signal model (14)g ( )g ( )

loz Time domain sample p1

2

0

1 Nj nk N

n kk

x X eN

πminus

minus= sum

Xk the frequency domain data at the k-th subcarrier0kN =

loz The multipath fading channel

( ) ( ) ( )1L

h t h t δminus

sum

loz Received time domain signal

( ) ( ) ( )0

l ll

h t h tτ τ δ τ τ=

= minussum

loz Received time-domain signal

( ) ( )1

L

n n ly h n l x w nminus

minus= +sum43

( ) ( )0l=sum

Synchronization2010722

Signal model (24)Signal model (24)g ( )g ( )

loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting

12

0

Nj kn N

k nn

Y y e πminus

minus

=

= sum

( )1 1

2

0

N Lj kn N

n l kn l

h n l x e Wπminus minus

minusminus

=

= +sumsum

loz The equation above could be expressed in matrix form

0n l

loz The equation above could be expressed in matrix form

= +Y HX W

44 Synchronization2010722

Signal model (34)Signal model (34)g ( )g ( )

H= = = = +Y Fy FCx FCF X HX W

loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal

y

H=H FCFcirculant matrix leading to be a diagonal matrix

=H FCF

( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0

h h L L hh h

⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥

( )( ) ( )

( ) ( )

1 10 1 0 0i

h Lh L h N L

⎢ ⎥⎢ ⎥minus minus⎢ ⎥

equiv= minus minus⎢ ⎥⎢ ⎥

C( ) ( )

( ) ( )

0 1 1 100

0 0 1 1 0

h L h N L

h N L L h N

⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

45

( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦

Synchronization2010722

Signal model (44)Signal model (44)g ( )g ( )

loz In a time-invariant channel the channel matrix H would be a diagonal matrix

k k k kY H X W= +

( )1 1

2

0 0

N Lj kl N

kn l

H h n l e πminus minus

minus

= =

= sumsum

loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI

k k k k kY H X I Wα= + +k k k k k

( ) ( )1 1 1

2 2 N N L

j m k Nj ml Nk mI X h n l e e ππ

minus minus minusminusminus⎛ ⎞

= ⎜ ⎟⎝ ⎠

sum sum sumICI

46

0 0 0m n lm k= = =ne

⎝ ⎠

Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

47 Synchronization2010722

Impact of timing offset Impact of timing offset p gp g

loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively

( )I f s

Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ

( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr

2 fj k Nk k k kY X H e Wπδminus= +

loz For integer part of timing offset k k k kY X H e W+

I sTδ ( ) 0g IN L δminus minus le leFFT

2 Ij k NY X H e Wπδminus= +

( ) ( ) 2 I sFFT

j f TI sh t T H f e π δδ minusminus rarr

48

k k k kY X H e W= +

Synchronization2010722

Impact of timing offset derivationImpact of timing offset derivationp gp g

49 Synchronization2010722

Discussion of symbol boundaryDiscussion of symbol boundaryy yy y

50 Synchronization2010722

Impact of timing offsetImpact of timing offsetp gp g

loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI

2 Ij k NIk k k k

NY X H e ISI W

Nπδδ minusminus

= + +

loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk

e e

p kloz Phase Shift can be resolved by Differential Encoding

Decoding or carrier recovery with pilots

51 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

52 Synchronization2010722

Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )

53 Synchronization2010722

Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )

( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s

i n t i N N T N T nTy y t e

= + + +=

( ) ( )( ) ( ) ( ) 12

sin

sin

g gI f f

I I

i N N N Nj jf N Ni k i k k

f

Y X H e eN

π ε ε π ε

ε ε

πεπε

+ + minus+

minus minus=⎛ ⎞⎜ ⎟

Faded signal attenuated and rotated by CFO

( )( ) ( ) ( ) ( )11 2

sin

sin g gI f I f

i N N N NN j j l kI f

NN

l k π ε ε π ε επ ε ε + + minusminus + + + minus

⎜ ⎟⎝ ⎠

+ + minussum

( )( )( )

( ) ( )2

0

sin

I f I f

I

j j l kI f N Ni l l

l l k I f

X H e el k

NN

π ε ε π ε ε

ε π ε ε

+ + +

= ne minus

+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠

sum

i k

N

W

⎜ ⎟⎝ ⎠

+ Inter-Carrier Interference (ICI)

54 Synchronization2010722

Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )

loz An easier representation of received signal with only p g yfractional CFO is

1N

( ) ( )1

0

0N

k k k l l kl l k

Y H X C H X C l k Wminus

= ne

= + minus +⎡ ⎤⎣ ⎦sum

where the ICI coefficient is given by

( )( )( ) ( )

sin 1exp 1fl k

C l k j l kπ ε

π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )

( )exp 1sin

f

f

C l k j l kNN l k

N

π επ ε

minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠

55 Synchronization2010722

Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )

loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality

loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift

56 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

57 Synchronization2010722

TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )

loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq

58 Synchronization2010722

TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )

loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below

loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and

rN2 can be expressed by

0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+

( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +

59

( )

Synchronization2010722

TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )

loz Without CFO the first half is identical to the second half (in time order)

loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below

( ) 12 in discretesft T f NT

NTφ π π π ε πε

⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟

⎝ ⎠

helliphellip

sNT⎝ ⎠

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)

60

Length=N2

Synchronization2010722

TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )

loz Received signal without CFOg0

0 0

jr a e θ=jθ 2 0

2 2 0

Nj j

N Nr a e a e

θ θ= =

loz Received signal with CFO0 0

0 0

j jr a e eθ φ= Extra phase rotation due to CFO0 0

2 2 20

2 2 0

N N Nj j jj

N Nr a e e a e e

θ φ φθ= =

due to CFO

Phase difference 2 2 0N N

( )2 02 20 2 0 0

Nj j T fN

r r a e a eφ φ πminus Δ= =

which contains the information about CFO

61

0 2 0 0Nr r a e a e

Synchronization2010722

TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )

loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be

loz Note that d is a time index corresponding to the first( ) ( ) 2 1

20

Nd m d m Nm

P d r rminus

+ + +== sum

loz Note that d is a time index corresponding to the first sample in a window of N samples

loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by

( )22 1

20

Nd m NR d rminus

+ += sumloz A timing metric can be defined as

))(()(

)( 2

2

dRdP

dM =

( ) 20 d m Nm + +=sum

62

))(( dR

Synchronization2010722

TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )

63 Synchronization2010722

TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )

loz If drsquo is the correct symbol timing offset

( )2

2 2

0 1 12

j T f j T f j T fN

P d a e a e a eπ π πΔ Δ Δ

minus= + + +

22

2 2

0 1 12

j T fN

e a a aπ Δ

minus

⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭

loz If drsquo falls behind the correct symbol timing offset 1 sample

2⎪ ⎪⎩ ⎭

64 Synchronization2010722

TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )

65 Synchronization2010722

TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )

loz Drawback Plateau effect

Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)

((N2) ((N2) ((N2)

helliphellip

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)

Sliding window (length = N)

66 Synchronization2010722

TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )

loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula

( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus

扣掉相加

( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +

loz The method also called delay correlator

67 Synchronization2010722

IEEE 80211aIEEE 80211a

68 Synchronization2010722

Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )

loz Minnrsquos Method [2][ ]B B －B －B

( )( )

( )

22

2Minn

P dM d =

1 4 1N minus

( )( )( )2

2

MinnR d

( ) ( ) ( ) ( )

1 4 1

2 2 2 40 0

N

d N m k d N m k Nm k

P d r r+ + + + += =

= sum sum

( ) ( ) ( )2

21 4 1

2 40 0

N

d N m k Nk

R d rminus

+ + += sum sum69

0 0m k= =

Synchronization2010722

Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )

loz Parkrsquos Method [3][ ]

where D is the symmetric version of CC D C D

where D is the symmetric version of C

( )( ) 2

3k

P dM d =( )

( )( )23

ParkM dR d

2N

( ) 2

30

N

d k d kk

P d r rminus +=

= sdotsum

( )3

22

0

N

d kk

R d r += sum70

0k=

Synchronization2010722

Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )

loz Renrsquos Method [4][ ]

AS AS

where S is the PN sequence weighted of the original bl 2preamble

( )( )( )( )

24

Re 24

n

P dM d

R d=

( )( )4R d

( ) 2 1

4 2 2

N

k k N d k d k NP d s s r rminus

+ + + += sum0k=sum

( )21

41 N

d kR d rminus

= sum71

( )402 d k

kR d r +

=sum

Synchronization2010722

Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )

72 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

73 Synchronization2010722

Fractional CFO estimation Fractional CFO estimation

loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected

11 Rminus⎛ ⎞sum

h L i th di t b t t id ti l bl k R i

0

12 d m d m L

ms

f r rLTπ + + +

=

⎛ ⎞= ang⎜ ⎟⎝ ⎠sum

where L is the distance between two identical block R is the block size

loz For exampleNN cp

L = N R = NL = N R = Ncp74 Synchronization2010722

Estimation rangeEstimation rangegg

loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval

f ( ) ( )s s⎣ ⎦

loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)

loz DVB-T (L=N) 802 11 (L N4)

[ ]05 05s sf fminus

[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)

(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus

loz Note that is the subcarrier spacing1sf NT=

75

sNT

Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz

plusmng q y

loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn

76 Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz

plusmng q y

loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for

subcarrier spacing is required

plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn

p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs

77 Synchronization2010722

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

SynchronizationSynchronization

Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offset (TO)loz Impact of timing offset (TO)loz Impact of carrier frequency offset (CFO)loz TO estimation algorithmloz CFO estimation algorithmg

38 Synchronization2010722

Introduction (13)Introduction (13)( )( )

loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods

loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead

loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the

cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of

computational complexity therefore it may be not be available in short-burst wireless communication

39 Synchronization2010722

Introduction (23)Introduction (23)( )( )

loz Transcevier

40 Synchronization2010722

Introduction (33)Introduction (33)( )( )

loz Synchronization erroryloz Symbol timing offset (TO)

loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)

loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect

41 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg

42 Synchronization2010722

Signal model (14)Signal model (14)g ( )g ( )

loz Time domain sample p1

2

0

1 Nj nk N

n kk

x X eN

πminus

minus= sum

Xk the frequency domain data at the k-th subcarrier0kN =

loz The multipath fading channel

( ) ( ) ( )1L

h t h t δminus

sum

loz Received time domain signal

( ) ( ) ( )0

l ll

h t h tτ τ δ τ τ=

= minussum

loz Received time-domain signal

( ) ( )1

L

n n ly h n l x w nminus

minus= +sum43

( ) ( )0l=sum

Synchronization2010722

Signal model (24)Signal model (24)g ( )g ( )

loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting

12

0

Nj kn N

k nn

Y y e πminus

minus

=

= sum

( )1 1

2

0

N Lj kn N

n l kn l

h n l x e Wπminus minus

minusminus

=

= +sumsum

loz The equation above could be expressed in matrix form

0n l

loz The equation above could be expressed in matrix form

= +Y HX W

44 Synchronization2010722

Signal model (34)Signal model (34)g ( )g ( )

H= = = = +Y Fy FCx FCF X HX W

loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal

y

H=H FCFcirculant matrix leading to be a diagonal matrix

=H FCF

( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0

h h L L hh h

⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥

( )( ) ( )

( ) ( )

1 10 1 0 0i

h Lh L h N L

⎢ ⎥⎢ ⎥minus minus⎢ ⎥

equiv= minus minus⎢ ⎥⎢ ⎥

C( ) ( )

( ) ( )

0 1 1 100

0 0 1 1 0

h L h N L

h N L L h N

⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

45

( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦

Synchronization2010722

Signal model (44)Signal model (44)g ( )g ( )

loz In a time-invariant channel the channel matrix H would be a diagonal matrix

k k k kY H X W= +

( )1 1

2

0 0

N Lj kl N

kn l

H h n l e πminus minus

minus

= =

= sumsum

loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI

k k k k kY H X I Wα= + +k k k k k

( ) ( )1 1 1

2 2 N N L

j m k Nj ml Nk mI X h n l e e ππ

minus minus minusminusminus⎛ ⎞

= ⎜ ⎟⎝ ⎠

sum sum sumICI

46

0 0 0m n lm k= = =ne

⎝ ⎠

Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

47 Synchronization2010722

Impact of timing offset Impact of timing offset p gp g

loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively

( )I f s

Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ

( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr

2 fj k Nk k k kY X H e Wπδminus= +

loz For integer part of timing offset k k k kY X H e W+

I sTδ ( ) 0g IN L δminus minus le leFFT

2 Ij k NY X H e Wπδminus= +

( ) ( ) 2 I sFFT

j f TI sh t T H f e π δδ minusminus rarr

48

k k k kY X H e W= +

Synchronization2010722

Impact of timing offset derivationImpact of timing offset derivationp gp g

49 Synchronization2010722

Discussion of symbol boundaryDiscussion of symbol boundaryy yy y

50 Synchronization2010722

Impact of timing offsetImpact of timing offsetp gp g

loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI

2 Ij k NIk k k k

NY X H e ISI W

Nπδδ minusminus

= + +

loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk

e e

p kloz Phase Shift can be resolved by Differential Encoding

Decoding or carrier recovery with pilots

51 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

52 Synchronization2010722

Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )

53 Synchronization2010722

Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )

( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s

i n t i N N T N T nTy y t e

= + + +=

( ) ( )( ) ( ) ( ) 12

sin

sin

g gI f f

I I

i N N N Nj jf N Ni k i k k

f

Y X H e eN

π ε ε π ε

ε ε

πεπε

+ + minus+

minus minus=⎛ ⎞⎜ ⎟

Faded signal attenuated and rotated by CFO

( )( ) ( ) ( ) ( )11 2

sin

sin g gI f I f

i N N N NN j j l kI f

NN

l k π ε ε π ε επ ε ε + + minusminus + + + minus

⎜ ⎟⎝ ⎠

+ + minussum

( )( )( )

( ) ( )2

0

sin

I f I f

I

j j l kI f N Ni l l

l l k I f

X H e el k

NN

π ε ε π ε ε

ε π ε ε

+ + +

= ne minus

+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠

sum

i k

N

W

⎜ ⎟⎝ ⎠

+ Inter-Carrier Interference (ICI)

54 Synchronization2010722

Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )

loz An easier representation of received signal with only p g yfractional CFO is

1N

( ) ( )1

0

0N

k k k l l kl l k

Y H X C H X C l k Wminus

= ne

= + minus +⎡ ⎤⎣ ⎦sum

where the ICI coefficient is given by

( )( )( ) ( )

sin 1exp 1fl k

C l k j l kπ ε

π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )

( )exp 1sin

f

f

C l k j l kNN l k

N

π επ ε

minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠

55 Synchronization2010722

Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )

loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality

loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift

56 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

57 Synchronization2010722

TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )

loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq

58 Synchronization2010722

TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )

loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below

loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and

rN2 can be expressed by

0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+

( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +

59

( )

Synchronization2010722

TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )

loz Without CFO the first half is identical to the second half (in time order)

loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below

( ) 12 in discretesft T f NT

NTφ π π π ε πε

⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟

⎝ ⎠

helliphellip

sNT⎝ ⎠

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)

60

Length=N2

Synchronization2010722

TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )

loz Received signal without CFOg0

0 0

jr a e θ=jθ 2 0

2 2 0

Nj j

N Nr a e a e

θ θ= =

loz Received signal with CFO0 0

0 0

j jr a e eθ φ= Extra phase rotation due to CFO0 0

2 2 20

2 2 0

N N Nj j jj

N Nr a e e a e e

θ φ φθ= =

due to CFO

Phase difference 2 2 0N N

( )2 02 20 2 0 0

Nj j T fN

r r a e a eφ φ πminus Δ= =

which contains the information about CFO

61

0 2 0 0Nr r a e a e

Synchronization2010722

TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )

loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be

loz Note that d is a time index corresponding to the first( ) ( ) 2 1

20

Nd m d m Nm

P d r rminus

+ + +== sum

loz Note that d is a time index corresponding to the first sample in a window of N samples

loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by

( )22 1

20

Nd m NR d rminus

+ += sumloz A timing metric can be defined as

))(()(

)( 2

2

dRdP

dM =

( ) 20 d m Nm + +=sum

62

))(( dR

Synchronization2010722

TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )

63 Synchronization2010722

TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )

loz If drsquo is the correct symbol timing offset

( )2

2 2

0 1 12

j T f j T f j T fN

P d a e a e a eπ π πΔ Δ Δ

minus= + + +

22

2 2

0 1 12

j T fN

e a a aπ Δ

minus

⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭

loz If drsquo falls behind the correct symbol timing offset 1 sample

2⎪ ⎪⎩ ⎭

64 Synchronization2010722

TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )

65 Synchronization2010722

TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )

loz Drawback Plateau effect

Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)

((N2) ((N2) ((N2)

helliphellip

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)

Sliding window (length = N)

66 Synchronization2010722

TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )

loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula

( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus

扣掉相加

( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +

loz The method also called delay correlator

67 Synchronization2010722

IEEE 80211aIEEE 80211a

68 Synchronization2010722

Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )

loz Minnrsquos Method [2][ ]B B －B －B

( )( )

( )

22

2Minn

P dM d =

1 4 1N minus

( )( )( )2

2

MinnR d

( ) ( ) ( ) ( )

1 4 1

2 2 2 40 0

N

d N m k d N m k Nm k

P d r r+ + + + += =

= sum sum

( ) ( ) ( )2

21 4 1

2 40 0

N

d N m k Nk

R d rminus

+ + += sum sum69

0 0m k= =

Synchronization2010722

Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )

loz Parkrsquos Method [3][ ]

where D is the symmetric version of CC D C D

where D is the symmetric version of C

( )( ) 2

3k

P dM d =( )

( )( )23

ParkM dR d

2N

( ) 2

30

N

d k d kk

P d r rminus +=

= sdotsum

( )3

22

0

N

d kk

R d r += sum70

0k=

Synchronization2010722

Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )

loz Renrsquos Method [4][ ]

AS AS

where S is the PN sequence weighted of the original bl 2preamble

( )( )( )( )

24

Re 24

n

P dM d

R d=

( )( )4R d

( ) 2 1

4 2 2

N

k k N d k d k NP d s s r rminus

+ + + += sum0k=sum

( )21

41 N

d kR d rminus

= sum71

( )402 d k

kR d r +

=sum

Synchronization2010722

Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )

72 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

73 Synchronization2010722

Fractional CFO estimation Fractional CFO estimation

loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected

11 Rminus⎛ ⎞sum

h L i th di t b t t id ti l bl k R i

0

12 d m d m L

ms

f r rLTπ + + +

=

⎛ ⎞= ang⎜ ⎟⎝ ⎠sum

where L is the distance between two identical block R is the block size

loz For exampleNN cp

L = N R = NL = N R = Ncp74 Synchronization2010722

Estimation rangeEstimation rangegg

loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval

f ( ) ( )s s⎣ ⎦

loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)

loz DVB-T (L=N) 802 11 (L N4)

[ ]05 05s sf fminus

[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)

(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus

loz Note that is the subcarrier spacing1sf NT=

75

sNT

Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz

plusmng q y

loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn

76 Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz

plusmng q y

loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for

subcarrier spacing is required

plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn

p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs

77 Synchronization2010722

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offset (TO)loz Impact of timing offset (TO)loz Impact of carrier frequency offset (CFO)loz TO estimation algorithmloz CFO estimation algorithmg

38 Synchronization2010722

Introduction (13)Introduction (13)( )( )

loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods

loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead

loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the

cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of

computational complexity therefore it may be not be available in short-burst wireless communication

39 Synchronization2010722

Introduction (23)Introduction (23)( )( )

loz Transcevier

40 Synchronization2010722

Introduction (33)Introduction (33)( )( )

loz Synchronization erroryloz Symbol timing offset (TO)

loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)

loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect

41 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg

42 Synchronization2010722

Signal model (14)Signal model (14)g ( )g ( )

loz Time domain sample p1

2

0

1 Nj nk N

n kk

x X eN

πminus

minus= sum

Xk the frequency domain data at the k-th subcarrier0kN =

loz The multipath fading channel

( ) ( ) ( )1L

h t h t δminus

sum

loz Received time domain signal

( ) ( ) ( )0

l ll

h t h tτ τ δ τ τ=

= minussum

loz Received time-domain signal

( ) ( )1

L

n n ly h n l x w nminus

minus= +sum43

( ) ( )0l=sum

Synchronization2010722

Signal model (24)Signal model (24)g ( )g ( )

loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting

12

0

Nj kn N

k nn

Y y e πminus

minus

=

= sum

( )1 1

2

0

N Lj kn N

n l kn l

h n l x e Wπminus minus

minusminus

=

= +sumsum

loz The equation above could be expressed in matrix form

0n l

loz The equation above could be expressed in matrix form

= +Y HX W

44 Synchronization2010722

Signal model (34)Signal model (34)g ( )g ( )

H= = = = +Y Fy FCx FCF X HX W

loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal

y

H=H FCFcirculant matrix leading to be a diagonal matrix

=H FCF

( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0

h h L L hh h

⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥

( )( ) ( )

( ) ( )

1 10 1 0 0i

h Lh L h N L

⎢ ⎥⎢ ⎥minus minus⎢ ⎥

equiv= minus minus⎢ ⎥⎢ ⎥

C( ) ( )

( ) ( )

0 1 1 100

0 0 1 1 0

h L h N L

h N L L h N

⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

45

( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦

Synchronization2010722

Signal model (44)Signal model (44)g ( )g ( )

loz In a time-invariant channel the channel matrix H would be a diagonal matrix

k k k kY H X W= +

( )1 1

2

0 0

N Lj kl N

kn l

H h n l e πminus minus

minus

= =

= sumsum

loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI

k k k k kY H X I Wα= + +k k k k k

( ) ( )1 1 1

2 2 N N L

j m k Nj ml Nk mI X h n l e e ππ

minus minus minusminusminus⎛ ⎞

= ⎜ ⎟⎝ ⎠

sum sum sumICI

46

0 0 0m n lm k= = =ne

⎝ ⎠

Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

47 Synchronization2010722

Impact of timing offset Impact of timing offset p gp g

loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively

( )I f s

Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ

( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr

2 fj k Nk k k kY X H e Wπδminus= +

loz For integer part of timing offset k k k kY X H e W+

I sTδ ( ) 0g IN L δminus minus le leFFT

2 Ij k NY X H e Wπδminus= +

( ) ( ) 2 I sFFT

j f TI sh t T H f e π δδ minusminus rarr

48

k k k kY X H e W= +

Synchronization2010722

Impact of timing offset derivationImpact of timing offset derivationp gp g

49 Synchronization2010722

Discussion of symbol boundaryDiscussion of symbol boundaryy yy y

50 Synchronization2010722

Impact of timing offsetImpact of timing offsetp gp g

loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI

2 Ij k NIk k k k

NY X H e ISI W

Nπδδ minusminus

= + +

loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk

e e

p kloz Phase Shift can be resolved by Differential Encoding

Decoding or carrier recovery with pilots

51 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

52 Synchronization2010722

Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )

53 Synchronization2010722

Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )

( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s

i n t i N N T N T nTy y t e

= + + +=

( ) ( )( ) ( ) ( ) 12

sin

sin

g gI f f

I I

i N N N Nj jf N Ni k i k k

f

Y X H e eN

π ε ε π ε

ε ε

πεπε

+ + minus+

minus minus=⎛ ⎞⎜ ⎟

Faded signal attenuated and rotated by CFO

( )( ) ( ) ( ) ( )11 2

sin

sin g gI f I f

i N N N NN j j l kI f

NN

l k π ε ε π ε επ ε ε + + minusminus + + + minus

⎜ ⎟⎝ ⎠

+ + minussum

( )( )( )

( ) ( )2

0

sin

I f I f

I

j j l kI f N Ni l l

l l k I f

X H e el k

NN

π ε ε π ε ε

ε π ε ε

+ + +

= ne minus

+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠

sum

i k

N

W

⎜ ⎟⎝ ⎠

+ Inter-Carrier Interference (ICI)

54 Synchronization2010722

Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )

loz An easier representation of received signal with only p g yfractional CFO is

1N

( ) ( )1

0

0N

k k k l l kl l k

Y H X C H X C l k Wminus

= ne

= + minus +⎡ ⎤⎣ ⎦sum

where the ICI coefficient is given by

( )( )( ) ( )

sin 1exp 1fl k

C l k j l kπ ε

π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )

( )exp 1sin

f

f

C l k j l kNN l k

N

π επ ε

minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠

55 Synchronization2010722

Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )

loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality

loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift

56 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

57 Synchronization2010722

TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )

loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq

58 Synchronization2010722

TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )

loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below

loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and

rN2 can be expressed by

0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+

( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +

59

( )

Synchronization2010722

TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )

loz Without CFO the first half is identical to the second half (in time order)

loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below

( ) 12 in discretesft T f NT

NTφ π π π ε πε

⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟

⎝ ⎠

helliphellip

sNT⎝ ⎠

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)

60

Length=N2

Synchronization2010722

TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )

loz Received signal without CFOg0

0 0

jr a e θ=jθ 2 0

2 2 0

Nj j

N Nr a e a e

θ θ= =

loz Received signal with CFO0 0

0 0

j jr a e eθ φ= Extra phase rotation due to CFO0 0

2 2 20

2 2 0

N N Nj j jj

N Nr a e e a e e

θ φ φθ= =

due to CFO

Phase difference 2 2 0N N

( )2 02 20 2 0 0

Nj j T fN

r r a e a eφ φ πminus Δ= =

which contains the information about CFO

61

0 2 0 0Nr r a e a e

Synchronization2010722

TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )

loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be

loz Note that d is a time index corresponding to the first( ) ( ) 2 1

20

Nd m d m Nm

P d r rminus

+ + +== sum

loz Note that d is a time index corresponding to the first sample in a window of N samples

loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by

( )22 1

20

Nd m NR d rminus

+ += sumloz A timing metric can be defined as

))(()(

)( 2

2

dRdP

dM =

( ) 20 d m Nm + +=sum

62

))(( dR

Synchronization2010722

TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )

63 Synchronization2010722

TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )

loz If drsquo is the correct symbol timing offset

( )2

2 2

0 1 12

j T f j T f j T fN

P d a e a e a eπ π πΔ Δ Δ

minus= + + +

22

2 2

0 1 12

j T fN

e a a aπ Δ

minus

⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭

loz If drsquo falls behind the correct symbol timing offset 1 sample

2⎪ ⎪⎩ ⎭

64 Synchronization2010722

TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )

65 Synchronization2010722

TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )

loz Drawback Plateau effect

Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)

((N2) ((N2) ((N2)

helliphellip

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)

Sliding window (length = N)

66 Synchronization2010722

TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )

loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula

( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus

扣掉相加

( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +

loz The method also called delay correlator

67 Synchronization2010722

IEEE 80211aIEEE 80211a

68 Synchronization2010722

Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )

loz Minnrsquos Method [2][ ]B B －B －B

( )( )

( )

22

2Minn

P dM d =

1 4 1N minus

( )( )( )2

2

MinnR d

( ) ( ) ( ) ( )

1 4 1

2 2 2 40 0

N

d N m k d N m k Nm k

P d r r+ + + + += =

= sum sum

( ) ( ) ( )2

21 4 1

2 40 0

N

d N m k Nk

R d rminus

+ + += sum sum69

0 0m k= =

Synchronization2010722

Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )

loz Parkrsquos Method [3][ ]

where D is the symmetric version of CC D C D

where D is the symmetric version of C

( )( ) 2

3k

P dM d =( )

( )( )23

ParkM dR d

2N

( ) 2

30

N

d k d kk

P d r rminus +=

= sdotsum

( )3

22

0

N

d kk

R d r += sum70

0k=

Synchronization2010722

Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )

loz Renrsquos Method [4][ ]

AS AS

where S is the PN sequence weighted of the original bl 2preamble

( )( )( )( )

24

Re 24

n

P dM d

R d=

( )( )4R d

( ) 2 1

4 2 2

N

k k N d k d k NP d s s r rminus

+ + + += sum0k=sum

( )21

41 N

d kR d rminus

= sum71

( )402 d k

kR d r +

=sum

Synchronization2010722

Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )

72 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

73 Synchronization2010722

Fractional CFO estimation Fractional CFO estimation

loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected

11 Rminus⎛ ⎞sum

h L i th di t b t t id ti l bl k R i

0

12 d m d m L

ms

f r rLTπ + + +

=

⎛ ⎞= ang⎜ ⎟⎝ ⎠sum

where L is the distance between two identical block R is the block size

loz For exampleNN cp

L = N R = NL = N R = Ncp74 Synchronization2010722

Estimation rangeEstimation rangegg

loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval

f ( ) ( )s s⎣ ⎦

loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)

loz DVB-T (L=N) 802 11 (L N4)

[ ]05 05s sf fminus

[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)

(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus

loz Note that is the subcarrier spacing1sf NT=

75

sNT

Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz

plusmng q y

loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn

76 Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz

plusmng q y

loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for

subcarrier spacing is required

plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn

p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs

77 Synchronization2010722

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

Introduction (13)Introduction (13)( )( )

loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods

loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead

loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the

cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of

computational complexity therefore it may be not be available in short-burst wireless communication

39 Synchronization2010722

Introduction (23)Introduction (23)( )( )

loz Transcevier

40 Synchronization2010722

Introduction (33)Introduction (33)( )( )

loz Synchronization erroryloz Symbol timing offset (TO)

loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)

loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect

41 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg

42 Synchronization2010722

Signal model (14)Signal model (14)g ( )g ( )

loz Time domain sample p1

2

0

1 Nj nk N

n kk

x X eN

πminus

minus= sum

Xk the frequency domain data at the k-th subcarrier0kN =

loz The multipath fading channel

( ) ( ) ( )1L

h t h t δminus

sum

loz Received time domain signal

( ) ( ) ( )0

l ll

h t h tτ τ δ τ τ=

= minussum

loz Received time-domain signal

( ) ( )1

L

n n ly h n l x w nminus

minus= +sum43

( ) ( )0l=sum

Synchronization2010722

Signal model (24)Signal model (24)g ( )g ( )

loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting

12

0

Nj kn N

k nn

Y y e πminus

minus

=

= sum

( )1 1

2

0

N Lj kn N

n l kn l

h n l x e Wπminus minus

minusminus

=

= +sumsum

loz The equation above could be expressed in matrix form

0n l

loz The equation above could be expressed in matrix form

= +Y HX W

44 Synchronization2010722

Signal model (34)Signal model (34)g ( )g ( )

H= = = = +Y Fy FCx FCF X HX W

loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal

y

H=H FCFcirculant matrix leading to be a diagonal matrix

=H FCF

( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0

h h L L hh h

⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥

( )( ) ( )

( ) ( )

1 10 1 0 0i

h Lh L h N L

⎢ ⎥⎢ ⎥minus minus⎢ ⎥

equiv= minus minus⎢ ⎥⎢ ⎥

C( ) ( )

( ) ( )

0 1 1 100

0 0 1 1 0

h L h N L

h N L L h N

⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

45

( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦

Synchronization2010722

Signal model (44)Signal model (44)g ( )g ( )

loz In a time-invariant channel the channel matrix H would be a diagonal matrix

k k k kY H X W= +

( )1 1

2

0 0

N Lj kl N

kn l

H h n l e πminus minus

minus

= =

= sumsum

loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI

k k k k kY H X I Wα= + +k k k k k

( ) ( )1 1 1

2 2 N N L

j m k Nj ml Nk mI X h n l e e ππ

minus minus minusminusminus⎛ ⎞

= ⎜ ⎟⎝ ⎠

sum sum sumICI

46

0 0 0m n lm k= = =ne

⎝ ⎠

Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

47 Synchronization2010722

Impact of timing offset Impact of timing offset p gp g

loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively

( )I f s

Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ

( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr

2 fj k Nk k k kY X H e Wπδminus= +

loz For integer part of timing offset k k k kY X H e W+

I sTδ ( ) 0g IN L δminus minus le leFFT

2 Ij k NY X H e Wπδminus= +

( ) ( ) 2 I sFFT

j f TI sh t T H f e π δδ minusminus rarr

48

k k k kY X H e W= +

Synchronization2010722

Impact of timing offset derivationImpact of timing offset derivationp gp g

49 Synchronization2010722

Discussion of symbol boundaryDiscussion of symbol boundaryy yy y

50 Synchronization2010722

Impact of timing offsetImpact of timing offsetp gp g

loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI

2 Ij k NIk k k k

NY X H e ISI W

Nπδδ minusminus

= + +

loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk

e e

p kloz Phase Shift can be resolved by Differential Encoding

Decoding or carrier recovery with pilots

51 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

52 Synchronization2010722

Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )

53 Synchronization2010722

Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )

( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s

i n t i N N T N T nTy y t e

= + + +=

( ) ( )( ) ( ) ( ) 12

sin

sin

g gI f f

I I

i N N N Nj jf N Ni k i k k

f

Y X H e eN

π ε ε π ε

ε ε

πεπε

+ + minus+

minus minus=⎛ ⎞⎜ ⎟

Faded signal attenuated and rotated by CFO

( )( ) ( ) ( ) ( )11 2

sin

sin g gI f I f

i N N N NN j j l kI f

NN

l k π ε ε π ε επ ε ε + + minusminus + + + minus

⎜ ⎟⎝ ⎠

+ + minussum

( )( )( )

( ) ( )2

0

sin

I f I f

I

j j l kI f N Ni l l

l l k I f

X H e el k

NN

π ε ε π ε ε

ε π ε ε

+ + +

= ne minus

+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠

sum

i k

N

W

⎜ ⎟⎝ ⎠

+ Inter-Carrier Interference (ICI)

54 Synchronization2010722

Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )

loz An easier representation of received signal with only p g yfractional CFO is

1N

( ) ( )1

0

0N

k k k l l kl l k

Y H X C H X C l k Wminus

= ne

= + minus +⎡ ⎤⎣ ⎦sum

where the ICI coefficient is given by

( )( )( ) ( )

sin 1exp 1fl k

C l k j l kπ ε

π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )

( )exp 1sin

f

f

C l k j l kNN l k

N

π επ ε

minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠

55 Synchronization2010722

Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )

loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality

loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift

56 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

57 Synchronization2010722

TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )

loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq

58 Synchronization2010722

TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )

loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below

loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and

rN2 can be expressed by

0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+

( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +

59

( )

Synchronization2010722

TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )

loz Without CFO the first half is identical to the second half (in time order)

loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below

( ) 12 in discretesft T f NT

NTφ π π π ε πε

⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟

⎝ ⎠

helliphellip

sNT⎝ ⎠

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)

60

Length=N2

Synchronization2010722

TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )

loz Received signal without CFOg0

0 0

jr a e θ=jθ 2 0

2 2 0

Nj j

N Nr a e a e

θ θ= =

loz Received signal with CFO0 0

0 0

j jr a e eθ φ= Extra phase rotation due to CFO0 0

2 2 20

2 2 0

N N Nj j jj

N Nr a e e a e e

θ φ φθ= =

due to CFO

Phase difference 2 2 0N N

( )2 02 20 2 0 0

Nj j T fN

r r a e a eφ φ πminus Δ= =

which contains the information about CFO

61

0 2 0 0Nr r a e a e

Synchronization2010722

TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )

loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be

loz Note that d is a time index corresponding to the first( ) ( ) 2 1

20

Nd m d m Nm

P d r rminus

+ + +== sum

loz Note that d is a time index corresponding to the first sample in a window of N samples

loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by

( )22 1

20

Nd m NR d rminus

+ += sumloz A timing metric can be defined as

))(()(

)( 2

2

dRdP

dM =

( ) 20 d m Nm + +=sum

62

))(( dR

Synchronization2010722

TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )

63 Synchronization2010722

TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )

loz If drsquo is the correct symbol timing offset

( )2

2 2

0 1 12

j T f j T f j T fN

P d a e a e a eπ π πΔ Δ Δ

minus= + + +

22

2 2

0 1 12

j T fN

e a a aπ Δ

minus

⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭

loz If drsquo falls behind the correct symbol timing offset 1 sample

2⎪ ⎪⎩ ⎭

64 Synchronization2010722

TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )

65 Synchronization2010722

TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )

loz Drawback Plateau effect

Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)

((N2) ((N2) ((N2)

helliphellip

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)

Sliding window (length = N)

66 Synchronization2010722

TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )

loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula

( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus

扣掉相加

( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +

loz The method also called delay correlator

67 Synchronization2010722

IEEE 80211aIEEE 80211a

68 Synchronization2010722

Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )

loz Minnrsquos Method [2][ ]B B －B －B

( )( )

( )

22

2Minn

P dM d =

1 4 1N minus

( )( )( )2

2

MinnR d

( ) ( ) ( ) ( )

1 4 1

2 2 2 40 0

N

d N m k d N m k Nm k

P d r r+ + + + += =

= sum sum

( ) ( ) ( )2

21 4 1

2 40 0

N

d N m k Nk

R d rminus

+ + += sum sum69

0 0m k= =

Synchronization2010722

Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )

loz Parkrsquos Method [3][ ]

where D is the symmetric version of CC D C D

where D is the symmetric version of C

( )( ) 2

3k

P dM d =( )

( )( )23

ParkM dR d

2N

( ) 2

30

N

d k d kk

P d r rminus +=

= sdotsum

( )3

22

0

N

d kk

R d r += sum70

0k=

Synchronization2010722

Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )

loz Renrsquos Method [4][ ]

AS AS

where S is the PN sequence weighted of the original bl 2preamble

( )( )( )( )

24

Re 24

n

P dM d

R d=

( )( )4R d

( ) 2 1

4 2 2

N

k k N d k d k NP d s s r rminus

+ + + += sum0k=sum

( )21

41 N

d kR d rminus

= sum71

( )402 d k

kR d r +

=sum

Synchronization2010722

Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )

72 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

73 Synchronization2010722

Fractional CFO estimation Fractional CFO estimation

loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected

11 Rminus⎛ ⎞sum

h L i th di t b t t id ti l bl k R i

0

12 d m d m L

ms

f r rLTπ + + +

=

⎛ ⎞= ang⎜ ⎟⎝ ⎠sum

where L is the distance between two identical block R is the block size

loz For exampleNN cp

L = N R = NL = N R = Ncp74 Synchronization2010722

Estimation rangeEstimation rangegg

loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval

f ( ) ( )s s⎣ ⎦

loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)

loz DVB-T (L=N) 802 11 (L N4)

[ ]05 05s sf fminus

[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)

(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus

loz Note that is the subcarrier spacing1sf NT=

75

sNT

Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz

plusmng q y

loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn

76 Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz

plusmng q y

loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for

subcarrier spacing is required

plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn

p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs

77 Synchronization2010722

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

Introduction (23)Introduction (23)( )( )

loz Transcevier

40 Synchronization2010722

Introduction (33)Introduction (33)( )( )

loz Synchronization erroryloz Symbol timing offset (TO)

loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)

loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect

41 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg

42 Synchronization2010722

Signal model (14)Signal model (14)g ( )g ( )

loz Time domain sample p1

2

0

1 Nj nk N

n kk

x X eN

πminus

minus= sum

Xk the frequency domain data at the k-th subcarrier0kN =

loz The multipath fading channel

( ) ( ) ( )1L

h t h t δminus

sum

loz Received time domain signal

( ) ( ) ( )0

l ll

h t h tτ τ δ τ τ=

= minussum

loz Received time-domain signal

( ) ( )1

L

n n ly h n l x w nminus

minus= +sum43

( ) ( )0l=sum

Synchronization2010722

Signal model (24)Signal model (24)g ( )g ( )

loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting

12

0

Nj kn N

k nn

Y y e πminus

minus

=

= sum

( )1 1

2

0

N Lj kn N

n l kn l

h n l x e Wπminus minus

minusminus

=

= +sumsum

loz The equation above could be expressed in matrix form

0n l

loz The equation above could be expressed in matrix form

= +Y HX W

44 Synchronization2010722

Signal model (34)Signal model (34)g ( )g ( )

H= = = = +Y Fy FCx FCF X HX W

loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal

y

H=H FCFcirculant matrix leading to be a diagonal matrix

=H FCF

( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0

h h L L hh h

⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥

( )( ) ( )

( ) ( )

1 10 1 0 0i

h Lh L h N L

⎢ ⎥⎢ ⎥minus minus⎢ ⎥

equiv= minus minus⎢ ⎥⎢ ⎥

C( ) ( )

( ) ( )

0 1 1 100

0 0 1 1 0

h L h N L

h N L L h N

⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

45

( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦

Synchronization2010722

Signal model (44)Signal model (44)g ( )g ( )

loz In a time-invariant channel the channel matrix H would be a diagonal matrix

k k k kY H X W= +

( )1 1

2

0 0

N Lj kl N

kn l

H h n l e πminus minus

minus

= =

= sumsum

loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI

k k k k kY H X I Wα= + +k k k k k

( ) ( )1 1 1

2 2 N N L

j m k Nj ml Nk mI X h n l e e ππ

minus minus minusminusminus⎛ ⎞

= ⎜ ⎟⎝ ⎠

sum sum sumICI

46

0 0 0m n lm k= = =ne

⎝ ⎠

Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

47 Synchronization2010722

Impact of timing offset Impact of timing offset p gp g

loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively

( )I f s

Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ

( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr

2 fj k Nk k k kY X H e Wπδminus= +

loz For integer part of timing offset k k k kY X H e W+

I sTδ ( ) 0g IN L δminus minus le leFFT

2 Ij k NY X H e Wπδminus= +

( ) ( ) 2 I sFFT

j f TI sh t T H f e π δδ minusminus rarr

48

k k k kY X H e W= +

Synchronization2010722

Impact of timing offset derivationImpact of timing offset derivationp gp g

49 Synchronization2010722

Discussion of symbol boundaryDiscussion of symbol boundaryy yy y

50 Synchronization2010722

Impact of timing offsetImpact of timing offsetp gp g

loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI

2 Ij k NIk k k k

NY X H e ISI W

Nπδδ minusminus

= + +

loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk

e e

p kloz Phase Shift can be resolved by Differential Encoding

Decoding or carrier recovery with pilots

51 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

52 Synchronization2010722

Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )

53 Synchronization2010722

Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )

( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s

i n t i N N T N T nTy y t e

= + + +=

( ) ( )( ) ( ) ( ) 12

sin

sin

g gI f f

I I

i N N N Nj jf N Ni k i k k

f

Y X H e eN

π ε ε π ε

ε ε

πεπε

+ + minus+

minus minus=⎛ ⎞⎜ ⎟

Faded signal attenuated and rotated by CFO

( )( ) ( ) ( ) ( )11 2

sin

sin g gI f I f

i N N N NN j j l kI f

NN

l k π ε ε π ε επ ε ε + + minusminus + + + minus

⎜ ⎟⎝ ⎠

+ + minussum

( )( )( )

( ) ( )2

0

sin

I f I f

I

j j l kI f N Ni l l

l l k I f

X H e el k

NN

π ε ε π ε ε

ε π ε ε

+ + +

= ne minus

+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠

sum

i k

N

W

⎜ ⎟⎝ ⎠

+ Inter-Carrier Interference (ICI)

54 Synchronization2010722

Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )

loz An easier representation of received signal with only p g yfractional CFO is

1N

( ) ( )1

0

0N

k k k l l kl l k

Y H X C H X C l k Wminus

= ne

= + minus +⎡ ⎤⎣ ⎦sum

where the ICI coefficient is given by

( )( )( ) ( )

sin 1exp 1fl k

C l k j l kπ ε

π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )

( )exp 1sin

f

f

C l k j l kNN l k

N

π επ ε

minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠

55 Synchronization2010722

Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )

loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality

loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift

56 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

57 Synchronization2010722

TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )

loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq

58 Synchronization2010722

TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )

loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below

loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and

rN2 can be expressed by

0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+

( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +

59

( )

Synchronization2010722

TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )

loz Without CFO the first half is identical to the second half (in time order)

loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below

( ) 12 in discretesft T f NT

NTφ π π π ε πε

⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟

⎝ ⎠

helliphellip

sNT⎝ ⎠

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)

60

Length=N2

Synchronization2010722

TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )

loz Received signal without CFOg0

0 0

jr a e θ=jθ 2 0

2 2 0

Nj j

N Nr a e a e

θ θ= =

loz Received signal with CFO0 0

0 0

j jr a e eθ φ= Extra phase rotation due to CFO0 0

2 2 20

2 2 0

N N Nj j jj

N Nr a e e a e e

θ φ φθ= =

due to CFO

Phase difference 2 2 0N N

( )2 02 20 2 0 0

Nj j T fN

r r a e a eφ φ πminus Δ= =

which contains the information about CFO

61

0 2 0 0Nr r a e a e

Synchronization2010722

TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )

loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be

loz Note that d is a time index corresponding to the first( ) ( ) 2 1

20

Nd m d m Nm

P d r rminus

+ + +== sum

loz Note that d is a time index corresponding to the first sample in a window of N samples

loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by

( )22 1

20

Nd m NR d rminus

+ += sumloz A timing metric can be defined as

))(()(

)( 2

2

dRdP

dM =

( ) 20 d m Nm + +=sum

62

))(( dR

Synchronization2010722

TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )

63 Synchronization2010722

TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )

loz If drsquo is the correct symbol timing offset

( )2

2 2

0 1 12

j T f j T f j T fN

P d a e a e a eπ π πΔ Δ Δ

minus= + + +

22

2 2

0 1 12

j T fN

e a a aπ Δ

minus

⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭

loz If drsquo falls behind the correct symbol timing offset 1 sample

2⎪ ⎪⎩ ⎭

64 Synchronization2010722

TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )

65 Synchronization2010722

TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )

loz Drawback Plateau effect

Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)

((N2) ((N2) ((N2)

helliphellip

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)

Sliding window (length = N)

66 Synchronization2010722

TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )

loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula

( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus

扣掉相加

( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +

loz The method also called delay correlator

67 Synchronization2010722

IEEE 80211aIEEE 80211a

68 Synchronization2010722

Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )

loz Minnrsquos Method [2][ ]B B －B －B

( )( )

( )

22

2Minn

P dM d =

1 4 1N minus

( )( )( )2

2

MinnR d

( ) ( ) ( ) ( )

1 4 1

2 2 2 40 0

N

d N m k d N m k Nm k

P d r r+ + + + += =

= sum sum

( ) ( ) ( )2

21 4 1

2 40 0

N

d N m k Nk

R d rminus

+ + += sum sum69

0 0m k= =

Synchronization2010722

Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )

loz Parkrsquos Method [3][ ]

where D is the symmetric version of CC D C D

where D is the symmetric version of C

( )( ) 2

3k

P dM d =( )

( )( )23

ParkM dR d

2N

( ) 2

30

N

d k d kk

P d r rminus +=

= sdotsum

( )3

22

0

N

d kk

R d r += sum70

0k=

Synchronization2010722

Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )

loz Renrsquos Method [4][ ]

AS AS

where S is the PN sequence weighted of the original bl 2preamble

( )( )( )( )

24

Re 24

n

P dM d

R d=

( )( )4R d

( ) 2 1

4 2 2

N

k k N d k d k NP d s s r rminus

+ + + += sum0k=sum

( )21

41 N

d kR d rminus

= sum71

( )402 d k

kR d r +

=sum

Synchronization2010722

Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )

72 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

73 Synchronization2010722

Fractional CFO estimation Fractional CFO estimation

loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected

11 Rminus⎛ ⎞sum

h L i th di t b t t id ti l bl k R i

0

12 d m d m L

ms

f r rLTπ + + +

=

⎛ ⎞= ang⎜ ⎟⎝ ⎠sum

where L is the distance between two identical block R is the block size

loz For exampleNN cp

L = N R = NL = N R = Ncp74 Synchronization2010722

Estimation rangeEstimation rangegg

loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval

f ( ) ( )s s⎣ ⎦

loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)

loz DVB-T (L=N) 802 11 (L N4)

[ ]05 05s sf fminus

[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)

(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus

loz Note that is the subcarrier spacing1sf NT=

75

sNT

Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz

plusmng q y

loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn

76 Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz

plusmng q y

loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for

subcarrier spacing is required

plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn

p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs

77 Synchronization2010722

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

Introduction (33)Introduction (33)( )( )

loz Synchronization erroryloz Symbol timing offset (TO)

loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)

loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect

41 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg

42 Synchronization2010722

Signal model (14)Signal model (14)g ( )g ( )

loz Time domain sample p1

2

0

1 Nj nk N

n kk

x X eN

πminus

minus= sum

Xk the frequency domain data at the k-th subcarrier0kN =

loz The multipath fading channel

( ) ( ) ( )1L

h t h t δminus

sum

loz Received time domain signal

( ) ( ) ( )0

l ll

h t h tτ τ δ τ τ=

= minussum

loz Received time-domain signal

( ) ( )1

L

n n ly h n l x w nminus

minus= +sum43

( ) ( )0l=sum

Synchronization2010722

Signal model (24)Signal model (24)g ( )g ( )

loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting

12

0

Nj kn N

k nn

Y y e πminus

minus

=

= sum

( )1 1

2

0

N Lj kn N

n l kn l

h n l x e Wπminus minus

minusminus

=

= +sumsum

loz The equation above could be expressed in matrix form

0n l

loz The equation above could be expressed in matrix form

= +Y HX W

44 Synchronization2010722

Signal model (34)Signal model (34)g ( )g ( )

H= = = = +Y Fy FCx FCF X HX W

loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal

y

H=H FCFcirculant matrix leading to be a diagonal matrix

=H FCF

( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0

h h L L hh h

⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥

( )( ) ( )

( ) ( )

1 10 1 0 0i

h Lh L h N L

⎢ ⎥⎢ ⎥minus minus⎢ ⎥

equiv= minus minus⎢ ⎥⎢ ⎥

C( ) ( )

( ) ( )

0 1 1 100

0 0 1 1 0

h L h N L

h N L L h N

⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

45

( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦

Synchronization2010722

Signal model (44)Signal model (44)g ( )g ( )

loz In a time-invariant channel the channel matrix H would be a diagonal matrix

k k k kY H X W= +

( )1 1

2

0 0

N Lj kl N

kn l

H h n l e πminus minus

minus

= =

= sumsum

loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI

k k k k kY H X I Wα= + +k k k k k

( ) ( )1 1 1

2 2 N N L

j m k Nj ml Nk mI X h n l e e ππ

minus minus minusminusminus⎛ ⎞

= ⎜ ⎟⎝ ⎠

sum sum sumICI

46

0 0 0m n lm k= = =ne

⎝ ⎠

Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

47 Synchronization2010722

Impact of timing offset Impact of timing offset p gp g

loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively

( )I f s

Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ

( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr

2 fj k Nk k k kY X H e Wπδminus= +

loz For integer part of timing offset k k k kY X H e W+

I sTδ ( ) 0g IN L δminus minus le leFFT

2 Ij k NY X H e Wπδminus= +

( ) ( ) 2 I sFFT

j f TI sh t T H f e π δδ minusminus rarr

48

k k k kY X H e W= +

Synchronization2010722

Impact of timing offset derivationImpact of timing offset derivationp gp g

49 Synchronization2010722

Discussion of symbol boundaryDiscussion of symbol boundaryy yy y

50 Synchronization2010722

Impact of timing offsetImpact of timing offsetp gp g

loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI

2 Ij k NIk k k k

NY X H e ISI W

Nπδδ minusminus

= + +

loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk

e e

p kloz Phase Shift can be resolved by Differential Encoding

Decoding or carrier recovery with pilots

51 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

52 Synchronization2010722

Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )

53 Synchronization2010722

Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )

( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s

i n t i N N T N T nTy y t e

= + + +=

( ) ( )( ) ( ) ( ) 12

sin

sin

g gI f f

I I

i N N N Nj jf N Ni k i k k

f

Y X H e eN

π ε ε π ε

ε ε

πεπε

+ + minus+

minus minus=⎛ ⎞⎜ ⎟

Faded signal attenuated and rotated by CFO

( )( ) ( ) ( ) ( )11 2

sin

sin g gI f I f

i N N N NN j j l kI f

NN

l k π ε ε π ε επ ε ε + + minusminus + + + minus

⎜ ⎟⎝ ⎠

+ + minussum

( )( )( )

( ) ( )2

0

sin

I f I f

I

j j l kI f N Ni l l

l l k I f

X H e el k

NN

π ε ε π ε ε

ε π ε ε

+ + +

= ne minus

+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠

sum

i k

N

W

⎜ ⎟⎝ ⎠

+ Inter-Carrier Interference (ICI)

54 Synchronization2010722

Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )

loz An easier representation of received signal with only p g yfractional CFO is

1N

( ) ( )1

0

0N

k k k l l kl l k

Y H X C H X C l k Wminus

= ne

= + minus +⎡ ⎤⎣ ⎦sum

where the ICI coefficient is given by

( )( )( ) ( )

sin 1exp 1fl k

C l k j l kπ ε

π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )

( )exp 1sin

f

f

C l k j l kNN l k

N

π επ ε

minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠

55 Synchronization2010722

Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )

loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality

loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift

56 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

57 Synchronization2010722

TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )

loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq

58 Synchronization2010722

TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )

loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below

loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and

rN2 can be expressed by

0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+

( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +

59

( )

Synchronization2010722

TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )

loz Without CFO the first half is identical to the second half (in time order)

loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below

( ) 12 in discretesft T f NT

NTφ π π π ε πε

⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟

⎝ ⎠

helliphellip

sNT⎝ ⎠

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)

60

Length=N2

Synchronization2010722

TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )

loz Received signal without CFOg0

0 0

jr a e θ=jθ 2 0

2 2 0

Nj j

N Nr a e a e

θ θ= =

loz Received signal with CFO0 0

0 0

j jr a e eθ φ= Extra phase rotation due to CFO0 0

2 2 20

2 2 0

N N Nj j jj

N Nr a e e a e e

θ φ φθ= =

due to CFO

Phase difference 2 2 0N N

( )2 02 20 2 0 0

Nj j T fN

r r a e a eφ φ πminus Δ= =

which contains the information about CFO

61

0 2 0 0Nr r a e a e

Synchronization2010722

TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )

loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be

loz Note that d is a time index corresponding to the first( ) ( ) 2 1

20

Nd m d m Nm

P d r rminus

+ + +== sum

loz Note that d is a time index corresponding to the first sample in a window of N samples

loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by

( )22 1

20

Nd m NR d rminus

+ += sumloz A timing metric can be defined as

))(()(

)( 2

2

dRdP

dM =

( ) 20 d m Nm + +=sum

62

))(( dR

Synchronization2010722

TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )

63 Synchronization2010722

TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )

loz If drsquo is the correct symbol timing offset

( )2

2 2

0 1 12

j T f j T f j T fN

P d a e a e a eπ π πΔ Δ Δ

minus= + + +

22

2 2

0 1 12

j T fN

e a a aπ Δ

minus

⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭

loz If drsquo falls behind the correct symbol timing offset 1 sample

2⎪ ⎪⎩ ⎭

64 Synchronization2010722

TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )

65 Synchronization2010722

TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )

loz Drawback Plateau effect

Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)

((N2) ((N2) ((N2)

helliphellip

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)

Sliding window (length = N)

66 Synchronization2010722

TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )

loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula

( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus

扣掉相加

( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +

loz The method also called delay correlator

67 Synchronization2010722

IEEE 80211aIEEE 80211a

68 Synchronization2010722

Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )

loz Minnrsquos Method [2][ ]B B －B －B

( )( )

( )

22

2Minn

P dM d =

1 4 1N minus

( )( )( )2

2

MinnR d

( ) ( ) ( ) ( )

1 4 1

2 2 2 40 0

N

d N m k d N m k Nm k

P d r r+ + + + += =

= sum sum

( ) ( ) ( )2

21 4 1

2 40 0

N

d N m k Nk

R d rminus

+ + += sum sum69

0 0m k= =

Synchronization2010722

Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )

loz Parkrsquos Method [3][ ]

where D is the symmetric version of CC D C D

where D is the symmetric version of C

( )( ) 2

3k

P dM d =( )

( )( )23

ParkM dR d

2N

( ) 2

30

N

d k d kk

P d r rminus +=

= sdotsum

( )3

22

0

N

d kk

R d r += sum70

0k=

Synchronization2010722

Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )

loz Renrsquos Method [4][ ]

AS AS

where S is the PN sequence weighted of the original bl 2preamble

( )( )( )( )

24

Re 24

n

P dM d

R d=

( )( )4R d

( ) 2 1

4 2 2

N

k k N d k d k NP d s s r rminus

+ + + += sum0k=sum

( )21

41 N

d kR d rminus

= sum71

( )402 d k

kR d r +

=sum

Synchronization2010722

Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )

72 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

73 Synchronization2010722

Fractional CFO estimation Fractional CFO estimation

loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected

11 Rminus⎛ ⎞sum

h L i th di t b t t id ti l bl k R i

0

12 d m d m L

ms

f r rLTπ + + +

=

⎛ ⎞= ang⎜ ⎟⎝ ⎠sum

where L is the distance between two identical block R is the block size

loz For exampleNN cp

L = N R = NL = N R = Ncp74 Synchronization2010722

Estimation rangeEstimation rangegg

loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval

f ( ) ( )s s⎣ ⎦

loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)

loz DVB-T (L=N) 802 11 (L N4)

[ ]05 05s sf fminus

[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)

(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus

loz Note that is the subcarrier spacing1sf NT=

75

sNT

Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz

plusmng q y

loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn

76 Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz

plusmng q y

loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for

subcarrier spacing is required

plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn

p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs

77 Synchronization2010722

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg

42 Synchronization2010722

Signal model (14)Signal model (14)g ( )g ( )

loz Time domain sample p1

2

0

1 Nj nk N

n kk

x X eN

πminus

minus= sum

Xk the frequency domain data at the k-th subcarrier0kN =

loz The multipath fading channel

( ) ( ) ( )1L

h t h t δminus

sum

loz Received time domain signal

( ) ( ) ( )0

l ll

h t h tτ τ δ τ τ=

= minussum

loz Received time-domain signal

( ) ( )1

L

n n ly h n l x w nminus

minus= +sum43

( ) ( )0l=sum

Synchronization2010722

Signal model (24)Signal model (24)g ( )g ( )

loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting

12

0

Nj kn N

k nn

Y y e πminus

minus

=

= sum

( )1 1

2

0

N Lj kn N

n l kn l

h n l x e Wπminus minus

minusminus

=

= +sumsum

loz The equation above could be expressed in matrix form

0n l

loz The equation above could be expressed in matrix form

= +Y HX W

44 Synchronization2010722

Signal model (34)Signal model (34)g ( )g ( )

H= = = = +Y Fy FCx FCF X HX W

loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal

y

H=H FCFcirculant matrix leading to be a diagonal matrix

=H FCF

( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0

h h L L hh h

⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥

( )( ) ( )

( ) ( )

1 10 1 0 0i

h Lh L h N L

⎢ ⎥⎢ ⎥minus minus⎢ ⎥

equiv= minus minus⎢ ⎥⎢ ⎥

C( ) ( )

( ) ( )

0 1 1 100

0 0 1 1 0

h L h N L

h N L L h N

⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

45

( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦

Synchronization2010722

Signal model (44)Signal model (44)g ( )g ( )

loz In a time-invariant channel the channel matrix H would be a diagonal matrix

k k k kY H X W= +

( )1 1

2

0 0

N Lj kl N

kn l

H h n l e πminus minus

minus

= =

= sumsum

loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI

k k k k kY H X I Wα= + +k k k k k

( ) ( )1 1 1

2 2 N N L

j m k Nj ml Nk mI X h n l e e ππ

minus minus minusminusminus⎛ ⎞

= ⎜ ⎟⎝ ⎠

sum sum sumICI

46

0 0 0m n lm k= = =ne

⎝ ⎠

Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

47 Synchronization2010722

Impact of timing offset Impact of timing offset p gp g

loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively

( )I f s

Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ

( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr

2 fj k Nk k k kY X H e Wπδminus= +

loz For integer part of timing offset k k k kY X H e W+

I sTδ ( ) 0g IN L δminus minus le leFFT

2 Ij k NY X H e Wπδminus= +

( ) ( ) 2 I sFFT

j f TI sh t T H f e π δδ minusminus rarr

48

k k k kY X H e W= +

Synchronization2010722

Impact of timing offset derivationImpact of timing offset derivationp gp g

49 Synchronization2010722

Discussion of symbol boundaryDiscussion of symbol boundaryy yy y

50 Synchronization2010722

Impact of timing offsetImpact of timing offsetp gp g

loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI

2 Ij k NIk k k k

NY X H e ISI W

Nπδδ minusminus

= + +

loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk

e e

p kloz Phase Shift can be resolved by Differential Encoding

Decoding or carrier recovery with pilots

51 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

52 Synchronization2010722

Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )

53 Synchronization2010722

Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )

( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s

i n t i N N T N T nTy y t e

= + + +=

( ) ( )( ) ( ) ( ) 12

sin

sin

g gI f f

I I

i N N N Nj jf N Ni k i k k

f

Y X H e eN

π ε ε π ε

ε ε

πεπε

+ + minus+

minus minus=⎛ ⎞⎜ ⎟

Faded signal attenuated and rotated by CFO

( )( ) ( ) ( ) ( )11 2

sin

sin g gI f I f

i N N N NN j j l kI f

NN

l k π ε ε π ε επ ε ε + + minusminus + + + minus

⎜ ⎟⎝ ⎠

+ + minussum

( )( )( )

( ) ( )2

0

sin

I f I f

I

j j l kI f N Ni l l

l l k I f

X H e el k

NN

π ε ε π ε ε

ε π ε ε

+ + +

= ne minus

+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠

sum

i k

N

W

⎜ ⎟⎝ ⎠

+ Inter-Carrier Interference (ICI)

54 Synchronization2010722

Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )

loz An easier representation of received signal with only p g yfractional CFO is

1N

( ) ( )1

0

0N

k k k l l kl l k

Y H X C H X C l k Wminus

= ne

= + minus +⎡ ⎤⎣ ⎦sum

where the ICI coefficient is given by

( )( )( ) ( )

sin 1exp 1fl k

C l k j l kπ ε

π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )

( )exp 1sin

f

f

C l k j l kNN l k

N

π επ ε

minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠

55 Synchronization2010722

Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )

loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality

loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift

56 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

57 Synchronization2010722

TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )

loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq

58 Synchronization2010722

TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )

loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below

loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and

rN2 can be expressed by

0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+

( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +

59

( )

Synchronization2010722

TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )

loz Without CFO the first half is identical to the second half (in time order)

loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below

( ) 12 in discretesft T f NT

NTφ π π π ε πε

⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟

⎝ ⎠

helliphellip

sNT⎝ ⎠

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)

60

Length=N2

Synchronization2010722

TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )

loz Received signal without CFOg0

0 0

jr a e θ=jθ 2 0

2 2 0

Nj j

N Nr a e a e

θ θ= =

loz Received signal with CFO0 0

0 0

j jr a e eθ φ= Extra phase rotation due to CFO0 0

2 2 20

2 2 0

N N Nj j jj

N Nr a e e a e e

θ φ φθ= =

due to CFO

Phase difference 2 2 0N N

( )2 02 20 2 0 0

Nj j T fN

r r a e a eφ φ πminus Δ= =

which contains the information about CFO

61

0 2 0 0Nr r a e a e

Synchronization2010722

TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )

loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be

loz Note that d is a time index corresponding to the first( ) ( ) 2 1

20

Nd m d m Nm

P d r rminus

+ + +== sum

loz Note that d is a time index corresponding to the first sample in a window of N samples

loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by

( )22 1

20

Nd m NR d rminus

+ += sumloz A timing metric can be defined as

))(()(

)( 2

2

dRdP

dM =

( ) 20 d m Nm + +=sum

62

))(( dR

Synchronization2010722

TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )

63 Synchronization2010722

TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )

loz If drsquo is the correct symbol timing offset

( )2

2 2

0 1 12

j T f j T f j T fN

P d a e a e a eπ π πΔ Δ Δ

minus= + + +

22

2 2

0 1 12

j T fN

e a a aπ Δ

minus

⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭

loz If drsquo falls behind the correct symbol timing offset 1 sample

2⎪ ⎪⎩ ⎭

64 Synchronization2010722

TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )

65 Synchronization2010722

TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )

loz Drawback Plateau effect

Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)

((N2) ((N2) ((N2)

helliphellip

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)

Sliding window (length = N)

66 Synchronization2010722

TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )

loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula

( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus

扣掉相加

( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +

loz The method also called delay correlator

67 Synchronization2010722

IEEE 80211aIEEE 80211a

68 Synchronization2010722

Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )

loz Minnrsquos Method [2][ ]B B －B －B

( )( )

( )

22

2Minn

P dM d =

1 4 1N minus

( )( )( )2

2

MinnR d

( ) ( ) ( ) ( )

1 4 1

2 2 2 40 0

N

d N m k d N m k Nm k

P d r r+ + + + += =

= sum sum

( ) ( ) ( )2

21 4 1

2 40 0

N

d N m k Nk

R d rminus

+ + += sum sum69

0 0m k= =

Synchronization2010722

Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )

loz Parkrsquos Method [3][ ]

where D is the symmetric version of CC D C D

where D is the symmetric version of C

( )( ) 2

3k

P dM d =( )

( )( )23

ParkM dR d

2N

( ) 2

30

N

d k d kk

P d r rminus +=

= sdotsum

( )3

22

0

N

d kk

R d r += sum70

0k=

Synchronization2010722

Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )

loz Renrsquos Method [4][ ]

AS AS

where S is the PN sequence weighted of the original bl 2preamble

( )( )( )( )

24

Re 24

n

P dM d

R d=

( )( )4R d

( ) 2 1

4 2 2

N

k k N d k d k NP d s s r rminus

+ + + += sum0k=sum

( )21

41 N

d kR d rminus

= sum71

( )402 d k

kR d r +

=sum

Synchronization2010722

Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )

72 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

73 Synchronization2010722

Fractional CFO estimation Fractional CFO estimation

loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected

11 Rminus⎛ ⎞sum

h L i th di t b t t id ti l bl k R i

0

12 d m d m L

ms

f r rLTπ + + +

=

⎛ ⎞= ang⎜ ⎟⎝ ⎠sum

where L is the distance between two identical block R is the block size

loz For exampleNN cp

L = N R = NL = N R = Ncp74 Synchronization2010722

Estimation rangeEstimation rangegg

loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval

f ( ) ( )s s⎣ ⎦

loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)

loz DVB-T (L=N) 802 11 (L N4)

[ ]05 05s sf fminus

[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)

(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus

loz Note that is the subcarrier spacing1sf NT=

75

sNT

Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz

plusmng q y

loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn

76 Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz

plusmng q y

loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for

subcarrier spacing is required

plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn

p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs

77 Synchronization2010722

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

Signal model (14)Signal model (14)g ( )g ( )

loz Time domain sample p1

2

0

1 Nj nk N

n kk

x X eN

πminus

minus= sum

Xk the frequency domain data at the k-th subcarrier0kN =

loz The multipath fading channel

( ) ( ) ( )1L

h t h t δminus

sum

loz Received time domain signal

( ) ( ) ( )0

l ll

h t h tτ τ δ τ τ=

= minussum

loz Received time-domain signal

( ) ( )1

L

n n ly h n l x w nminus

minus= +sum43

( ) ( )0l=sum

Synchronization2010722

Signal model (24)Signal model (24)g ( )g ( )

loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting

12

0

Nj kn N

k nn

Y y e πminus

minus

=

= sum

( )1 1

2

0

N Lj kn N

n l kn l

h n l x e Wπminus minus

minusminus

=

= +sumsum

loz The equation above could be expressed in matrix form

0n l

loz The equation above could be expressed in matrix form

= +Y HX W

44 Synchronization2010722

Signal model (34)Signal model (34)g ( )g ( )

H= = = = +Y Fy FCx FCF X HX W

loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal

y

H=H FCFcirculant matrix leading to be a diagonal matrix

=H FCF

( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0

h h L L hh h

⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥

( )( ) ( )

( ) ( )

1 10 1 0 0i

h Lh L h N L

⎢ ⎥⎢ ⎥minus minus⎢ ⎥

equiv= minus minus⎢ ⎥⎢ ⎥

C( ) ( )

( ) ( )

0 1 1 100

0 0 1 1 0

h L h N L

h N L L h N

⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

45

( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦

Synchronization2010722

Signal model (44)Signal model (44)g ( )g ( )

loz In a time-invariant channel the channel matrix H would be a diagonal matrix

k k k kY H X W= +

( )1 1

2

0 0

N Lj kl N

kn l

H h n l e πminus minus

minus

= =

= sumsum

loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI

k k k k kY H X I Wα= + +k k k k k

( ) ( )1 1 1

2 2 N N L

j m k Nj ml Nk mI X h n l e e ππ

minus minus minusminusminus⎛ ⎞

= ⎜ ⎟⎝ ⎠

sum sum sumICI

46

0 0 0m n lm k= = =ne

⎝ ⎠

Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

47 Synchronization2010722

Impact of timing offset Impact of timing offset p gp g

loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively

( )I f s

Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ

( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr

2 fj k Nk k k kY X H e Wπδminus= +

loz For integer part of timing offset k k k kY X H e W+

I sTδ ( ) 0g IN L δminus minus le leFFT

2 Ij k NY X H e Wπδminus= +

( ) ( ) 2 I sFFT

j f TI sh t T H f e π δδ minusminus rarr

48

k k k kY X H e W= +

Synchronization2010722

Impact of timing offset derivationImpact of timing offset derivationp gp g

49 Synchronization2010722

Discussion of symbol boundaryDiscussion of symbol boundaryy yy y

50 Synchronization2010722

Impact of timing offsetImpact of timing offsetp gp g

loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI

2 Ij k NIk k k k

NY X H e ISI W

Nπδδ minusminus

= + +

loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk

e e

p kloz Phase Shift can be resolved by Differential Encoding

Decoding or carrier recovery with pilots

51 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

52 Synchronization2010722

Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )

53 Synchronization2010722

Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )

( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s

i n t i N N T N T nTy y t e

= + + +=

( ) ( )( ) ( ) ( ) 12

sin

sin

g gI f f

I I

i N N N Nj jf N Ni k i k k

f

Y X H e eN

π ε ε π ε

ε ε

πεπε

+ + minus+

minus minus=⎛ ⎞⎜ ⎟

Faded signal attenuated and rotated by CFO

( )( ) ( ) ( ) ( )11 2

sin

sin g gI f I f

i N N N NN j j l kI f

NN

l k π ε ε π ε επ ε ε + + minusminus + + + minus

⎜ ⎟⎝ ⎠

+ + minussum

( )( )( )

( ) ( )2

0

sin

I f I f

I

j j l kI f N Ni l l

l l k I f

X H e el k

NN

π ε ε π ε ε

ε π ε ε

+ + +

= ne minus

+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠

sum

i k

N

W

⎜ ⎟⎝ ⎠

+ Inter-Carrier Interference (ICI)

54 Synchronization2010722

Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )

loz An easier representation of received signal with only p g yfractional CFO is

1N

( ) ( )1

0

0N

k k k l l kl l k

Y H X C H X C l k Wminus

= ne

= + minus +⎡ ⎤⎣ ⎦sum

where the ICI coefficient is given by

( )( )( ) ( )

sin 1exp 1fl k

C l k j l kπ ε

π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )

( )exp 1sin

f

f

C l k j l kNN l k

N

π επ ε

minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠

55 Synchronization2010722

Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )

loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality

loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift

56 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

57 Synchronization2010722

TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )

loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq

58 Synchronization2010722

TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )

loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below

loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and

rN2 can be expressed by

0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+

( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +

59

( )

Synchronization2010722

TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )

loz Without CFO the first half is identical to the second half (in time order)

loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below

( ) 12 in discretesft T f NT

NTφ π π π ε πε

⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟

⎝ ⎠

helliphellip

sNT⎝ ⎠

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)

60

Length=N2

Synchronization2010722

TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )

loz Received signal without CFOg0

0 0

jr a e θ=jθ 2 0

2 2 0

Nj j

N Nr a e a e

θ θ= =

loz Received signal with CFO0 0

0 0

j jr a e eθ φ= Extra phase rotation due to CFO0 0

2 2 20

2 2 0

N N Nj j jj

N Nr a e e a e e

θ φ φθ= =

due to CFO

Phase difference 2 2 0N N

( )2 02 20 2 0 0

Nj j T fN

r r a e a eφ φ πminus Δ= =

which contains the information about CFO

61

0 2 0 0Nr r a e a e

Synchronization2010722

TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )

loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be

loz Note that d is a time index corresponding to the first( ) ( ) 2 1

20

Nd m d m Nm

P d r rminus

+ + +== sum

loz Note that d is a time index corresponding to the first sample in a window of N samples

loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by

( )22 1

20

Nd m NR d rminus

+ += sumloz A timing metric can be defined as

))(()(

)( 2

2

dRdP

dM =

( ) 20 d m Nm + +=sum

62

))(( dR

Synchronization2010722

TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )

63 Synchronization2010722

TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )

loz If drsquo is the correct symbol timing offset

( )2

2 2

0 1 12

j T f j T f j T fN

P d a e a e a eπ π πΔ Δ Δ

minus= + + +

22

2 2

0 1 12

j T fN

e a a aπ Δ

minus

⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭

loz If drsquo falls behind the correct symbol timing offset 1 sample

2⎪ ⎪⎩ ⎭

64 Synchronization2010722

TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )

65 Synchronization2010722

TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )

loz Drawback Plateau effect

Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)

((N2) ((N2) ((N2)

helliphellip

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)

Sliding window (length = N)

66 Synchronization2010722

TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )

loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula

( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus

扣掉相加

( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +

loz The method also called delay correlator

67 Synchronization2010722

IEEE 80211aIEEE 80211a

68 Synchronization2010722

Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )

loz Minnrsquos Method [2][ ]B B －B －B

( )( )

( )

22

2Minn

P dM d =

1 4 1N minus

( )( )( )2

2

MinnR d

( ) ( ) ( ) ( )

1 4 1

2 2 2 40 0

N

d N m k d N m k Nm k

P d r r+ + + + += =

= sum sum

( ) ( ) ( )2

21 4 1

2 40 0

N

d N m k Nk

R d rminus

+ + += sum sum69

0 0m k= =

Synchronization2010722

Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )

loz Parkrsquos Method [3][ ]

where D is the symmetric version of CC D C D

where D is the symmetric version of C

( )( ) 2

3k

P dM d =( )

( )( )23

ParkM dR d

2N

( ) 2

30

N

d k d kk

P d r rminus +=

= sdotsum

( )3

22

0

N

d kk

R d r += sum70

0k=

Synchronization2010722

Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )

loz Renrsquos Method [4][ ]

AS AS

where S is the PN sequence weighted of the original bl 2preamble

( )( )( )( )

24

Re 24

n

P dM d

R d=

( )( )4R d

( ) 2 1

4 2 2

N

k k N d k d k NP d s s r rminus

+ + + += sum0k=sum

( )21

41 N

d kR d rminus

= sum71

( )402 d k

kR d r +

=sum

Synchronization2010722

Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )

72 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

73 Synchronization2010722

Fractional CFO estimation Fractional CFO estimation

loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected

11 Rminus⎛ ⎞sum

h L i th di t b t t id ti l bl k R i

0

12 d m d m L

ms

f r rLTπ + + +

=

⎛ ⎞= ang⎜ ⎟⎝ ⎠sum

where L is the distance between two identical block R is the block size

loz For exampleNN cp

L = N R = NL = N R = Ncp74 Synchronization2010722

Estimation rangeEstimation rangegg

loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval

f ( ) ( )s s⎣ ⎦

loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)

loz DVB-T (L=N) 802 11 (L N4)

[ ]05 05s sf fminus

[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)

(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus

loz Note that is the subcarrier spacing1sf NT=

75

sNT

Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz

plusmng q y

loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn

76 Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz

plusmng q y

loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for

subcarrier spacing is required

plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn

p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs

77 Synchronization2010722

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

Signal model (24)Signal model (24)g ( )g ( )

loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting

12

0

Nj kn N

k nn

Y y e πminus

minus

=

= sum

( )1 1

2

0

N Lj kn N

n l kn l

h n l x e Wπminus minus

minusminus

=

= +sumsum

loz The equation above could be expressed in matrix form

0n l

loz The equation above could be expressed in matrix form

= +Y HX W

44 Synchronization2010722

Signal model (34)Signal model (34)g ( )g ( )

H= = = = +Y Fy FCx FCF X HX W

loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal

y

H=H FCFcirculant matrix leading to be a diagonal matrix

=H FCF

( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0

h h L L hh h

⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥

( )( ) ( )

( ) ( )

1 10 1 0 0i

h Lh L h N L

⎢ ⎥⎢ ⎥minus minus⎢ ⎥

equiv= minus minus⎢ ⎥⎢ ⎥

C( ) ( )

( ) ( )

0 1 1 100

0 0 1 1 0

h L h N L

h N L L h N

⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

45

( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦

Synchronization2010722

Signal model (44)Signal model (44)g ( )g ( )

loz In a time-invariant channel the channel matrix H would be a diagonal matrix

k k k kY H X W= +

( )1 1

2

0 0

N Lj kl N

kn l

H h n l e πminus minus

minus

= =

= sumsum

loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI

k k k k kY H X I Wα= + +k k k k k

( ) ( )1 1 1

2 2 N N L

j m k Nj ml Nk mI X h n l e e ππ

minus minus minusminusminus⎛ ⎞

= ⎜ ⎟⎝ ⎠

sum sum sumICI

46

0 0 0m n lm k= = =ne

⎝ ⎠

Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

47 Synchronization2010722

Impact of timing offset Impact of timing offset p gp g

loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively

( )I f s

Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ

( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr

2 fj k Nk k k kY X H e Wπδminus= +

loz For integer part of timing offset k k k kY X H e W+

I sTδ ( ) 0g IN L δminus minus le leFFT

2 Ij k NY X H e Wπδminus= +

( ) ( ) 2 I sFFT

j f TI sh t T H f e π δδ minusminus rarr

48

k k k kY X H e W= +

Synchronization2010722

Impact of timing offset derivationImpact of timing offset derivationp gp g

49 Synchronization2010722

Discussion of symbol boundaryDiscussion of symbol boundaryy yy y

50 Synchronization2010722

Impact of timing offsetImpact of timing offsetp gp g

loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI

2 Ij k NIk k k k

NY X H e ISI W

Nπδδ minusminus

= + +

loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk

e e

p kloz Phase Shift can be resolved by Differential Encoding

Decoding or carrier recovery with pilots

51 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

52 Synchronization2010722

Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )

53 Synchronization2010722

Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )

( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s

i n t i N N T N T nTy y t e

= + + +=

( ) ( )( ) ( ) ( ) 12

sin

sin

g gI f f

I I

i N N N Nj jf N Ni k i k k

f

Y X H e eN

π ε ε π ε

ε ε

πεπε

+ + minus+

minus minus=⎛ ⎞⎜ ⎟

Faded signal attenuated and rotated by CFO

( )( ) ( ) ( ) ( )11 2

sin

sin g gI f I f

i N N N NN j j l kI f

NN

l k π ε ε π ε επ ε ε + + minusminus + + + minus

⎜ ⎟⎝ ⎠

+ + minussum

( )( )( )

( ) ( )2

0

sin

I f I f

I

j j l kI f N Ni l l

l l k I f

X H e el k

NN

π ε ε π ε ε

ε π ε ε

+ + +

= ne minus

+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠

sum

i k

N

W

⎜ ⎟⎝ ⎠

+ Inter-Carrier Interference (ICI)

54 Synchronization2010722

Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )

loz An easier representation of received signal with only p g yfractional CFO is

1N

( ) ( )1

0

0N

k k k l l kl l k

Y H X C H X C l k Wminus

= ne

= + minus +⎡ ⎤⎣ ⎦sum

where the ICI coefficient is given by

( )( )( ) ( )

sin 1exp 1fl k

C l k j l kπ ε

π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )

( )exp 1sin

f

f

C l k j l kNN l k

N

π επ ε

minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠

55 Synchronization2010722

Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )

loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality

loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift

56 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

57 Synchronization2010722

TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )

loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq

58 Synchronization2010722

TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )

loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below

loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and

rN2 can be expressed by

0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+

( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +

59

( )

Synchronization2010722

TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )

loz Without CFO the first half is identical to the second half (in time order)

loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below

( ) 12 in discretesft T f NT

NTφ π π π ε πε

⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟

⎝ ⎠

helliphellip

sNT⎝ ⎠

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)

60

Length=N2

Synchronization2010722

TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )

loz Received signal without CFOg0

0 0

jr a e θ=jθ 2 0

2 2 0

Nj j

N Nr a e a e

θ θ= =

loz Received signal with CFO0 0

0 0

j jr a e eθ φ= Extra phase rotation due to CFO0 0

2 2 20

2 2 0

N N Nj j jj

N Nr a e e a e e

θ φ φθ= =

due to CFO

Phase difference 2 2 0N N

( )2 02 20 2 0 0

Nj j T fN

r r a e a eφ φ πminus Δ= =

which contains the information about CFO

61

0 2 0 0Nr r a e a e

Synchronization2010722

TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )

loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be

loz Note that d is a time index corresponding to the first( ) ( ) 2 1

20

Nd m d m Nm

P d r rminus

+ + +== sum

loz Note that d is a time index corresponding to the first sample in a window of N samples

loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by

( )22 1

20

Nd m NR d rminus

+ += sumloz A timing metric can be defined as

))(()(

)( 2

2

dRdP

dM =

( ) 20 d m Nm + +=sum

62

))(( dR

Synchronization2010722

TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )

63 Synchronization2010722

TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )

loz If drsquo is the correct symbol timing offset

( )2

2 2

0 1 12

j T f j T f j T fN

P d a e a e a eπ π πΔ Δ Δ

minus= + + +

22

2 2

0 1 12

j T fN

e a a aπ Δ

minus

⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭

loz If drsquo falls behind the correct symbol timing offset 1 sample

2⎪ ⎪⎩ ⎭

64 Synchronization2010722

TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )

65 Synchronization2010722

TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )

loz Drawback Plateau effect

Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)

((N2) ((N2) ((N2)

helliphellip

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)

Sliding window (length = N)

66 Synchronization2010722

TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )

loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula

( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus

扣掉相加

( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +

loz The method also called delay correlator

67 Synchronization2010722

IEEE 80211aIEEE 80211a

68 Synchronization2010722

Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )

loz Minnrsquos Method [2][ ]B B －B －B

( )( )

( )

22

2Minn

P dM d =

1 4 1N minus

( )( )( )2

2

MinnR d

( ) ( ) ( ) ( )

1 4 1

2 2 2 40 0

N

d N m k d N m k Nm k

P d r r+ + + + += =

= sum sum

( ) ( ) ( )2

21 4 1

2 40 0

N

d N m k Nk

R d rminus

+ + += sum sum69

0 0m k= =

Synchronization2010722

Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )

loz Parkrsquos Method [3][ ]

where D is the symmetric version of CC D C D

where D is the symmetric version of C

( )( ) 2

3k

P dM d =( )

( )( )23

ParkM dR d

2N

( ) 2

30

N

d k d kk

P d r rminus +=

= sdotsum

( )3

22

0

N

d kk

R d r += sum70

0k=

Synchronization2010722

Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )

loz Renrsquos Method [4][ ]

AS AS

where S is the PN sequence weighted of the original bl 2preamble

( )( )( )( )

24

Re 24

n

P dM d

R d=

( )( )4R d

( ) 2 1

4 2 2

N

k k N d k d k NP d s s r rminus

+ + + += sum0k=sum

( )21

41 N

d kR d rminus

= sum71

( )402 d k

kR d r +

=sum

Synchronization2010722

Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )

72 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

73 Synchronization2010722

Fractional CFO estimation Fractional CFO estimation

loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected

11 Rminus⎛ ⎞sum

h L i th di t b t t id ti l bl k R i

0

12 d m d m L

ms

f r rLTπ + + +

=

⎛ ⎞= ang⎜ ⎟⎝ ⎠sum

where L is the distance between two identical block R is the block size

loz For exampleNN cp

L = N R = NL = N R = Ncp74 Synchronization2010722

Estimation rangeEstimation rangegg

loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval

f ( ) ( )s s⎣ ⎦

loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)

loz DVB-T (L=N) 802 11 (L N4)

[ ]05 05s sf fminus

[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)

(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus

loz Note that is the subcarrier spacing1sf NT=

75

sNT

Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz

plusmng q y

loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn

76 Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz

plusmng q y

loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for

subcarrier spacing is required

plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn

p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs

77 Synchronization2010722

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

Signal model (34)Signal model (34)g ( )g ( )

H= = = = +Y Fy FCx FCF X HX W

loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal

y

H=H FCFcirculant matrix leading to be a diagonal matrix

=H FCF

( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0

h h L L hh h

⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥

( )( ) ( )

( ) ( )

1 10 1 0 0i

h Lh L h N L

⎢ ⎥⎢ ⎥minus minus⎢ ⎥

equiv= minus minus⎢ ⎥⎢ ⎥

C( ) ( )

( ) ( )

0 1 1 100

0 0 1 1 0

h L h N L

h N L L h N

⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

45

( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦

Synchronization2010722

Signal model (44)Signal model (44)g ( )g ( )

loz In a time-invariant channel the channel matrix H would be a diagonal matrix

k k k kY H X W= +

( )1 1

2

0 0

N Lj kl N

kn l

H h n l e πminus minus

minus

= =

= sumsum

loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI

k k k k kY H X I Wα= + +k k k k k

( ) ( )1 1 1

2 2 N N L

j m k Nj ml Nk mI X h n l e e ππ

minus minus minusminusminus⎛ ⎞

= ⎜ ⎟⎝ ⎠

sum sum sumICI

46

0 0 0m n lm k= = =ne

⎝ ⎠

Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

47 Synchronization2010722

Impact of timing offset Impact of timing offset p gp g

loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively

( )I f s

Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ

( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr

2 fj k Nk k k kY X H e Wπδminus= +

loz For integer part of timing offset k k k kY X H e W+

I sTδ ( ) 0g IN L δminus minus le leFFT

2 Ij k NY X H e Wπδminus= +

( ) ( ) 2 I sFFT

j f TI sh t T H f e π δδ minusminus rarr

48

k k k kY X H e W= +

Synchronization2010722

Impact of timing offset derivationImpact of timing offset derivationp gp g

49 Synchronization2010722

Discussion of symbol boundaryDiscussion of symbol boundaryy yy y

50 Synchronization2010722

Impact of timing offsetImpact of timing offsetp gp g

loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI

2 Ij k NIk k k k

NY X H e ISI W

Nπδδ minusminus

= + +

loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk

e e

p kloz Phase Shift can be resolved by Differential Encoding

Decoding or carrier recovery with pilots

51 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

52 Synchronization2010722

Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )

53 Synchronization2010722

Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )

( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s

i n t i N N T N T nTy y t e

= + + +=

( ) ( )( ) ( ) ( ) 12

sin

sin

g gI f f

I I

i N N N Nj jf N Ni k i k k

f

Y X H e eN

π ε ε π ε

ε ε

πεπε

+ + minus+

minus minus=⎛ ⎞⎜ ⎟

Faded signal attenuated and rotated by CFO

( )( ) ( ) ( ) ( )11 2

sin

sin g gI f I f

i N N N NN j j l kI f

NN

l k π ε ε π ε επ ε ε + + minusminus + + + minus

⎜ ⎟⎝ ⎠

+ + minussum

( )( )( )

( ) ( )2

0

sin

I f I f

I

j j l kI f N Ni l l

l l k I f

X H e el k

NN

π ε ε π ε ε

ε π ε ε

+ + +

= ne minus

+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠

sum

i k

N

W

⎜ ⎟⎝ ⎠

+ Inter-Carrier Interference (ICI)

54 Synchronization2010722

Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )

loz An easier representation of received signal with only p g yfractional CFO is

1N

( ) ( )1

0

0N

k k k l l kl l k

Y H X C H X C l k Wminus

= ne

= + minus +⎡ ⎤⎣ ⎦sum

where the ICI coefficient is given by

( )( )( ) ( )

sin 1exp 1fl k

C l k j l kπ ε

π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )

( )exp 1sin

f

f

C l k j l kNN l k

N

π επ ε

minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠

55 Synchronization2010722

Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )

loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality

loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift

56 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

57 Synchronization2010722

TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )

loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq

58 Synchronization2010722

TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )

loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below

loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and

rN2 can be expressed by

0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+

( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +

59

( )

Synchronization2010722

TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )

loz Without CFO the first half is identical to the second half (in time order)

loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below

( ) 12 in discretesft T f NT

NTφ π π π ε πε

⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟

⎝ ⎠

helliphellip

sNT⎝ ⎠

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)

60

Length=N2

Synchronization2010722

TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )

loz Received signal without CFOg0

0 0

jr a e θ=jθ 2 0

2 2 0

Nj j

N Nr a e a e

θ θ= =

loz Received signal with CFO0 0

0 0

j jr a e eθ φ= Extra phase rotation due to CFO0 0

2 2 20

2 2 0

N N Nj j jj

N Nr a e e a e e

θ φ φθ= =

due to CFO

Phase difference 2 2 0N N

( )2 02 20 2 0 0

Nj j T fN

r r a e a eφ φ πminus Δ= =

which contains the information about CFO

61

0 2 0 0Nr r a e a e

Synchronization2010722

TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )

loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be

loz Note that d is a time index corresponding to the first( ) ( ) 2 1

20

Nd m d m Nm

P d r rminus

+ + +== sum

loz Note that d is a time index corresponding to the first sample in a window of N samples

loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by

( )22 1

20

Nd m NR d rminus

+ += sumloz A timing metric can be defined as

))(()(

)( 2

2

dRdP

dM =

( ) 20 d m Nm + +=sum

62

))(( dR

Synchronization2010722

TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )

63 Synchronization2010722

TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )

loz If drsquo is the correct symbol timing offset

( )2

2 2

0 1 12

j T f j T f j T fN

P d a e a e a eπ π πΔ Δ Δ

minus= + + +

22

2 2

0 1 12

j T fN

e a a aπ Δ

minus

⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭

loz If drsquo falls behind the correct symbol timing offset 1 sample

2⎪ ⎪⎩ ⎭

64 Synchronization2010722

TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )

65 Synchronization2010722

TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )

loz Drawback Plateau effect

Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)

((N2) ((N2) ((N2)

helliphellip

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)

Sliding window (length = N)

66 Synchronization2010722

TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )

loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula

( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus

扣掉相加

( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +

loz The method also called delay correlator

67 Synchronization2010722

IEEE 80211aIEEE 80211a

68 Synchronization2010722

Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )

loz Minnrsquos Method [2][ ]B B －B －B

( )( )

( )

22

2Minn

P dM d =

1 4 1N minus

( )( )( )2

2

MinnR d

( ) ( ) ( ) ( )

1 4 1

2 2 2 40 0

N

d N m k d N m k Nm k

P d r r+ + + + += =

= sum sum

( ) ( ) ( )2

21 4 1

2 40 0

N

d N m k Nk

R d rminus

+ + += sum sum69

0 0m k= =

Synchronization2010722

Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )

loz Parkrsquos Method [3][ ]

where D is the symmetric version of CC D C D

where D is the symmetric version of C

( )( ) 2

3k

P dM d =( )

( )( )23

ParkM dR d

2N

( ) 2

30

N

d k d kk

P d r rminus +=

= sdotsum

( )3

22

0

N

d kk

R d r += sum70

0k=

Synchronization2010722

Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )

loz Renrsquos Method [4][ ]

AS AS

where S is the PN sequence weighted of the original bl 2preamble

( )( )( )( )

24

Re 24

n

P dM d

R d=

( )( )4R d

( ) 2 1

4 2 2

N

k k N d k d k NP d s s r rminus

+ + + += sum0k=sum

( )21

41 N

d kR d rminus

= sum71

( )402 d k

kR d r +

=sum

Synchronization2010722

Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )

72 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

73 Synchronization2010722

Fractional CFO estimation Fractional CFO estimation

loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected

11 Rminus⎛ ⎞sum

h L i th di t b t t id ti l bl k R i

0

12 d m d m L

ms

f r rLTπ + + +

=

⎛ ⎞= ang⎜ ⎟⎝ ⎠sum

where L is the distance between two identical block R is the block size

loz For exampleNN cp

L = N R = NL = N R = Ncp74 Synchronization2010722

Estimation rangeEstimation rangegg

loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval

f ( ) ( )s s⎣ ⎦

loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)

loz DVB-T (L=N) 802 11 (L N4)

[ ]05 05s sf fminus

[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)

(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus

loz Note that is the subcarrier spacing1sf NT=

75

sNT

Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz

plusmng q y

loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn

76 Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz

plusmng q y

loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for

subcarrier spacing is required

plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn

p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs

77 Synchronization2010722

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

Signal model (44)Signal model (44)g ( )g ( )

loz In a time-invariant channel the channel matrix H would be a diagonal matrix

k k k kY H X W= +

( )1 1

2

0 0

N Lj kl N

kn l

H h n l e πminus minus

minus

= =

= sumsum

loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI

k k k k kY H X I Wα= + +k k k k k

( ) ( )1 1 1

2 2 N N L

j m k Nj ml Nk mI X h n l e e ππ

minus minus minusminusminus⎛ ⎞

= ⎜ ⎟⎝ ⎠

sum sum sumICI

46

0 0 0m n lm k= = =ne

⎝ ⎠

Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

47 Synchronization2010722

Impact of timing offset Impact of timing offset p gp g

loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively

( )I f s

Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ

( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr

2 fj k Nk k k kY X H e Wπδminus= +

loz For integer part of timing offset k k k kY X H e W+

I sTδ ( ) 0g IN L δminus minus le leFFT

2 Ij k NY X H e Wπδminus= +

( ) ( ) 2 I sFFT

j f TI sh t T H f e π δδ minusminus rarr

48

k k k kY X H e W= +

Synchronization2010722

Impact of timing offset derivationImpact of timing offset derivationp gp g

49 Synchronization2010722

Discussion of symbol boundaryDiscussion of symbol boundaryy yy y

50 Synchronization2010722

Impact of timing offsetImpact of timing offsetp gp g

loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI

2 Ij k NIk k k k

NY X H e ISI W

Nπδδ minusminus

= + +

loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk

e e

p kloz Phase Shift can be resolved by Differential Encoding

Decoding or carrier recovery with pilots

51 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

52 Synchronization2010722

Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )

53 Synchronization2010722

Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )

( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s

i n t i N N T N T nTy y t e

= + + +=

( ) ( )( ) ( ) ( ) 12

sin

sin

g gI f f

I I

i N N N Nj jf N Ni k i k k

f

Y X H e eN

π ε ε π ε

ε ε

πεπε

+ + minus+

minus minus=⎛ ⎞⎜ ⎟

Faded signal attenuated and rotated by CFO

( )( ) ( ) ( ) ( )11 2

sin

sin g gI f I f

i N N N NN j j l kI f

NN

l k π ε ε π ε επ ε ε + + minusminus + + + minus

⎜ ⎟⎝ ⎠

+ + minussum

( )( )( )

( ) ( )2

0

sin

I f I f

I

j j l kI f N Ni l l

l l k I f

X H e el k

NN

π ε ε π ε ε

ε π ε ε

+ + +

= ne minus

+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠

sum

i k

N

W

⎜ ⎟⎝ ⎠

+ Inter-Carrier Interference (ICI)

54 Synchronization2010722

Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )

loz An easier representation of received signal with only p g yfractional CFO is

1N

( ) ( )1

0

0N

k k k l l kl l k

Y H X C H X C l k Wminus

= ne

= + minus +⎡ ⎤⎣ ⎦sum

where the ICI coefficient is given by

( )( )( ) ( )

sin 1exp 1fl k

C l k j l kπ ε

π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )

( )exp 1sin

f

f

C l k j l kNN l k

N

π επ ε

minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠

55 Synchronization2010722

Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )

loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality

loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift

56 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

57 Synchronization2010722

TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )

loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq

58 Synchronization2010722

TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )

loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below

loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and

rN2 can be expressed by

0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+

( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +

59

( )

Synchronization2010722

TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )

loz Without CFO the first half is identical to the second half (in time order)

loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below

( ) 12 in discretesft T f NT

NTφ π π π ε πε

⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟

⎝ ⎠

helliphellip

sNT⎝ ⎠

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)

60

Length=N2

Synchronization2010722

TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )

loz Received signal without CFOg0

0 0

jr a e θ=jθ 2 0

2 2 0

Nj j

N Nr a e a e

θ θ= =

loz Received signal with CFO0 0

0 0

j jr a e eθ φ= Extra phase rotation due to CFO0 0

2 2 20

2 2 0

N N Nj j jj

N Nr a e e a e e

θ φ φθ= =

due to CFO

Phase difference 2 2 0N N

( )2 02 20 2 0 0

Nj j T fN

r r a e a eφ φ πminus Δ= =

which contains the information about CFO

61

0 2 0 0Nr r a e a e

Synchronization2010722

TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )

loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be

loz Note that d is a time index corresponding to the first( ) ( ) 2 1

20

Nd m d m Nm

P d r rminus

+ + +== sum

loz Note that d is a time index corresponding to the first sample in a window of N samples

loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by

( )22 1

20

Nd m NR d rminus

+ += sumloz A timing metric can be defined as

))(()(

)( 2

2

dRdP

dM =

( ) 20 d m Nm + +=sum

62

))(( dR

Synchronization2010722

TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )

63 Synchronization2010722

TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )

loz If drsquo is the correct symbol timing offset

( )2

2 2

0 1 12

j T f j T f j T fN

P d a e a e a eπ π πΔ Δ Δ

minus= + + +

22

2 2

0 1 12

j T fN

e a a aπ Δ

minus

⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭

loz If drsquo falls behind the correct symbol timing offset 1 sample

2⎪ ⎪⎩ ⎭

64 Synchronization2010722

TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )

65 Synchronization2010722

TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )

loz Drawback Plateau effect

Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)

((N2) ((N2) ((N2)

helliphellip

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)

Sliding window (length = N)

66 Synchronization2010722

TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )

loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula

( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus

扣掉相加

( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +

loz The method also called delay correlator

67 Synchronization2010722

IEEE 80211aIEEE 80211a

68 Synchronization2010722

Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )

loz Minnrsquos Method [2][ ]B B －B －B

( )( )

( )

22

2Minn

P dM d =

1 4 1N minus

( )( )( )2

2

MinnR d

( ) ( ) ( ) ( )

1 4 1

2 2 2 40 0

N

d N m k d N m k Nm k

P d r r+ + + + += =

= sum sum

( ) ( ) ( )2

21 4 1

2 40 0

N

d N m k Nk

R d rminus

+ + += sum sum69

0 0m k= =

Synchronization2010722

Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )

loz Parkrsquos Method [3][ ]

where D is the symmetric version of CC D C D

where D is the symmetric version of C

( )( ) 2

3k

P dM d =( )

( )( )23

ParkM dR d

2N

( ) 2

30

N

d k d kk

P d r rminus +=

= sdotsum

( )3

22

0

N

d kk

R d r += sum70

0k=

Synchronization2010722

Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )

loz Renrsquos Method [4][ ]

AS AS

where S is the PN sequence weighted of the original bl 2preamble

( )( )( )( )

24

Re 24

n

P dM d

R d=

( )( )4R d

( ) 2 1

4 2 2

N

k k N d k d k NP d s s r rminus

+ + + += sum0k=sum

( )21

41 N

d kR d rminus

= sum71

( )402 d k

kR d r +

=sum

Synchronization2010722

Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )

72 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

73 Synchronization2010722

Fractional CFO estimation Fractional CFO estimation

loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected

11 Rminus⎛ ⎞sum

h L i th di t b t t id ti l bl k R i

0

12 d m d m L

ms

f r rLTπ + + +

=

⎛ ⎞= ang⎜ ⎟⎝ ⎠sum

where L is the distance between two identical block R is the block size

loz For exampleNN cp

L = N R = NL = N R = Ncp74 Synchronization2010722

Estimation rangeEstimation rangegg

loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval

f ( ) ( )s s⎣ ⎦

loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)

loz DVB-T (L=N) 802 11 (L N4)

[ ]05 05s sf fminus

[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)

(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus

loz Note that is the subcarrier spacing1sf NT=

75

sNT

Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz

plusmng q y

loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn

76 Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz

plusmng q y

loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for

subcarrier spacing is required

plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn

p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs

77 Synchronization2010722

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

47 Synchronization2010722

Impact of timing offset Impact of timing offset p gp g

loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively

( )I f s

Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ

( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr

2 fj k Nk k k kY X H e Wπδminus= +

loz For integer part of timing offset k k k kY X H e W+

I sTδ ( ) 0g IN L δminus minus le leFFT

2 Ij k NY X H e Wπδminus= +

( ) ( ) 2 I sFFT

j f TI sh t T H f e π δδ minusminus rarr

48

k k k kY X H e W= +

Synchronization2010722

Impact of timing offset derivationImpact of timing offset derivationp gp g

49 Synchronization2010722

Discussion of symbol boundaryDiscussion of symbol boundaryy yy y

50 Synchronization2010722

Impact of timing offsetImpact of timing offsetp gp g

loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI

2 Ij k NIk k k k

NY X H e ISI W

Nπδδ minusminus

= + +

loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk

e e

p kloz Phase Shift can be resolved by Differential Encoding

Decoding or carrier recovery with pilots

51 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

52 Synchronization2010722

Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )

53 Synchronization2010722

Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )

( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s

i n t i N N T N T nTy y t e

= + + +=

( ) ( )( ) ( ) ( ) 12

sin

sin

g gI f f

I I

i N N N Nj jf N Ni k i k k

f

Y X H e eN

π ε ε π ε

ε ε

πεπε

+ + minus+

minus minus=⎛ ⎞⎜ ⎟

Faded signal attenuated and rotated by CFO

( )( ) ( ) ( ) ( )11 2

sin

sin g gI f I f

i N N N NN j j l kI f

NN

l k π ε ε π ε επ ε ε + + minusminus + + + minus

⎜ ⎟⎝ ⎠

+ + minussum

( )( )( )

( ) ( )2

0

sin

I f I f

I

j j l kI f N Ni l l

l l k I f

X H e el k

NN

π ε ε π ε ε

ε π ε ε

+ + +

= ne minus

+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠

sum

i k

N

W

⎜ ⎟⎝ ⎠

+ Inter-Carrier Interference (ICI)

54 Synchronization2010722

Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )

loz An easier representation of received signal with only p g yfractional CFO is

1N

( ) ( )1

0

0N

k k k l l kl l k

Y H X C H X C l k Wminus

= ne

= + minus +⎡ ⎤⎣ ⎦sum

where the ICI coefficient is given by

( )( )( ) ( )

sin 1exp 1fl k

C l k j l kπ ε

π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )

( )exp 1sin

f

f

C l k j l kNN l k

N

π επ ε

minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠

55 Synchronization2010722

Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )

loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality

loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift

56 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

57 Synchronization2010722

TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )

loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq

58 Synchronization2010722

TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )

loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below

loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and

rN2 can be expressed by

0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+

( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +

59

( )

Synchronization2010722

TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )

loz Without CFO the first half is identical to the second half (in time order)

loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below

( ) 12 in discretesft T f NT

NTφ π π π ε πε

⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟

⎝ ⎠

helliphellip

sNT⎝ ⎠

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)

60

Length=N2

Synchronization2010722

TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )

loz Received signal without CFOg0

0 0

jr a e θ=jθ 2 0

2 2 0

Nj j

N Nr a e a e

θ θ= =

loz Received signal with CFO0 0

0 0

j jr a e eθ φ= Extra phase rotation due to CFO0 0

2 2 20

2 2 0

N N Nj j jj

N Nr a e e a e e

θ φ φθ= =

due to CFO

Phase difference 2 2 0N N

( )2 02 20 2 0 0

Nj j T fN

r r a e a eφ φ πminus Δ= =

which contains the information about CFO

61

0 2 0 0Nr r a e a e

Synchronization2010722

TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )

loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be

loz Note that d is a time index corresponding to the first( ) ( ) 2 1

20

Nd m d m Nm

P d r rminus

+ + +== sum

loz Note that d is a time index corresponding to the first sample in a window of N samples

loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by

( )22 1

20

Nd m NR d rminus

+ += sumloz A timing metric can be defined as

))(()(

)( 2

2

dRdP

dM =

( ) 20 d m Nm + +=sum

62

))(( dR

Synchronization2010722

TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )

63 Synchronization2010722

TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )

loz If drsquo is the correct symbol timing offset

( )2

2 2

0 1 12

j T f j T f j T fN

P d a e a e a eπ π πΔ Δ Δ

minus= + + +

22

2 2

0 1 12

j T fN

e a a aπ Δ

minus

⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭

loz If drsquo falls behind the correct symbol timing offset 1 sample

2⎪ ⎪⎩ ⎭

64 Synchronization2010722

TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )

65 Synchronization2010722

TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )

loz Drawback Plateau effect

Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)

((N2) ((N2) ((N2)

helliphellip

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)

Sliding window (length = N)

66 Synchronization2010722

TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )

loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula

( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus

扣掉相加

( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +

loz The method also called delay correlator

67 Synchronization2010722

IEEE 80211aIEEE 80211a

68 Synchronization2010722

Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )

loz Minnrsquos Method [2][ ]B B －B －B

( )( )

( )

22

2Minn

P dM d =

1 4 1N minus

( )( )( )2

2

MinnR d

( ) ( ) ( ) ( )

1 4 1

2 2 2 40 0

N

d N m k d N m k Nm k

P d r r+ + + + += =

= sum sum

( ) ( ) ( )2

21 4 1

2 40 0

N

d N m k Nk

R d rminus

+ + += sum sum69

0 0m k= =

Synchronization2010722

Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )

loz Parkrsquos Method [3][ ]

where D is the symmetric version of CC D C D

where D is the symmetric version of C

( )( ) 2

3k

P dM d =( )

( )( )23

ParkM dR d

2N

( ) 2

30

N

d k d kk

P d r rminus +=

= sdotsum

( )3

22

0

N

d kk

R d r += sum70

0k=

Synchronization2010722

Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )

loz Renrsquos Method [4][ ]

AS AS

where S is the PN sequence weighted of the original bl 2preamble

( )( )( )( )

24

Re 24

n

P dM d

R d=

( )( )4R d

( ) 2 1

4 2 2

N

k k N d k d k NP d s s r rminus

+ + + += sum0k=sum

( )21

41 N

d kR d rminus

= sum71

( )402 d k

kR d r +

=sum

Synchronization2010722

Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )

72 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

73 Synchronization2010722

Fractional CFO estimation Fractional CFO estimation

loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected

11 Rminus⎛ ⎞sum

h L i th di t b t t id ti l bl k R i

0

12 d m d m L

ms

f r rLTπ + + +

=

⎛ ⎞= ang⎜ ⎟⎝ ⎠sum

where L is the distance between two identical block R is the block size

loz For exampleNN cp

L = N R = NL = N R = Ncp74 Synchronization2010722

Estimation rangeEstimation rangegg

loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval

f ( ) ( )s s⎣ ⎦

loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)

loz DVB-T (L=N) 802 11 (L N4)

[ ]05 05s sf fminus

[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)

(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus

loz Note that is the subcarrier spacing1sf NT=

75

sNT

Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz

plusmng q y

loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn

76 Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz

plusmng q y

loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for

subcarrier spacing is required

plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn

p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs

77 Synchronization2010722

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

Impact of timing offset Impact of timing offset p gp g

loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively

( )I f s

Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ

( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr

2 fj k Nk k k kY X H e Wπδminus= +

loz For integer part of timing offset k k k kY X H e W+

I sTδ ( ) 0g IN L δminus minus le leFFT

2 Ij k NY X H e Wπδminus= +

( ) ( ) 2 I sFFT

j f TI sh t T H f e π δδ minusminus rarr

48

k k k kY X H e W= +

Synchronization2010722

Impact of timing offset derivationImpact of timing offset derivationp gp g

49 Synchronization2010722

Discussion of symbol boundaryDiscussion of symbol boundaryy yy y

50 Synchronization2010722

Impact of timing offsetImpact of timing offsetp gp g

loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI

2 Ij k NIk k k k

NY X H e ISI W

Nπδδ minusminus

= + +

loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk

e e

p kloz Phase Shift can be resolved by Differential Encoding

Decoding or carrier recovery with pilots

51 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

52 Synchronization2010722

Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )

53 Synchronization2010722

Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )

( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s

i n t i N N T N T nTy y t e

= + + +=

( ) ( )( ) ( ) ( ) 12

sin

sin

g gI f f

I I

i N N N Nj jf N Ni k i k k

f

Y X H e eN

π ε ε π ε

ε ε

πεπε

+ + minus+

minus minus=⎛ ⎞⎜ ⎟

Faded signal attenuated and rotated by CFO

( )( ) ( ) ( ) ( )11 2

sin

sin g gI f I f

i N N N NN j j l kI f

NN

l k π ε ε π ε επ ε ε + + minusminus + + + minus

⎜ ⎟⎝ ⎠

+ + minussum

( )( )( )

( ) ( )2

0

sin

I f I f

I

j j l kI f N Ni l l

l l k I f

X H e el k

NN

π ε ε π ε ε

ε π ε ε

+ + +

= ne minus

+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠

sum

i k

N

W

⎜ ⎟⎝ ⎠

+ Inter-Carrier Interference (ICI)

54 Synchronization2010722

Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )

loz An easier representation of received signal with only p g yfractional CFO is

1N

( ) ( )1

0

0N

k k k l l kl l k

Y H X C H X C l k Wminus

= ne

= + minus +⎡ ⎤⎣ ⎦sum

where the ICI coefficient is given by

( )( )( ) ( )

sin 1exp 1fl k

C l k j l kπ ε

π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )

( )exp 1sin

f

f

C l k j l kNN l k

N

π επ ε

minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠

55 Synchronization2010722

Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )

loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality

loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift

56 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

57 Synchronization2010722

TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )

loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq

58 Synchronization2010722

TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )

loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below

loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and

rN2 can be expressed by

0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+

( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +

59

( )

Synchronization2010722

TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )

loz Without CFO the first half is identical to the second half (in time order)

loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below

( ) 12 in discretesft T f NT

NTφ π π π ε πε

⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟

⎝ ⎠

helliphellip

sNT⎝ ⎠

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)

60

Length=N2

Synchronization2010722

TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )

loz Received signal without CFOg0

0 0

jr a e θ=jθ 2 0

2 2 0

Nj j

N Nr a e a e

θ θ= =

loz Received signal with CFO0 0

0 0

j jr a e eθ φ= Extra phase rotation due to CFO0 0

2 2 20

2 2 0

N N Nj j jj

N Nr a e e a e e

θ φ φθ= =

due to CFO

Phase difference 2 2 0N N

( )2 02 20 2 0 0

Nj j T fN

r r a e a eφ φ πminus Δ= =

which contains the information about CFO

61

0 2 0 0Nr r a e a e

Synchronization2010722

TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )

loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be

loz Note that d is a time index corresponding to the first( ) ( ) 2 1

20

Nd m d m Nm

P d r rminus

+ + +== sum

loz Note that d is a time index corresponding to the first sample in a window of N samples

loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by

( )22 1

20

Nd m NR d rminus

+ += sumloz A timing metric can be defined as

))(()(

)( 2

2

dRdP

dM =

( ) 20 d m Nm + +=sum

62

))(( dR

Synchronization2010722

TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )

63 Synchronization2010722

TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )

loz If drsquo is the correct symbol timing offset

( )2

2 2

0 1 12

j T f j T f j T fN

P d a e a e a eπ π πΔ Δ Δ

minus= + + +

22

2 2

0 1 12

j T fN

e a a aπ Δ

minus

⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭

loz If drsquo falls behind the correct symbol timing offset 1 sample

2⎪ ⎪⎩ ⎭

64 Synchronization2010722

TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )

65 Synchronization2010722

TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )

loz Drawback Plateau effect

Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)

((N2) ((N2) ((N2)

helliphellip

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)

Sliding window (length = N)

66 Synchronization2010722

TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )

loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula

( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus

扣掉相加

( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +

loz The method also called delay correlator

67 Synchronization2010722

IEEE 80211aIEEE 80211a

68 Synchronization2010722

Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )

loz Minnrsquos Method [2][ ]B B －B －B

( )( )

( )

22

2Minn

P dM d =

1 4 1N minus

( )( )( )2

2

MinnR d

( ) ( ) ( ) ( )

1 4 1

2 2 2 40 0

N

d N m k d N m k Nm k

P d r r+ + + + += =

= sum sum

( ) ( ) ( )2

21 4 1

2 40 0

N

d N m k Nk

R d rminus

+ + += sum sum69

0 0m k= =

Synchronization2010722

Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )

loz Parkrsquos Method [3][ ]

where D is the symmetric version of CC D C D

where D is the symmetric version of C

( )( ) 2

3k

P dM d =( )

( )( )23

ParkM dR d

2N

( ) 2

30

N

d k d kk

P d r rminus +=

= sdotsum

( )3

22

0

N

d kk

R d r += sum70

0k=

Synchronization2010722

Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )

loz Renrsquos Method [4][ ]

AS AS

where S is the PN sequence weighted of the original bl 2preamble

( )( )( )( )

24

Re 24

n

P dM d

R d=

( )( )4R d

( ) 2 1

4 2 2

N

k k N d k d k NP d s s r rminus

+ + + += sum0k=sum

( )21

41 N

d kR d rminus

= sum71

( )402 d k

kR d r +

=sum

Synchronization2010722

Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )

72 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

73 Synchronization2010722

Fractional CFO estimation Fractional CFO estimation

loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected

11 Rminus⎛ ⎞sum

h L i th di t b t t id ti l bl k R i

0

12 d m d m L

ms

f r rLTπ + + +

=

⎛ ⎞= ang⎜ ⎟⎝ ⎠sum

where L is the distance between two identical block R is the block size

loz For exampleNN cp

L = N R = NL = N R = Ncp74 Synchronization2010722

Estimation rangeEstimation rangegg

loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval

f ( ) ( )s s⎣ ⎦

loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)

loz DVB-T (L=N) 802 11 (L N4)

[ ]05 05s sf fminus

[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)

(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus

loz Note that is the subcarrier spacing1sf NT=

75

sNT

Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz

plusmng q y

loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn

76 Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz

plusmng q y

loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for

subcarrier spacing is required

plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn

p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs

77 Synchronization2010722

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

Impact of timing offset derivationImpact of timing offset derivationp gp g

49 Synchronization2010722

Discussion of symbol boundaryDiscussion of symbol boundaryy yy y

50 Synchronization2010722

Impact of timing offsetImpact of timing offsetp gp g

loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI

2 Ij k NIk k k k

NY X H e ISI W

Nπδδ minusminus

= + +

loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk

e e

p kloz Phase Shift can be resolved by Differential Encoding

Decoding or carrier recovery with pilots

51 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

52 Synchronization2010722

Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )

53 Synchronization2010722

Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )

( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s

i n t i N N T N T nTy y t e

= + + +=

( ) ( )( ) ( ) ( ) 12

sin

sin

g gI f f

I I

i N N N Nj jf N Ni k i k k

f

Y X H e eN

π ε ε π ε

ε ε

πεπε

+ + minus+

minus minus=⎛ ⎞⎜ ⎟

Faded signal attenuated and rotated by CFO

( )( ) ( ) ( ) ( )11 2

sin

sin g gI f I f

i N N N NN j j l kI f

NN

l k π ε ε π ε επ ε ε + + minusminus + + + minus

⎜ ⎟⎝ ⎠

+ + minussum

( )( )( )

( ) ( )2

0

sin

I f I f

I

j j l kI f N Ni l l

l l k I f

X H e el k

NN

π ε ε π ε ε

ε π ε ε

+ + +

= ne minus

+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠

sum

i k

N

W

⎜ ⎟⎝ ⎠

+ Inter-Carrier Interference (ICI)

54 Synchronization2010722

Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )

loz An easier representation of received signal with only p g yfractional CFO is

1N

( ) ( )1

0

0N

k k k l l kl l k

Y H X C H X C l k Wminus

= ne

= + minus +⎡ ⎤⎣ ⎦sum

where the ICI coefficient is given by

( )( )( ) ( )

sin 1exp 1fl k

C l k j l kπ ε

π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )

( )exp 1sin

f

f

C l k j l kNN l k

N

π επ ε

minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠

55 Synchronization2010722

Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )

loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality

loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift

56 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

57 Synchronization2010722

TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )

loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq

58 Synchronization2010722

TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )

loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below

loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and

rN2 can be expressed by

0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+

( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +

59

( )

Synchronization2010722

TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )

loz Without CFO the first half is identical to the second half (in time order)

loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below

( ) 12 in discretesft T f NT

NTφ π π π ε πε

⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟

⎝ ⎠

helliphellip

sNT⎝ ⎠

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)

60

Length=N2

Synchronization2010722

TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )

loz Received signal without CFOg0

0 0

jr a e θ=jθ 2 0

2 2 0

Nj j

N Nr a e a e

θ θ= =

loz Received signal with CFO0 0

0 0

j jr a e eθ φ= Extra phase rotation due to CFO0 0

2 2 20

2 2 0

N N Nj j jj

N Nr a e e a e e

θ φ φθ= =

due to CFO

Phase difference 2 2 0N N

( )2 02 20 2 0 0

Nj j T fN

r r a e a eφ φ πminus Δ= =

which contains the information about CFO

61

0 2 0 0Nr r a e a e

Synchronization2010722

TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )

loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be

loz Note that d is a time index corresponding to the first( ) ( ) 2 1

20

Nd m d m Nm

P d r rminus

+ + +== sum

loz Note that d is a time index corresponding to the first sample in a window of N samples

loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by

( )22 1

20

Nd m NR d rminus

+ += sumloz A timing metric can be defined as

))(()(

)( 2

2

dRdP

dM =

( ) 20 d m Nm + +=sum

62

))(( dR

Synchronization2010722

TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )

63 Synchronization2010722

TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )

loz If drsquo is the correct symbol timing offset

( )2

2 2

0 1 12

j T f j T f j T fN

P d a e a e a eπ π πΔ Δ Δ

minus= + + +

22

2 2

0 1 12

j T fN

e a a aπ Δ

minus

⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭

loz If drsquo falls behind the correct symbol timing offset 1 sample

2⎪ ⎪⎩ ⎭

64 Synchronization2010722

TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )

65 Synchronization2010722

TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )

loz Drawback Plateau effect

Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)

((N2) ((N2) ((N2)

helliphellip

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)

Sliding window (length = N)

66 Synchronization2010722

TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )

loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula

( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus

扣掉相加

( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +

loz The method also called delay correlator

67 Synchronization2010722

IEEE 80211aIEEE 80211a

68 Synchronization2010722

Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )

loz Minnrsquos Method [2][ ]B B －B －B

( )( )

( )

22

2Minn

P dM d =

1 4 1N minus

( )( )( )2

2

MinnR d

( ) ( ) ( ) ( )

1 4 1

2 2 2 40 0

N

d N m k d N m k Nm k

P d r r+ + + + += =

= sum sum

( ) ( ) ( )2

21 4 1

2 40 0

N

d N m k Nk

R d rminus

+ + += sum sum69

0 0m k= =

Synchronization2010722

Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )

loz Parkrsquos Method [3][ ]

where D is the symmetric version of CC D C D

where D is the symmetric version of C

( )( ) 2

3k

P dM d =( )

( )( )23

ParkM dR d

2N

( ) 2

30

N

d k d kk

P d r rminus +=

= sdotsum

( )3

22

0

N

d kk

R d r += sum70

0k=

Synchronization2010722

Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )

loz Renrsquos Method [4][ ]

AS AS

where S is the PN sequence weighted of the original bl 2preamble

( )( )( )( )

24

Re 24

n

P dM d

R d=

( )( )4R d

( ) 2 1

4 2 2

N

k k N d k d k NP d s s r rminus

+ + + += sum0k=sum

( )21

41 N

d kR d rminus

= sum71

( )402 d k

kR d r +

=sum

Synchronization2010722

Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )

72 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

73 Synchronization2010722

Fractional CFO estimation Fractional CFO estimation

loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected

11 Rminus⎛ ⎞sum

h L i th di t b t t id ti l bl k R i

0

12 d m d m L

ms

f r rLTπ + + +

=

⎛ ⎞= ang⎜ ⎟⎝ ⎠sum

where L is the distance between two identical block R is the block size

loz For exampleNN cp

L = N R = NL = N R = Ncp74 Synchronization2010722

Estimation rangeEstimation rangegg

loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval

f ( ) ( )s s⎣ ⎦

loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)

loz DVB-T (L=N) 802 11 (L N4)

[ ]05 05s sf fminus

[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)

(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus

loz Note that is the subcarrier spacing1sf NT=

75

sNT

Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz

plusmng q y

loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn

76 Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz

plusmng q y

loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for

subcarrier spacing is required

plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn

p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs

77 Synchronization2010722

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

Discussion of symbol boundaryDiscussion of symbol boundaryy yy y

50 Synchronization2010722

Impact of timing offsetImpact of timing offsetp gp g

loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI

2 Ij k NIk k k k

NY X H e ISI W

Nπδδ minusminus

= + +

loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk

e e

p kloz Phase Shift can be resolved by Differential Encoding

Decoding or carrier recovery with pilots

51 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

52 Synchronization2010722

Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )

53 Synchronization2010722

Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )

( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s

i n t i N N T N T nTy y t e

= + + +=

( ) ( )( ) ( ) ( ) 12

sin

sin

g gI f f

I I

i N N N Nj jf N Ni k i k k

f

Y X H e eN

π ε ε π ε

ε ε

πεπε

+ + minus+

minus minus=⎛ ⎞⎜ ⎟

Faded signal attenuated and rotated by CFO

( )( ) ( ) ( ) ( )11 2

sin

sin g gI f I f

i N N N NN j j l kI f

NN

l k π ε ε π ε επ ε ε + + minusminus + + + minus

⎜ ⎟⎝ ⎠

+ + minussum

( )( )( )

( ) ( )2

0

sin

I f I f

I

j j l kI f N Ni l l

l l k I f

X H e el k

NN

π ε ε π ε ε

ε π ε ε

+ + +

= ne minus

+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠

sum

i k

N

W

⎜ ⎟⎝ ⎠

+ Inter-Carrier Interference (ICI)

54 Synchronization2010722

Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )

loz An easier representation of received signal with only p g yfractional CFO is

1N

( ) ( )1

0

0N

k k k l l kl l k

Y H X C H X C l k Wminus

= ne

= + minus +⎡ ⎤⎣ ⎦sum

where the ICI coefficient is given by

( )( )( ) ( )

sin 1exp 1fl k

C l k j l kπ ε

π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )

( )exp 1sin

f

f

C l k j l kNN l k

N

π επ ε

minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠

55 Synchronization2010722

Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )

loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality

loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift

56 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

57 Synchronization2010722

TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )

loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq

58 Synchronization2010722

TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )

loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below

loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and

rN2 can be expressed by

0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+

( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +

59

( )

Synchronization2010722

TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )

loz Without CFO the first half is identical to the second half (in time order)

loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below

( ) 12 in discretesft T f NT

NTφ π π π ε πε

⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟

⎝ ⎠

helliphellip

sNT⎝ ⎠

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)

60

Length=N2

Synchronization2010722

TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )

loz Received signal without CFOg0

0 0

jr a e θ=jθ 2 0

2 2 0

Nj j

N Nr a e a e

θ θ= =

loz Received signal with CFO0 0

0 0

j jr a e eθ φ= Extra phase rotation due to CFO0 0

2 2 20

2 2 0

N N Nj j jj

N Nr a e e a e e

θ φ φθ= =

due to CFO

Phase difference 2 2 0N N

( )2 02 20 2 0 0

Nj j T fN

r r a e a eφ φ πminus Δ= =

which contains the information about CFO

61

0 2 0 0Nr r a e a e

Synchronization2010722

TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )

loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be

loz Note that d is a time index corresponding to the first( ) ( ) 2 1

20

Nd m d m Nm

P d r rminus

+ + +== sum

loz Note that d is a time index corresponding to the first sample in a window of N samples

loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by

( )22 1

20

Nd m NR d rminus

+ += sumloz A timing metric can be defined as

))(()(

)( 2

2

dRdP

dM =

( ) 20 d m Nm + +=sum

62

))(( dR

Synchronization2010722

TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )

63 Synchronization2010722

TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )

loz If drsquo is the correct symbol timing offset

( )2

2 2

0 1 12

j T f j T f j T fN

P d a e a e a eπ π πΔ Δ Δ

minus= + + +

22

2 2

0 1 12

j T fN

e a a aπ Δ

minus

⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭

loz If drsquo falls behind the correct symbol timing offset 1 sample

2⎪ ⎪⎩ ⎭

64 Synchronization2010722

TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )

65 Synchronization2010722

TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )

loz Drawback Plateau effect

Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)

((N2) ((N2) ((N2)

helliphellip

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)

Sliding window (length = N)

66 Synchronization2010722

TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )

loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula

( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus

扣掉相加

( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +

loz The method also called delay correlator

67 Synchronization2010722

IEEE 80211aIEEE 80211a

68 Synchronization2010722

Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )

loz Minnrsquos Method [2][ ]B B －B －B

( )( )

( )

22

2Minn

P dM d =

1 4 1N minus

( )( )( )2

2

MinnR d

( ) ( ) ( ) ( )

1 4 1

2 2 2 40 0

N

d N m k d N m k Nm k

P d r r+ + + + += =

= sum sum

( ) ( ) ( )2

21 4 1

2 40 0

N

d N m k Nk

R d rminus

+ + += sum sum69

0 0m k= =

Synchronization2010722

Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )

loz Parkrsquos Method [3][ ]

where D is the symmetric version of CC D C D

where D is the symmetric version of C

( )( ) 2

3k

P dM d =( )

( )( )23

ParkM dR d

2N

( ) 2

30

N

d k d kk

P d r rminus +=

= sdotsum

( )3

22

0

N

d kk

R d r += sum70

0k=

Synchronization2010722

Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )

loz Renrsquos Method [4][ ]

AS AS

where S is the PN sequence weighted of the original bl 2preamble

( )( )( )( )

24

Re 24

n

P dM d

R d=

( )( )4R d

( ) 2 1

4 2 2

N

k k N d k d k NP d s s r rminus

+ + + += sum0k=sum

( )21

41 N

d kR d rminus

= sum71

( )402 d k

kR d r +

=sum

Synchronization2010722

Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )

72 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

73 Synchronization2010722

Fractional CFO estimation Fractional CFO estimation

loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected

11 Rminus⎛ ⎞sum

h L i th di t b t t id ti l bl k R i

0

12 d m d m L

ms

f r rLTπ + + +

=

⎛ ⎞= ang⎜ ⎟⎝ ⎠sum

where L is the distance between two identical block R is the block size

loz For exampleNN cp

L = N R = NL = N R = Ncp74 Synchronization2010722

Estimation rangeEstimation rangegg

loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval

f ( ) ( )s s⎣ ⎦

loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)

loz DVB-T (L=N) 802 11 (L N4)

[ ]05 05s sf fminus

[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)

(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus

loz Note that is the subcarrier spacing1sf NT=

75

sNT

Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz

plusmng q y

loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn

76 Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz

plusmng q y

loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for

subcarrier spacing is required

plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn

p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs

77 Synchronization2010722

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

Impact of timing offsetImpact of timing offsetp gp g

loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI

2 Ij k NIk k k k

NY X H e ISI W

Nπδδ minusminus

= + +

loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk

e e

p kloz Phase Shift can be resolved by Differential Encoding

Decoding or carrier recovery with pilots

51 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

52 Synchronization2010722

Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )

53 Synchronization2010722

Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )

( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s

i n t i N N T N T nTy y t e

= + + +=

( ) ( )( ) ( ) ( ) 12

sin

sin

g gI f f

I I

i N N N Nj jf N Ni k i k k

f

Y X H e eN

π ε ε π ε

ε ε

πεπε

+ + minus+

minus minus=⎛ ⎞⎜ ⎟

Faded signal attenuated and rotated by CFO

( )( ) ( ) ( ) ( )11 2

sin

sin g gI f I f

i N N N NN j j l kI f

NN

l k π ε ε π ε επ ε ε + + minusminus + + + minus

⎜ ⎟⎝ ⎠

+ + minussum

( )( )( )

( ) ( )2

0

sin

I f I f

I

j j l kI f N Ni l l

l l k I f

X H e el k

NN

π ε ε π ε ε

ε π ε ε

+ + +

= ne minus

+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠

sum

i k

N

W

⎜ ⎟⎝ ⎠

+ Inter-Carrier Interference (ICI)

54 Synchronization2010722

Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )

loz An easier representation of received signal with only p g yfractional CFO is

1N

( ) ( )1

0

0N

k k k l l kl l k

Y H X C H X C l k Wminus

= ne

= + minus +⎡ ⎤⎣ ⎦sum

where the ICI coefficient is given by

( )( )( ) ( )

sin 1exp 1fl k

C l k j l kπ ε

π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )

( )exp 1sin

f

f

C l k j l kNN l k

N

π επ ε

minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠

55 Synchronization2010722

Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )

loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality

loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift

56 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

57 Synchronization2010722

TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )

loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq

58 Synchronization2010722

TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )

loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below

loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and

rN2 can be expressed by

0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+

( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +

59

( )

Synchronization2010722

TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )

loz Without CFO the first half is identical to the second half (in time order)

loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below

( ) 12 in discretesft T f NT

NTφ π π π ε πε

⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟

⎝ ⎠

helliphellip

sNT⎝ ⎠

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)

60

Length=N2

Synchronization2010722

TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )

loz Received signal without CFOg0

0 0

jr a e θ=jθ 2 0

2 2 0

Nj j

N Nr a e a e

θ θ= =

loz Received signal with CFO0 0

0 0

j jr a e eθ φ= Extra phase rotation due to CFO0 0

2 2 20

2 2 0

N N Nj j jj

N Nr a e e a e e

θ φ φθ= =

due to CFO

Phase difference 2 2 0N N

( )2 02 20 2 0 0

Nj j T fN

r r a e a eφ φ πminus Δ= =

which contains the information about CFO

61

0 2 0 0Nr r a e a e

Synchronization2010722

TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )

loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be

loz Note that d is a time index corresponding to the first( ) ( ) 2 1

20

Nd m d m Nm

P d r rminus

+ + +== sum

loz Note that d is a time index corresponding to the first sample in a window of N samples

loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by

( )22 1

20

Nd m NR d rminus

+ += sumloz A timing metric can be defined as

))(()(

)( 2

2

dRdP

dM =

( ) 20 d m Nm + +=sum

62

))(( dR

Synchronization2010722

TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )

63 Synchronization2010722

TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )

loz If drsquo is the correct symbol timing offset

( )2

2 2

0 1 12

j T f j T f j T fN

P d a e a e a eπ π πΔ Δ Δ

minus= + + +

22

2 2

0 1 12

j T fN

e a a aπ Δ

minus

⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭

loz If drsquo falls behind the correct symbol timing offset 1 sample

2⎪ ⎪⎩ ⎭

64 Synchronization2010722

TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )

65 Synchronization2010722

TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )

loz Drawback Plateau effect

Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)

((N2) ((N2) ((N2)

helliphellip

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)

Sliding window (length = N)

66 Synchronization2010722

TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )

loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula

( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus

扣掉相加

( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +

loz The method also called delay correlator

67 Synchronization2010722

IEEE 80211aIEEE 80211a

68 Synchronization2010722

Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )

loz Minnrsquos Method [2][ ]B B －B －B

( )( )

( )

22

2Minn

P dM d =

1 4 1N minus

( )( )( )2

2

MinnR d

( ) ( ) ( ) ( )

1 4 1

2 2 2 40 0

N

d N m k d N m k Nm k

P d r r+ + + + += =

= sum sum

( ) ( ) ( )2

21 4 1

2 40 0

N

d N m k Nk

R d rminus

+ + += sum sum69

0 0m k= =

Synchronization2010722

Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )

loz Parkrsquos Method [3][ ]

where D is the symmetric version of CC D C D

where D is the symmetric version of C

( )( ) 2

3k

P dM d =( )

( )( )23

ParkM dR d

2N

( ) 2

30

N

d k d kk

P d r rminus +=

= sdotsum

( )3

22

0

N

d kk

R d r += sum70

0k=

Synchronization2010722

Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )

loz Renrsquos Method [4][ ]

AS AS

where S is the PN sequence weighted of the original bl 2preamble

( )( )( )( )

24

Re 24

n

P dM d

R d=

( )( )4R d

( ) 2 1

4 2 2

N

k k N d k d k NP d s s r rminus

+ + + += sum0k=sum

( )21

41 N

d kR d rminus

= sum71

( )402 d k

kR d r +

=sum

Synchronization2010722

Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )

72 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

73 Synchronization2010722

Fractional CFO estimation Fractional CFO estimation

loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected

11 Rminus⎛ ⎞sum

h L i th di t b t t id ti l bl k R i

0

12 d m d m L

ms

f r rLTπ + + +

=

⎛ ⎞= ang⎜ ⎟⎝ ⎠sum

where L is the distance between two identical block R is the block size

loz For exampleNN cp

L = N R = NL = N R = Ncp74 Synchronization2010722

Estimation rangeEstimation rangegg

loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval

f ( ) ( )s s⎣ ⎦

loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)

loz DVB-T (L=N) 802 11 (L N4)

[ ]05 05s sf fminus

[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)

(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus

loz Note that is the subcarrier spacing1sf NT=

75

sNT

Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz

plusmng q y

loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn

76 Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz

plusmng q y

loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for

subcarrier spacing is required

plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn

p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs

77 Synchronization2010722

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

52 Synchronization2010722

Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )

53 Synchronization2010722

Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )

( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s

i n t i N N T N T nTy y t e

= + + +=

( ) ( )( ) ( ) ( ) 12

sin

sin

g gI f f

I I

i N N N Nj jf N Ni k i k k

f

Y X H e eN

π ε ε π ε

ε ε

πεπε

+ + minus+

minus minus=⎛ ⎞⎜ ⎟

Faded signal attenuated and rotated by CFO

( )( ) ( ) ( ) ( )11 2

sin

sin g gI f I f

i N N N NN j j l kI f

NN

l k π ε ε π ε επ ε ε + + minusminus + + + minus

⎜ ⎟⎝ ⎠

+ + minussum

( )( )( )

( ) ( )2

0

sin

I f I f

I

j j l kI f N Ni l l

l l k I f

X H e el k

NN

π ε ε π ε ε

ε π ε ε

+ + +

= ne minus

+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠

sum

i k

N

W

⎜ ⎟⎝ ⎠

+ Inter-Carrier Interference (ICI)

54 Synchronization2010722

Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )

loz An easier representation of received signal with only p g yfractional CFO is

1N

( ) ( )1

0

0N

k k k l l kl l k

Y H X C H X C l k Wminus

= ne

= + minus +⎡ ⎤⎣ ⎦sum

where the ICI coefficient is given by

( )( )( ) ( )

sin 1exp 1fl k

C l k j l kπ ε

π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )

( )exp 1sin

f

f

C l k j l kNN l k

N

π επ ε

minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠

55 Synchronization2010722

Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )

loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality

loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift

56 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

57 Synchronization2010722

TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )

loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq

58 Synchronization2010722

TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )

loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below

loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and

rN2 can be expressed by

0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+

( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +

59

( )

Synchronization2010722

TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )

loz Without CFO the first half is identical to the second half (in time order)

loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below

( ) 12 in discretesft T f NT

NTφ π π π ε πε

⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟

⎝ ⎠

helliphellip

sNT⎝ ⎠

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)

60

Length=N2

Synchronization2010722

TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )

loz Received signal without CFOg0

0 0

jr a e θ=jθ 2 0

2 2 0

Nj j

N Nr a e a e

θ θ= =

loz Received signal with CFO0 0

0 0

j jr a e eθ φ= Extra phase rotation due to CFO0 0

2 2 20

2 2 0

N N Nj j jj

N Nr a e e a e e

θ φ φθ= =

due to CFO

Phase difference 2 2 0N N

( )2 02 20 2 0 0

Nj j T fN

r r a e a eφ φ πminus Δ= =

which contains the information about CFO

61

0 2 0 0Nr r a e a e

Synchronization2010722

TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )

loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be

loz Note that d is a time index corresponding to the first( ) ( ) 2 1

20

Nd m d m Nm

P d r rminus

+ + +== sum

loz Note that d is a time index corresponding to the first sample in a window of N samples

loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by

( )22 1

20

Nd m NR d rminus

+ += sumloz A timing metric can be defined as

))(()(

)( 2

2

dRdP

dM =

( ) 20 d m Nm + +=sum

62

))(( dR

Synchronization2010722

TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )

63 Synchronization2010722

TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )

loz If drsquo is the correct symbol timing offset

( )2

2 2

0 1 12

j T f j T f j T fN

P d a e a e a eπ π πΔ Δ Δ

minus= + + +

22

2 2

0 1 12

j T fN

e a a aπ Δ

minus

⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭

loz If drsquo falls behind the correct symbol timing offset 1 sample

2⎪ ⎪⎩ ⎭

64 Synchronization2010722

TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )

65 Synchronization2010722

TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )

loz Drawback Plateau effect

Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)

((N2) ((N2) ((N2)

helliphellip

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)

Sliding window (length = N)

66 Synchronization2010722

TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )

loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula

( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus

扣掉相加

( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +

loz The method also called delay correlator

67 Synchronization2010722

IEEE 80211aIEEE 80211a

68 Synchronization2010722

Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )

loz Minnrsquos Method [2][ ]B B －B －B

( )( )

( )

22

2Minn

P dM d =

1 4 1N minus

( )( )( )2

2

MinnR d

( ) ( ) ( ) ( )

1 4 1

2 2 2 40 0

N

d N m k d N m k Nm k

P d r r+ + + + += =

= sum sum

( ) ( ) ( )2

21 4 1

2 40 0

N

d N m k Nk

R d rminus

+ + += sum sum69

0 0m k= =

Synchronization2010722

Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )

loz Parkrsquos Method [3][ ]

where D is the symmetric version of CC D C D

where D is the symmetric version of C

( )( ) 2

3k

P dM d =( )

( )( )23

ParkM dR d

2N

( ) 2

30

N

d k d kk

P d r rminus +=

= sdotsum

( )3

22

0

N

d kk

R d r += sum70

0k=

Synchronization2010722

Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )

loz Renrsquos Method [4][ ]

AS AS

where S is the PN sequence weighted of the original bl 2preamble

( )( )( )( )

24

Re 24

n

P dM d

R d=

( )( )4R d

( ) 2 1

4 2 2

N

k k N d k d k NP d s s r rminus

+ + + += sum0k=sum

( )21

41 N

d kR d rminus

= sum71

( )402 d k

kR d r +

=sum

Synchronization2010722

Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )

72 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

73 Synchronization2010722

Fractional CFO estimation Fractional CFO estimation

loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected

11 Rminus⎛ ⎞sum

h L i th di t b t t id ti l bl k R i

0

12 d m d m L

ms

f r rLTπ + + +

=

⎛ ⎞= ang⎜ ⎟⎝ ⎠sum

where L is the distance between two identical block R is the block size

loz For exampleNN cp

L = N R = NL = N R = Ncp74 Synchronization2010722

Estimation rangeEstimation rangegg

loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval

f ( ) ( )s s⎣ ⎦

loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)

loz DVB-T (L=N) 802 11 (L N4)

[ ]05 05s sf fminus

[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)

(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus

loz Note that is the subcarrier spacing1sf NT=

75

sNT

Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz

plusmng q y

loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn

76 Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz

plusmng q y

loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for

subcarrier spacing is required

plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn

p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs

77 Synchronization2010722

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )

53 Synchronization2010722

Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )

( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s

i n t i N N T N T nTy y t e

= + + +=

( ) ( )( ) ( ) ( ) 12

sin

sin

g gI f f

I I

i N N N Nj jf N Ni k i k k

f

Y X H e eN

π ε ε π ε

ε ε

πεπε

+ + minus+

minus minus=⎛ ⎞⎜ ⎟

Faded signal attenuated and rotated by CFO

( )( ) ( ) ( ) ( )11 2

sin

sin g gI f I f

i N N N NN j j l kI f

NN

l k π ε ε π ε επ ε ε + + minusminus + + + minus

⎜ ⎟⎝ ⎠

+ + minussum

( )( )( )

( ) ( )2

0

sin

I f I f

I

j j l kI f N Ni l l

l l k I f

X H e el k

NN

π ε ε π ε ε

ε π ε ε

+ + +

= ne minus

+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠

sum

i k

N

W

⎜ ⎟⎝ ⎠

+ Inter-Carrier Interference (ICI)

54 Synchronization2010722

Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )

loz An easier representation of received signal with only p g yfractional CFO is

1N

( ) ( )1

0

0N

k k k l l kl l k

Y H X C H X C l k Wminus

= ne

= + minus +⎡ ⎤⎣ ⎦sum

where the ICI coefficient is given by

( )( )( ) ( )

sin 1exp 1fl k

C l k j l kπ ε

π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )

( )exp 1sin

f

f

C l k j l kNN l k

N

π επ ε

minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠

55 Synchronization2010722

Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )

loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality

loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift

56 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

57 Synchronization2010722

TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )

loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq

58 Synchronization2010722

TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )

loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below

loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and

rN2 can be expressed by

0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+

( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +

59

( )

Synchronization2010722

TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )

loz Without CFO the first half is identical to the second half (in time order)

loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below

( ) 12 in discretesft T f NT

NTφ π π π ε πε

⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟

⎝ ⎠

helliphellip

sNT⎝ ⎠

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)

60

Length=N2

Synchronization2010722

TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )

loz Received signal without CFOg0

0 0

jr a e θ=jθ 2 0

2 2 0

Nj j

N Nr a e a e

θ θ= =

loz Received signal with CFO0 0

0 0

j jr a e eθ φ= Extra phase rotation due to CFO0 0

2 2 20

2 2 0

N N Nj j jj

N Nr a e e a e e

θ φ φθ= =

due to CFO

Phase difference 2 2 0N N

( )2 02 20 2 0 0

Nj j T fN

r r a e a eφ φ πminus Δ= =

which contains the information about CFO

61

0 2 0 0Nr r a e a e

Synchronization2010722

TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )

loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be

loz Note that d is a time index corresponding to the first( ) ( ) 2 1

20

Nd m d m Nm

P d r rminus

+ + +== sum

loz Note that d is a time index corresponding to the first sample in a window of N samples

loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by

( )22 1

20

Nd m NR d rminus

+ += sumloz A timing metric can be defined as

))(()(

)( 2

2

dRdP

dM =

( ) 20 d m Nm + +=sum

62

))(( dR

Synchronization2010722

TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )

63 Synchronization2010722

TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )

loz If drsquo is the correct symbol timing offset

( )2

2 2

0 1 12

j T f j T f j T fN

P d a e a e a eπ π πΔ Δ Δ

minus= + + +

22

2 2

0 1 12

j T fN

e a a aπ Δ

minus

⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭

loz If drsquo falls behind the correct symbol timing offset 1 sample

2⎪ ⎪⎩ ⎭

64 Synchronization2010722

TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )

65 Synchronization2010722

TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )

loz Drawback Plateau effect

Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)

((N2) ((N2) ((N2)

helliphellip

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)

Sliding window (length = N)

66 Synchronization2010722

TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )

loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula

( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus

扣掉相加

( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +

loz The method also called delay correlator

67 Synchronization2010722

IEEE 80211aIEEE 80211a

68 Synchronization2010722

Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )

loz Minnrsquos Method [2][ ]B B －B －B

( )( )

( )

22

2Minn

P dM d =

1 4 1N minus

( )( )( )2

2

MinnR d

( ) ( ) ( ) ( )

1 4 1

2 2 2 40 0

N

d N m k d N m k Nm k

P d r r+ + + + += =

= sum sum

( ) ( ) ( )2

21 4 1

2 40 0

N

d N m k Nk

R d rminus

+ + += sum sum69

0 0m k= =

Synchronization2010722

Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )

loz Parkrsquos Method [3][ ]

where D is the symmetric version of CC D C D

where D is the symmetric version of C

( )( ) 2

3k

P dM d =( )

( )( )23

ParkM dR d

2N

( ) 2

30

N

d k d kk

P d r rminus +=

= sdotsum

( )3

22

0

N

d kk

R d r += sum70

0k=

Synchronization2010722

Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )

loz Renrsquos Method [4][ ]

AS AS

where S is the PN sequence weighted of the original bl 2preamble

( )( )( )( )

24

Re 24

n

P dM d

R d=

( )( )4R d

( ) 2 1

4 2 2

N

k k N d k d k NP d s s r rminus

+ + + += sum0k=sum

( )21

41 N

d kR d rminus

= sum71

( )402 d k

kR d r +

=sum

Synchronization2010722

Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )

72 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

73 Synchronization2010722

Fractional CFO estimation Fractional CFO estimation

loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected

11 Rminus⎛ ⎞sum

h L i th di t b t t id ti l bl k R i

0

12 d m d m L

ms

f r rLTπ + + +

=

⎛ ⎞= ang⎜ ⎟⎝ ⎠sum

where L is the distance between two identical block R is the block size

loz For exampleNN cp

L = N R = NL = N R = Ncp74 Synchronization2010722

Estimation rangeEstimation rangegg

loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval

f ( ) ( )s s⎣ ⎦

loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)

loz DVB-T (L=N) 802 11 (L N4)

[ ]05 05s sf fminus

[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)

(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus

loz Note that is the subcarrier spacing1sf NT=

75

sNT

Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz

plusmng q y

loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn

76 Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz

plusmng q y

loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for

subcarrier spacing is required

plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn

p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs

77 Synchronization2010722

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )

( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s

i n t i N N T N T nTy y t e

= + + +=

( ) ( )( ) ( ) ( ) 12

sin

sin

g gI f f

I I

i N N N Nj jf N Ni k i k k

f

Y X H e eN

π ε ε π ε

ε ε

πεπε

+ + minus+

minus minus=⎛ ⎞⎜ ⎟

Faded signal attenuated and rotated by CFO

( )( ) ( ) ( ) ( )11 2

sin

sin g gI f I f

i N N N NN j j l kI f

NN

l k π ε ε π ε επ ε ε + + minusminus + + + minus

⎜ ⎟⎝ ⎠

+ + minussum

( )( )( )

( ) ( )2

0

sin

I f I f

I

j j l kI f N Ni l l

l l k I f

X H e el k

NN

π ε ε π ε ε

ε π ε ε

+ + +

= ne minus

+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠

sum

i k

N

W

⎜ ⎟⎝ ⎠

+ Inter-Carrier Interference (ICI)

54 Synchronization2010722

Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )

loz An easier representation of received signal with only p g yfractional CFO is

1N

( ) ( )1

0

0N

k k k l l kl l k

Y H X C H X C l k Wminus

= ne

= + minus +⎡ ⎤⎣ ⎦sum

where the ICI coefficient is given by

( )( )( ) ( )

sin 1exp 1fl k

C l k j l kπ ε

π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )

( )exp 1sin

f

f

C l k j l kNN l k

N

π επ ε

minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠

55 Synchronization2010722

Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )

loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality

loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift

56 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

57 Synchronization2010722

TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )

loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq

58 Synchronization2010722

TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )

loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below

loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and

rN2 can be expressed by

0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+

( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +

59

( )

Synchronization2010722

TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )

loz Without CFO the first half is identical to the second half (in time order)

loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below

( ) 12 in discretesft T f NT

NTφ π π π ε πε

⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟

⎝ ⎠

helliphellip

sNT⎝ ⎠

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)

60

Length=N2

Synchronization2010722

TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )

loz Received signal without CFOg0

0 0

jr a e θ=jθ 2 0

2 2 0

Nj j

N Nr a e a e

θ θ= =

loz Received signal with CFO0 0

0 0

j jr a e eθ φ= Extra phase rotation due to CFO0 0

2 2 20

2 2 0

N N Nj j jj

N Nr a e e a e e

θ φ φθ= =

due to CFO

Phase difference 2 2 0N N

( )2 02 20 2 0 0

Nj j T fN

r r a e a eφ φ πminus Δ= =

which contains the information about CFO

61

0 2 0 0Nr r a e a e

Synchronization2010722

TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )

loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be

loz Note that d is a time index corresponding to the first( ) ( ) 2 1

20

Nd m d m Nm

P d r rminus

+ + +== sum

loz Note that d is a time index corresponding to the first sample in a window of N samples

loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by

( )22 1

20

Nd m NR d rminus

+ += sumloz A timing metric can be defined as

))(()(

)( 2

2

dRdP

dM =

( ) 20 d m Nm + +=sum

62

))(( dR

Synchronization2010722

TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )

63 Synchronization2010722

TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )

loz If drsquo is the correct symbol timing offset

( )2

2 2

0 1 12

j T f j T f j T fN

P d a e a e a eπ π πΔ Δ Δ

minus= + + +

22

2 2

0 1 12

j T fN

e a a aπ Δ

minus

⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭

loz If drsquo falls behind the correct symbol timing offset 1 sample

2⎪ ⎪⎩ ⎭

64 Synchronization2010722

TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )

65 Synchronization2010722

TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )

loz Drawback Plateau effect

Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)

((N2) ((N2) ((N2)

helliphellip

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)

Sliding window (length = N)

66 Synchronization2010722

TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )

loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula

( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus

扣掉相加

( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +

loz The method also called delay correlator

67 Synchronization2010722

IEEE 80211aIEEE 80211a

68 Synchronization2010722

Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )

loz Minnrsquos Method [2][ ]B B －B －B

( )( )

( )

22

2Minn

P dM d =

1 4 1N minus

( )( )( )2

2

MinnR d

( ) ( ) ( ) ( )

1 4 1

2 2 2 40 0

N

d N m k d N m k Nm k

P d r r+ + + + += =

= sum sum

( ) ( ) ( )2

21 4 1

2 40 0

N

d N m k Nk

R d rminus

+ + += sum sum69

0 0m k= =

Synchronization2010722

Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )

loz Parkrsquos Method [3][ ]

where D is the symmetric version of CC D C D

where D is the symmetric version of C

( )( ) 2

3k

P dM d =( )

( )( )23

ParkM dR d

2N

( ) 2

30

N

d k d kk

P d r rminus +=

= sdotsum

( )3

22

0

N

d kk

R d r += sum70

0k=

Synchronization2010722

Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )

loz Renrsquos Method [4][ ]

AS AS

where S is the PN sequence weighted of the original bl 2preamble

( )( )( )( )

24

Re 24

n

P dM d

R d=

( )( )4R d

( ) 2 1

4 2 2

N

k k N d k d k NP d s s r rminus

+ + + += sum0k=sum

( )21

41 N

d kR d rminus

= sum71

( )402 d k

kR d r +

=sum

Synchronization2010722

Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )

72 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

73 Synchronization2010722

Fractional CFO estimation Fractional CFO estimation

loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected

11 Rminus⎛ ⎞sum

h L i th di t b t t id ti l bl k R i

0

12 d m d m L

ms

f r rLTπ + + +

=

⎛ ⎞= ang⎜ ⎟⎝ ⎠sum

where L is the distance between two identical block R is the block size

loz For exampleNN cp

L = N R = NL = N R = Ncp74 Synchronization2010722

Estimation rangeEstimation rangegg

loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval

f ( ) ( )s s⎣ ⎦

loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)

loz DVB-T (L=N) 802 11 (L N4)

[ ]05 05s sf fminus

[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)

(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus

loz Note that is the subcarrier spacing1sf NT=

75

sNT

Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz

plusmng q y

loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn

76 Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz

plusmng q y

loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for

subcarrier spacing is required

plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn

p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs

77 Synchronization2010722

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )

loz An easier representation of received signal with only p g yfractional CFO is

1N

( ) ( )1

0

0N

k k k l l kl l k

Y H X C H X C l k Wminus

= ne

= + minus +⎡ ⎤⎣ ⎦sum

where the ICI coefficient is given by

( )( )( ) ( )

sin 1exp 1fl k

C l k j l kπ ε

π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )

( )exp 1sin

f

f

C l k j l kNN l k

N

π επ ε

minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠

55 Synchronization2010722

Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )

loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality

loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift

56 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

57 Synchronization2010722

TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )

loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq

58 Synchronization2010722

TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )

loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below

loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and

rN2 can be expressed by

0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+

( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +

59

( )

Synchronization2010722

TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )

loz Without CFO the first half is identical to the second half (in time order)

loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below

( ) 12 in discretesft T f NT

NTφ π π π ε πε

⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟

⎝ ⎠

helliphellip

sNT⎝ ⎠

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)

60

Length=N2

Synchronization2010722

TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )

loz Received signal without CFOg0

0 0

jr a e θ=jθ 2 0

2 2 0

Nj j

N Nr a e a e

θ θ= =

loz Received signal with CFO0 0

0 0

j jr a e eθ φ= Extra phase rotation due to CFO0 0

2 2 20

2 2 0

N N Nj j jj

N Nr a e e a e e

θ φ φθ= =

due to CFO

Phase difference 2 2 0N N

( )2 02 20 2 0 0

Nj j T fN

r r a e a eφ φ πminus Δ= =

which contains the information about CFO

61

0 2 0 0Nr r a e a e

Synchronization2010722

TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )

loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be

loz Note that d is a time index corresponding to the first( ) ( ) 2 1

20

Nd m d m Nm

P d r rminus

+ + +== sum

loz Note that d is a time index corresponding to the first sample in a window of N samples

loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by

( )22 1

20

Nd m NR d rminus

+ += sumloz A timing metric can be defined as

))(()(

)( 2

2

dRdP

dM =

( ) 20 d m Nm + +=sum

62

))(( dR

Synchronization2010722

TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )

63 Synchronization2010722

TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )

loz If drsquo is the correct symbol timing offset

( )2

2 2

0 1 12

j T f j T f j T fN

P d a e a e a eπ π πΔ Δ Δ

minus= + + +

22

2 2

0 1 12

j T fN

e a a aπ Δ

minus

⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭

loz If drsquo falls behind the correct symbol timing offset 1 sample

2⎪ ⎪⎩ ⎭

64 Synchronization2010722

TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )

65 Synchronization2010722

TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )

loz Drawback Plateau effect

Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)

((N2) ((N2) ((N2)

helliphellip

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)

Sliding window (length = N)

66 Synchronization2010722

TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )

loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula

( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus

扣掉相加

( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +

loz The method also called delay correlator

67 Synchronization2010722

IEEE 80211aIEEE 80211a

68 Synchronization2010722

Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )

loz Minnrsquos Method [2][ ]B B －B －B

( )( )

( )

22

2Minn

P dM d =

1 4 1N minus

( )( )( )2

2

MinnR d

( ) ( ) ( ) ( )

1 4 1

2 2 2 40 0

N

d N m k d N m k Nm k

P d r r+ + + + += =

= sum sum

( ) ( ) ( )2

21 4 1

2 40 0

N

d N m k Nk

R d rminus

+ + += sum sum69

0 0m k= =

Synchronization2010722

Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )

loz Parkrsquos Method [3][ ]

where D is the symmetric version of CC D C D

where D is the symmetric version of C

( )( ) 2

3k

P dM d =( )

( )( )23

ParkM dR d

2N

( ) 2

30

N

d k d kk

P d r rminus +=

= sdotsum

( )3

22

0

N

d kk

R d r += sum70

0k=

Synchronization2010722

Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )

loz Renrsquos Method [4][ ]

AS AS

where S is the PN sequence weighted of the original bl 2preamble

( )( )( )( )

24

Re 24

n

P dM d

R d=

( )( )4R d

( ) 2 1

4 2 2

N

k k N d k d k NP d s s r rminus

+ + + += sum0k=sum

( )21

41 N

d kR d rminus

= sum71

( )402 d k

kR d r +

=sum

Synchronization2010722

Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )

72 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

73 Synchronization2010722

Fractional CFO estimation Fractional CFO estimation

loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected

11 Rminus⎛ ⎞sum

h L i th di t b t t id ti l bl k R i

0

12 d m d m L

ms

f r rLTπ + + +

=

⎛ ⎞= ang⎜ ⎟⎝ ⎠sum

where L is the distance between two identical block R is the block size

loz For exampleNN cp

L = N R = NL = N R = Ncp74 Synchronization2010722

Estimation rangeEstimation rangegg

loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval

f ( ) ( )s s⎣ ⎦

loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)

loz DVB-T (L=N) 802 11 (L N4)

[ ]05 05s sf fminus

[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)

(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus

loz Note that is the subcarrier spacing1sf NT=

75

sNT

Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz

plusmng q y

loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn

76 Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz

plusmng q y

loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for

subcarrier spacing is required

plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn

p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs

77 Synchronization2010722

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )

loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality

loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift

56 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

57 Synchronization2010722

TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )

loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq

58 Synchronization2010722

TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )

loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below

loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and

rN2 can be expressed by

0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+

( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +

59

( )

Synchronization2010722

TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )

loz Without CFO the first half is identical to the second half (in time order)

loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below

( ) 12 in discretesft T f NT

NTφ π π π ε πε

⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟

⎝ ⎠

helliphellip

sNT⎝ ⎠

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)

60

Length=N2

Synchronization2010722

TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )

loz Received signal without CFOg0

0 0

jr a e θ=jθ 2 0

2 2 0

Nj j

N Nr a e a e

θ θ= =

loz Received signal with CFO0 0

0 0

j jr a e eθ φ= Extra phase rotation due to CFO0 0

2 2 20

2 2 0

N N Nj j jj

N Nr a e e a e e

θ φ φθ= =

due to CFO

Phase difference 2 2 0N N

( )2 02 20 2 0 0

Nj j T fN

r r a e a eφ φ πminus Δ= =

which contains the information about CFO

61

0 2 0 0Nr r a e a e

Synchronization2010722

TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )

loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be

loz Note that d is a time index corresponding to the first( ) ( ) 2 1

20

Nd m d m Nm

P d r rminus

+ + +== sum

loz Note that d is a time index corresponding to the first sample in a window of N samples

loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by

( )22 1

20

Nd m NR d rminus

+ += sumloz A timing metric can be defined as

))(()(

)( 2

2

dRdP

dM =

( ) 20 d m Nm + +=sum

62

))(( dR

Synchronization2010722

TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )

63 Synchronization2010722

TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )

loz If drsquo is the correct symbol timing offset

( )2

2 2

0 1 12

j T f j T f j T fN

P d a e a e a eπ π πΔ Δ Δ

minus= + + +

22

2 2

0 1 12

j T fN

e a a aπ Δ

minus

⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭

loz If drsquo falls behind the correct symbol timing offset 1 sample

2⎪ ⎪⎩ ⎭

64 Synchronization2010722

TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )

65 Synchronization2010722

TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )

loz Drawback Plateau effect

Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)

((N2) ((N2) ((N2)

helliphellip

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)

Sliding window (length = N)

66 Synchronization2010722

TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )

loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula

( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus

扣掉相加

( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +

loz The method also called delay correlator

67 Synchronization2010722

IEEE 80211aIEEE 80211a

68 Synchronization2010722

Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )

loz Minnrsquos Method [2][ ]B B －B －B

( )( )

( )

22

2Minn

P dM d =

1 4 1N minus

( )( )( )2

2

MinnR d

( ) ( ) ( ) ( )

1 4 1

2 2 2 40 0

N

d N m k d N m k Nm k

P d r r+ + + + += =

= sum sum

( ) ( ) ( )2

21 4 1

2 40 0

N

d N m k Nk

R d rminus

+ + += sum sum69

0 0m k= =

Synchronization2010722

Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )

loz Parkrsquos Method [3][ ]

where D is the symmetric version of CC D C D

where D is the symmetric version of C

( )( ) 2

3k

P dM d =( )

( )( )23

ParkM dR d

2N

( ) 2

30

N

d k d kk

P d r rminus +=

= sdotsum

( )3

22

0

N

d kk

R d r += sum70

0k=

Synchronization2010722

Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )

loz Renrsquos Method [4][ ]

AS AS

where S is the PN sequence weighted of the original bl 2preamble

( )( )( )( )

24

Re 24

n

P dM d

R d=

( )( )4R d

( ) 2 1

4 2 2

N

k k N d k d k NP d s s r rminus

+ + + += sum0k=sum

( )21

41 N

d kR d rminus

= sum71

( )402 d k

kR d r +

=sum

Synchronization2010722

Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )

72 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

73 Synchronization2010722

Fractional CFO estimation Fractional CFO estimation

loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected

11 Rminus⎛ ⎞sum

h L i th di t b t t id ti l bl k R i

0

12 d m d m L

ms

f r rLTπ + + +

=

⎛ ⎞= ang⎜ ⎟⎝ ⎠sum

where L is the distance between two identical block R is the block size

loz For exampleNN cp

L = N R = NL = N R = Ncp74 Synchronization2010722

Estimation rangeEstimation rangegg

loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval

f ( ) ( )s s⎣ ⎦

loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)

loz DVB-T (L=N) 802 11 (L N4)

[ ]05 05s sf fminus

[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)

(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus

loz Note that is the subcarrier spacing1sf NT=

75

sNT

Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz

plusmng q y

loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn

76 Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz

plusmng q y

loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for

subcarrier spacing is required

plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn

p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs

77 Synchronization2010722

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

57 Synchronization2010722

TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )

loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq

58 Synchronization2010722

TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )

loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below

loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and

rN2 can be expressed by

0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+

( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +

59

( )

Synchronization2010722

TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )

loz Without CFO the first half is identical to the second half (in time order)

loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below

( ) 12 in discretesft T f NT

NTφ π π π ε πε

⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟

⎝ ⎠

helliphellip

sNT⎝ ⎠

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)

60

Length=N2

Synchronization2010722

TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )

loz Received signal without CFOg0

0 0

jr a e θ=jθ 2 0

2 2 0

Nj j

N Nr a e a e

θ θ= =

loz Received signal with CFO0 0

0 0

j jr a e eθ φ= Extra phase rotation due to CFO0 0

2 2 20

2 2 0

N N Nj j jj

N Nr a e e a e e

θ φ φθ= =

due to CFO

Phase difference 2 2 0N N

( )2 02 20 2 0 0

Nj j T fN

r r a e a eφ φ πminus Δ= =

which contains the information about CFO

61

0 2 0 0Nr r a e a e

Synchronization2010722

TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )

loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be

loz Note that d is a time index corresponding to the first( ) ( ) 2 1

20

Nd m d m Nm

P d r rminus

+ + +== sum

loz Note that d is a time index corresponding to the first sample in a window of N samples

loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by

( )22 1

20

Nd m NR d rminus

+ += sumloz A timing metric can be defined as

))(()(

)( 2

2

dRdP

dM =

( ) 20 d m Nm + +=sum

62

))(( dR

Synchronization2010722

TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )

63 Synchronization2010722

TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )

loz If drsquo is the correct symbol timing offset

( )2

2 2

0 1 12

j T f j T f j T fN

P d a e a e a eπ π πΔ Δ Δ

minus= + + +

22

2 2

0 1 12

j T fN

e a a aπ Δ

minus

⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭

loz If drsquo falls behind the correct symbol timing offset 1 sample

2⎪ ⎪⎩ ⎭

64 Synchronization2010722

TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )

65 Synchronization2010722

TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )

loz Drawback Plateau effect

Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)

((N2) ((N2) ((N2)

helliphellip

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)

Sliding window (length = N)

66 Synchronization2010722

TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )

loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula

( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus

扣掉相加

( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +

loz The method also called delay correlator

67 Synchronization2010722

IEEE 80211aIEEE 80211a

68 Synchronization2010722

Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )

loz Minnrsquos Method [2][ ]B B －B －B

( )( )

( )

22

2Minn

P dM d =

1 4 1N minus

( )( )( )2

2

MinnR d

( ) ( ) ( ) ( )

1 4 1

2 2 2 40 0

N

d N m k d N m k Nm k

P d r r+ + + + += =

= sum sum

( ) ( ) ( )2

21 4 1

2 40 0

N

d N m k Nk

R d rminus

+ + += sum sum69

0 0m k= =

Synchronization2010722

Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )

loz Parkrsquos Method [3][ ]

where D is the symmetric version of CC D C D

where D is the symmetric version of C

( )( ) 2

3k

P dM d =( )

( )( )23

ParkM dR d

2N

( ) 2

30

N

d k d kk

P d r rminus +=

= sdotsum

( )3

22

0

N

d kk

R d r += sum70

0k=

Synchronization2010722

Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )

loz Renrsquos Method [4][ ]

AS AS

where S is the PN sequence weighted of the original bl 2preamble

( )( )( )( )

24

Re 24

n

P dM d

R d=

( )( )4R d

( ) 2 1

4 2 2

N

k k N d k d k NP d s s r rminus

+ + + += sum0k=sum

( )21

41 N

d kR d rminus

= sum71

( )402 d k

kR d r +

=sum

Synchronization2010722

Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )

72 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

73 Synchronization2010722

Fractional CFO estimation Fractional CFO estimation

loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected

11 Rminus⎛ ⎞sum

h L i th di t b t t id ti l bl k R i

0

12 d m d m L

ms

f r rLTπ + + +

=

⎛ ⎞= ang⎜ ⎟⎝ ⎠sum

where L is the distance between two identical block R is the block size

loz For exampleNN cp

L = N R = NL = N R = Ncp74 Synchronization2010722

Estimation rangeEstimation rangegg

loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval

f ( ) ( )s s⎣ ⎦

loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)

loz DVB-T (L=N) 802 11 (L N4)

[ ]05 05s sf fminus

[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)

(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus

loz Note that is the subcarrier spacing1sf NT=

75

sNT

Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz

plusmng q y

loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn

76 Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz

plusmng q y

loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for

subcarrier spacing is required

plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn

p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs

77 Synchronization2010722

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )

loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq

58 Synchronization2010722

TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )

loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below

loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and

rN2 can be expressed by

0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+

( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +

59

( )

Synchronization2010722

TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )

loz Without CFO the first half is identical to the second half (in time order)

loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below

( ) 12 in discretesft T f NT

NTφ π π π ε πε

⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟

⎝ ⎠

helliphellip

sNT⎝ ⎠

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)

60

Length=N2

Synchronization2010722

TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )

loz Received signal without CFOg0

0 0

jr a e θ=jθ 2 0

2 2 0

Nj j

N Nr a e a e

θ θ= =

loz Received signal with CFO0 0

0 0

j jr a e eθ φ= Extra phase rotation due to CFO0 0

2 2 20

2 2 0

N N Nj j jj

N Nr a e e a e e

θ φ φθ= =

due to CFO

Phase difference 2 2 0N N

( )2 02 20 2 0 0

Nj j T fN

r r a e a eφ φ πminus Δ= =

which contains the information about CFO

61

0 2 0 0Nr r a e a e

Synchronization2010722

TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )

loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be

loz Note that d is a time index corresponding to the first( ) ( ) 2 1

20

Nd m d m Nm

P d r rminus

+ + +== sum

loz Note that d is a time index corresponding to the first sample in a window of N samples

loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by

( )22 1

20

Nd m NR d rminus

+ += sumloz A timing metric can be defined as

))(()(

)( 2

2

dRdP

dM =

( ) 20 d m Nm + +=sum

62

))(( dR

Synchronization2010722

TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )

63 Synchronization2010722

TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )

loz If drsquo is the correct symbol timing offset

( )2

2 2

0 1 12

j T f j T f j T fN

P d a e a e a eπ π πΔ Δ Δ

minus= + + +

22

2 2

0 1 12

j T fN

e a a aπ Δ

minus

⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭

loz If drsquo falls behind the correct symbol timing offset 1 sample

2⎪ ⎪⎩ ⎭

64 Synchronization2010722

TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )

65 Synchronization2010722

TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )

loz Drawback Plateau effect

Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)

((N2) ((N2) ((N2)

helliphellip

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)

Sliding window (length = N)

66 Synchronization2010722

TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )

loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula

( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus

扣掉相加

( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +

loz The method also called delay correlator

67 Synchronization2010722

IEEE 80211aIEEE 80211a

68 Synchronization2010722

Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )

loz Minnrsquos Method [2][ ]B B －B －B

( )( )

( )

22

2Minn

P dM d =

1 4 1N minus

( )( )( )2

2

MinnR d

( ) ( ) ( ) ( )

1 4 1

2 2 2 40 0

N

d N m k d N m k Nm k

P d r r+ + + + += =

= sum sum

( ) ( ) ( )2

21 4 1

2 40 0

N

d N m k Nk

R d rminus

+ + += sum sum69

0 0m k= =

Synchronization2010722

Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )

loz Parkrsquos Method [3][ ]

where D is the symmetric version of CC D C D

where D is the symmetric version of C

( )( ) 2

3k

P dM d =( )

( )( )23

ParkM dR d

2N

( ) 2

30

N

d k d kk

P d r rminus +=

= sdotsum

( )3

22

0

N

d kk

R d r += sum70

0k=

Synchronization2010722

Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )

loz Renrsquos Method [4][ ]

AS AS

where S is the PN sequence weighted of the original bl 2preamble

( )( )( )( )

24

Re 24

n

P dM d

R d=

( )( )4R d

( ) 2 1

4 2 2

N

k k N d k d k NP d s s r rminus

+ + + += sum0k=sum

( )21

41 N

d kR d rminus

= sum71

( )402 d k

kR d r +

=sum

Synchronization2010722

Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )

72 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

73 Synchronization2010722

Fractional CFO estimation Fractional CFO estimation

loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected

11 Rminus⎛ ⎞sum

h L i th di t b t t id ti l bl k R i

0

12 d m d m L

ms

f r rLTπ + + +

=

⎛ ⎞= ang⎜ ⎟⎝ ⎠sum

where L is the distance between two identical block R is the block size

loz For exampleNN cp

L = N R = NL = N R = Ncp74 Synchronization2010722

Estimation rangeEstimation rangegg

loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval

f ( ) ( )s s⎣ ⎦

loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)

loz DVB-T (L=N) 802 11 (L N4)

[ ]05 05s sf fminus

[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)

(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus

loz Note that is the subcarrier spacing1sf NT=

75

sNT

Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz

plusmng q y

loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn

76 Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz

plusmng q y

loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for

subcarrier spacing is required

plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn

p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs

77 Synchronization2010722

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )

loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below

loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and

rN2 can be expressed by

0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+

( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +

59

( )

Synchronization2010722

TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )

loz Without CFO the first half is identical to the second half (in time order)

loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below

( ) 12 in discretesft T f NT

NTφ π π π ε πε

⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟

⎝ ⎠

helliphellip

sNT⎝ ⎠

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)

60

Length=N2

Synchronization2010722

TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )

loz Received signal without CFOg0

0 0

jr a e θ=jθ 2 0

2 2 0

Nj j

N Nr a e a e

θ θ= =

loz Received signal with CFO0 0

0 0

j jr a e eθ φ= Extra phase rotation due to CFO0 0

2 2 20

2 2 0

N N Nj j jj

N Nr a e e a e e

θ φ φθ= =

due to CFO

Phase difference 2 2 0N N

( )2 02 20 2 0 0

Nj j T fN

r r a e a eφ φ πminus Δ= =

which contains the information about CFO

61

0 2 0 0Nr r a e a e

Synchronization2010722

TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )

loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be

loz Note that d is a time index corresponding to the first( ) ( ) 2 1

20

Nd m d m Nm

P d r rminus

+ + +== sum

loz Note that d is a time index corresponding to the first sample in a window of N samples

loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by

( )22 1

20

Nd m NR d rminus

+ += sumloz A timing metric can be defined as

))(()(

)( 2

2

dRdP

dM =

( ) 20 d m Nm + +=sum

62

))(( dR

Synchronization2010722

TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )

63 Synchronization2010722

TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )

loz If drsquo is the correct symbol timing offset

( )2

2 2

0 1 12

j T f j T f j T fN

P d a e a e a eπ π πΔ Δ Δ

minus= + + +

22

2 2

0 1 12

j T fN

e a a aπ Δ

minus

⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭

loz If drsquo falls behind the correct symbol timing offset 1 sample

2⎪ ⎪⎩ ⎭

64 Synchronization2010722

TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )

65 Synchronization2010722

TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )

loz Drawback Plateau effect

Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)

((N2) ((N2) ((N2)

helliphellip

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)

Sliding window (length = N)

66 Synchronization2010722

TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )

loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula

( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus

扣掉相加

( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +

loz The method also called delay correlator

67 Synchronization2010722

IEEE 80211aIEEE 80211a

68 Synchronization2010722

Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )

loz Minnrsquos Method [2][ ]B B －B －B

( )( )

( )

22

2Minn

P dM d =

1 4 1N minus

( )( )( )2

2

MinnR d

( ) ( ) ( ) ( )

1 4 1

2 2 2 40 0

N

d N m k d N m k Nm k

P d r r+ + + + += =

= sum sum

( ) ( ) ( )2

21 4 1

2 40 0

N

d N m k Nk

R d rminus

+ + += sum sum69

0 0m k= =

Synchronization2010722

Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )

loz Parkrsquos Method [3][ ]

where D is the symmetric version of CC D C D

where D is the symmetric version of C

( )( ) 2

3k

P dM d =( )

( )( )23

ParkM dR d

2N

( ) 2

30

N

d k d kk

P d r rminus +=

= sdotsum

( )3

22

0

N

d kk

R d r += sum70

0k=

Synchronization2010722

Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )

loz Renrsquos Method [4][ ]

AS AS

where S is the PN sequence weighted of the original bl 2preamble

( )( )( )( )

24

Re 24

n

P dM d

R d=

( )( )4R d

( ) 2 1

4 2 2

N

k k N d k d k NP d s s r rminus

+ + + += sum0k=sum

( )21

41 N

d kR d rminus

= sum71

( )402 d k

kR d r +

=sum

Synchronization2010722

Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )

72 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

73 Synchronization2010722

Fractional CFO estimation Fractional CFO estimation

loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected

11 Rminus⎛ ⎞sum

h L i th di t b t t id ti l bl k R i

0

12 d m d m L

ms

f r rLTπ + + +

=

⎛ ⎞= ang⎜ ⎟⎝ ⎠sum

where L is the distance between two identical block R is the block size

loz For exampleNN cp

L = N R = NL = N R = Ncp74 Synchronization2010722

Estimation rangeEstimation rangegg

loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval

f ( ) ( )s s⎣ ⎦

loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)

loz DVB-T (L=N) 802 11 (L N4)

[ ]05 05s sf fminus

[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)

(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus

loz Note that is the subcarrier spacing1sf NT=

75

sNT

Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz

plusmng q y

loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn

76 Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz

plusmng q y

loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for

subcarrier spacing is required

plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn

p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs

77 Synchronization2010722

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )

loz Without CFO the first half is identical to the second half (in time order)

loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below

( ) 12 in discretesft T f NT

NTφ π π π ε πε

⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟

⎝ ⎠

helliphellip

sNT⎝ ⎠

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)

60

Length=N2

Synchronization2010722

TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )

loz Received signal without CFOg0

0 0

jr a e θ=jθ 2 0

2 2 0

Nj j

N Nr a e a e

θ θ= =

loz Received signal with CFO0 0

0 0

j jr a e eθ φ= Extra phase rotation due to CFO0 0

2 2 20

2 2 0

N N Nj j jj

N Nr a e e a e e

θ φ φθ= =

due to CFO

Phase difference 2 2 0N N

( )2 02 20 2 0 0

Nj j T fN

r r a e a eφ φ πminus Δ= =

which contains the information about CFO

61

0 2 0 0Nr r a e a e

Synchronization2010722

TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )

loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be

loz Note that d is a time index corresponding to the first( ) ( ) 2 1

20

Nd m d m Nm

P d r rminus

+ + +== sum

loz Note that d is a time index corresponding to the first sample in a window of N samples

loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by

( )22 1

20

Nd m NR d rminus

+ += sumloz A timing metric can be defined as

))(()(

)( 2

2

dRdP

dM =

( ) 20 d m Nm + +=sum

62

))(( dR

Synchronization2010722

TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )

63 Synchronization2010722

TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )

loz If drsquo is the correct symbol timing offset

( )2

2 2

0 1 12

j T f j T f j T fN

P d a e a e a eπ π πΔ Δ Δ

minus= + + +

22

2 2

0 1 12

j T fN

e a a aπ Δ

minus

⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭

loz If drsquo falls behind the correct symbol timing offset 1 sample

2⎪ ⎪⎩ ⎭

64 Synchronization2010722

TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )

65 Synchronization2010722

TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )

loz Drawback Plateau effect

Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)

((N2) ((N2) ((N2)

helliphellip

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)

Sliding window (length = N)

66 Synchronization2010722

TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )

loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula

( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus

扣掉相加

( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +

loz The method also called delay correlator

67 Synchronization2010722

IEEE 80211aIEEE 80211a

68 Synchronization2010722

Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )

loz Minnrsquos Method [2][ ]B B －B －B

( )( )

( )

22

2Minn

P dM d =

1 4 1N minus

( )( )( )2

2

MinnR d

( ) ( ) ( ) ( )

1 4 1

2 2 2 40 0

N

d N m k d N m k Nm k

P d r r+ + + + += =

= sum sum

( ) ( ) ( )2

21 4 1

2 40 0

N

d N m k Nk

R d rminus

+ + += sum sum69

0 0m k= =

Synchronization2010722

Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )

loz Parkrsquos Method [3][ ]

where D is the symmetric version of CC D C D

where D is the symmetric version of C

( )( ) 2

3k

P dM d =( )

( )( )23

ParkM dR d

2N

( ) 2

30

N

d k d kk

P d r rminus +=

= sdotsum

( )3

22

0

N

d kk

R d r += sum70

0k=

Synchronization2010722

Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )

loz Renrsquos Method [4][ ]

AS AS

where S is the PN sequence weighted of the original bl 2preamble

( )( )( )( )

24

Re 24

n

P dM d

R d=

( )( )4R d

( ) 2 1

4 2 2

N

k k N d k d k NP d s s r rminus

+ + + += sum0k=sum

( )21

41 N

d kR d rminus

= sum71

( )402 d k

kR d r +

=sum

Synchronization2010722

Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )

72 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

73 Synchronization2010722

Fractional CFO estimation Fractional CFO estimation

loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected

11 Rminus⎛ ⎞sum

h L i th di t b t t id ti l bl k R i

0

12 d m d m L

ms

f r rLTπ + + +

=

⎛ ⎞= ang⎜ ⎟⎝ ⎠sum

where L is the distance between two identical block R is the block size

loz For exampleNN cp

L = N R = NL = N R = Ncp74 Synchronization2010722

Estimation rangeEstimation rangegg

loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval

f ( ) ( )s s⎣ ⎦

loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)

loz DVB-T (L=N) 802 11 (L N4)

[ ]05 05s sf fminus

[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)

(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus

loz Note that is the subcarrier spacing1sf NT=

75

sNT

Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz

plusmng q y

loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn

76 Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz

plusmng q y

loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for

subcarrier spacing is required

plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn

p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs

77 Synchronization2010722

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )

loz Received signal without CFOg0

0 0

jr a e θ=jθ 2 0

2 2 0

Nj j

N Nr a e a e

θ θ= =

loz Received signal with CFO0 0

0 0

j jr a e eθ φ= Extra phase rotation due to CFO0 0

2 2 20

2 2 0

N N Nj j jj

N Nr a e e a e e

θ φ φθ= =

due to CFO

Phase difference 2 2 0N N

( )2 02 20 2 0 0

Nj j T fN

r r a e a eφ φ πminus Δ= =

which contains the information about CFO

61

0 2 0 0Nr r a e a e

Synchronization2010722

TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )

loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be

loz Note that d is a time index corresponding to the first( ) ( ) 2 1

20

Nd m d m Nm

P d r rminus

+ + +== sum

loz Note that d is a time index corresponding to the first sample in a window of N samples

loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by

( )22 1

20

Nd m NR d rminus

+ += sumloz A timing metric can be defined as

))(()(

)( 2

2

dRdP

dM =

( ) 20 d m Nm + +=sum

62

))(( dR

Synchronization2010722

TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )

63 Synchronization2010722

TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )

loz If drsquo is the correct symbol timing offset

( )2

2 2

0 1 12

j T f j T f j T fN

P d a e a e a eπ π πΔ Δ Δ

minus= + + +

22

2 2

0 1 12

j T fN

e a a aπ Δ

minus

⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭

loz If drsquo falls behind the correct symbol timing offset 1 sample

2⎪ ⎪⎩ ⎭

64 Synchronization2010722

TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )

65 Synchronization2010722

TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )

loz Drawback Plateau effect

Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)

((N2) ((N2) ((N2)

helliphellip

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)

Sliding window (length = N)

66 Synchronization2010722

TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )

loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula

( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus

扣掉相加

( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +

loz The method also called delay correlator

67 Synchronization2010722

IEEE 80211aIEEE 80211a

68 Synchronization2010722

Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )

loz Minnrsquos Method [2][ ]B B －B －B

( )( )

( )

22

2Minn

P dM d =

1 4 1N minus

( )( )( )2

2

MinnR d

( ) ( ) ( ) ( )

1 4 1

2 2 2 40 0

N

d N m k d N m k Nm k

P d r r+ + + + += =

= sum sum

( ) ( ) ( )2

21 4 1

2 40 0

N

d N m k Nk

R d rminus

+ + += sum sum69

0 0m k= =

Synchronization2010722

Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )

loz Parkrsquos Method [3][ ]

where D is the symmetric version of CC D C D

where D is the symmetric version of C

( )( ) 2

3k

P dM d =( )

( )( )23

ParkM dR d

2N

( ) 2

30

N

d k d kk

P d r rminus +=

= sdotsum

( )3

22

0

N

d kk

R d r += sum70

0k=

Synchronization2010722

Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )

loz Renrsquos Method [4][ ]

AS AS

where S is the PN sequence weighted of the original bl 2preamble

( )( )( )( )

24

Re 24

n

P dM d

R d=

( )( )4R d

( ) 2 1

4 2 2

N

k k N d k d k NP d s s r rminus

+ + + += sum0k=sum

( )21

41 N

d kR d rminus

= sum71

( )402 d k

kR d r +

=sum

Synchronization2010722

Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )

72 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

73 Synchronization2010722

Fractional CFO estimation Fractional CFO estimation

loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected

11 Rminus⎛ ⎞sum

h L i th di t b t t id ti l bl k R i

0

12 d m d m L

ms

f r rLTπ + + +

=

⎛ ⎞= ang⎜ ⎟⎝ ⎠sum

where L is the distance between two identical block R is the block size

loz For exampleNN cp

L = N R = NL = N R = Ncp74 Synchronization2010722

Estimation rangeEstimation rangegg

loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval

f ( ) ( )s s⎣ ⎦

loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)

loz DVB-T (L=N) 802 11 (L N4)

[ ]05 05s sf fminus

[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)

(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus

loz Note that is the subcarrier spacing1sf NT=

75

sNT

Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz

plusmng q y

loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn

76 Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz

plusmng q y

loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for

subcarrier spacing is required

plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn

p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs

77 Synchronization2010722

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )

loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be

loz Note that d is a time index corresponding to the first( ) ( ) 2 1

20

Nd m d m Nm

P d r rminus

+ + +== sum

loz Note that d is a time index corresponding to the first sample in a window of N samples

loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by

( )22 1

20

Nd m NR d rminus

+ += sumloz A timing metric can be defined as

))(()(

)( 2

2

dRdP

dM =

( ) 20 d m Nm + +=sum

62

))(( dR

Synchronization2010722

TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )

63 Synchronization2010722

TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )

loz If drsquo is the correct symbol timing offset

( )2

2 2

0 1 12

j T f j T f j T fN

P d a e a e a eπ π πΔ Δ Δ

minus= + + +

22

2 2

0 1 12

j T fN

e a a aπ Δ

minus

⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭

loz If drsquo falls behind the correct symbol timing offset 1 sample

2⎪ ⎪⎩ ⎭

64 Synchronization2010722

TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )

65 Synchronization2010722

TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )

loz Drawback Plateau effect

Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)

((N2) ((N2) ((N2)

helliphellip

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)

Sliding window (length = N)

66 Synchronization2010722

TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )

loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula

( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus

扣掉相加

( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +

loz The method also called delay correlator

67 Synchronization2010722

IEEE 80211aIEEE 80211a

68 Synchronization2010722

Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )

loz Minnrsquos Method [2][ ]B B －B －B

( )( )

( )

22

2Minn

P dM d =

1 4 1N minus

( )( )( )2

2

MinnR d

( ) ( ) ( ) ( )

1 4 1

2 2 2 40 0

N

d N m k d N m k Nm k

P d r r+ + + + += =

= sum sum

( ) ( ) ( )2

21 4 1

2 40 0

N

d N m k Nk

R d rminus

+ + += sum sum69

0 0m k= =

Synchronization2010722

Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )

loz Parkrsquos Method [3][ ]

where D is the symmetric version of CC D C D

where D is the symmetric version of C

( )( ) 2

3k

P dM d =( )

( )( )23

ParkM dR d

2N

( ) 2

30

N

d k d kk

P d r rminus +=

= sdotsum

( )3

22

0

N

d kk

R d r += sum70

0k=

Synchronization2010722

Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )

loz Renrsquos Method [4][ ]

AS AS

where S is the PN sequence weighted of the original bl 2preamble

( )( )( )( )

24

Re 24

n

P dM d

R d=

( )( )4R d

( ) 2 1

4 2 2

N

k k N d k d k NP d s s r rminus

+ + + += sum0k=sum

( )21

41 N

d kR d rminus

= sum71

( )402 d k

kR d r +

=sum

Synchronization2010722

Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )

72 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

73 Synchronization2010722

Fractional CFO estimation Fractional CFO estimation

loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected

11 Rminus⎛ ⎞sum

h L i th di t b t t id ti l bl k R i

0

12 d m d m L

ms

f r rLTπ + + +

=

⎛ ⎞= ang⎜ ⎟⎝ ⎠sum

where L is the distance between two identical block R is the block size

loz For exampleNN cp

L = N R = NL = N R = Ncp74 Synchronization2010722

Estimation rangeEstimation rangegg

loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval

f ( ) ( )s s⎣ ⎦

loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)

loz DVB-T (L=N) 802 11 (L N4)

[ ]05 05s sf fminus

[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)

(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus

loz Note that is the subcarrier spacing1sf NT=

75

sNT

Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz

plusmng q y

loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn

76 Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz

plusmng q y

loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for

subcarrier spacing is required

plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn

p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs

77 Synchronization2010722

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )

63 Synchronization2010722

TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )

loz If drsquo is the correct symbol timing offset

( )2

2 2

0 1 12

j T f j T f j T fN

P d a e a e a eπ π πΔ Δ Δ

minus= + + +

22

2 2

0 1 12

j T fN

e a a aπ Δ

minus

⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭

loz If drsquo falls behind the correct symbol timing offset 1 sample

2⎪ ⎪⎩ ⎭

64 Synchronization2010722

TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )

65 Synchronization2010722

TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )

loz Drawback Plateau effect

Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)

((N2) ((N2) ((N2)

helliphellip

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)

Sliding window (length = N)

66 Synchronization2010722

TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )

loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula

( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus

扣掉相加

( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +

loz The method also called delay correlator

67 Synchronization2010722

IEEE 80211aIEEE 80211a

68 Synchronization2010722

Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )

loz Minnrsquos Method [2][ ]B B －B －B

( )( )

( )

22

2Minn

P dM d =

1 4 1N minus

( )( )( )2

2

MinnR d

( ) ( ) ( ) ( )

1 4 1

2 2 2 40 0

N

d N m k d N m k Nm k

P d r r+ + + + += =

= sum sum

( ) ( ) ( )2

21 4 1

2 40 0

N

d N m k Nk

R d rminus

+ + += sum sum69

0 0m k= =

Synchronization2010722

Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )

loz Parkrsquos Method [3][ ]

where D is the symmetric version of CC D C D

where D is the symmetric version of C

( )( ) 2

3k

P dM d =( )

( )( )23

ParkM dR d

2N

( ) 2

30

N

d k d kk

P d r rminus +=

= sdotsum

( )3

22

0

N

d kk

R d r += sum70

0k=

Synchronization2010722

Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )

loz Renrsquos Method [4][ ]

AS AS

where S is the PN sequence weighted of the original bl 2preamble

( )( )( )( )

24

Re 24

n

P dM d

R d=

( )( )4R d

( ) 2 1

4 2 2

N

k k N d k d k NP d s s r rminus

+ + + += sum0k=sum

( )21

41 N

d kR d rminus

= sum71

( )402 d k

kR d r +

=sum

Synchronization2010722

Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )

72 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

73 Synchronization2010722

Fractional CFO estimation Fractional CFO estimation

loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected

11 Rminus⎛ ⎞sum

h L i th di t b t t id ti l bl k R i

0

12 d m d m L

ms

f r rLTπ + + +

=

⎛ ⎞= ang⎜ ⎟⎝ ⎠sum

where L is the distance between two identical block R is the block size

loz For exampleNN cp

L = N R = NL = N R = Ncp74 Synchronization2010722

Estimation rangeEstimation rangegg

loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval

f ( ) ( )s s⎣ ⎦

loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)

loz DVB-T (L=N) 802 11 (L N4)

[ ]05 05s sf fminus

[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)

(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus

loz Note that is the subcarrier spacing1sf NT=

75

sNT

Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz

plusmng q y

loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn

76 Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz

plusmng q y

loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for

subcarrier spacing is required

plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn

p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs

77 Synchronization2010722

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )

loz If drsquo is the correct symbol timing offset

( )2

2 2

0 1 12

j T f j T f j T fN

P d a e a e a eπ π πΔ Δ Δ

minus= + + +

22

2 2

0 1 12

j T fN

e a a aπ Δ

minus

⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭

loz If drsquo falls behind the correct symbol timing offset 1 sample

2⎪ ⎪⎩ ⎭

64 Synchronization2010722

TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )

65 Synchronization2010722

TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )

loz Drawback Plateau effect

Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)

((N2) ((N2) ((N2)

helliphellip

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)

Sliding window (length = N)

66 Synchronization2010722

TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )

loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula

( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus

扣掉相加

( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +

loz The method also called delay correlator

67 Synchronization2010722

IEEE 80211aIEEE 80211a

68 Synchronization2010722

Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )

loz Minnrsquos Method [2][ ]B B －B －B

( )( )

( )

22

2Minn

P dM d =

1 4 1N minus

( )( )( )2

2

MinnR d

( ) ( ) ( ) ( )

1 4 1

2 2 2 40 0

N

d N m k d N m k Nm k

P d r r+ + + + += =

= sum sum

( ) ( ) ( )2

21 4 1

2 40 0

N

d N m k Nk

R d rminus

+ + += sum sum69

0 0m k= =

Synchronization2010722

Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )

loz Parkrsquos Method [3][ ]

where D is the symmetric version of CC D C D

where D is the symmetric version of C

( )( ) 2

3k

P dM d =( )

( )( )23

ParkM dR d

2N

( ) 2

30

N

d k d kk

P d r rminus +=

= sdotsum

( )3

22

0

N

d kk

R d r += sum70

0k=

Synchronization2010722

Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )

loz Renrsquos Method [4][ ]

AS AS

where S is the PN sequence weighted of the original bl 2preamble

( )( )( )( )

24

Re 24

n

P dM d

R d=

( )( )4R d

( ) 2 1

4 2 2

N

k k N d k d k NP d s s r rminus

+ + + += sum0k=sum

( )21

41 N

d kR d rminus

= sum71

( )402 d k

kR d r +

=sum

Synchronization2010722

Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )

72 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

73 Synchronization2010722

Fractional CFO estimation Fractional CFO estimation

loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected

11 Rminus⎛ ⎞sum

h L i th di t b t t id ti l bl k R i

0

12 d m d m L

ms

f r rLTπ + + +

=

⎛ ⎞= ang⎜ ⎟⎝ ⎠sum

where L is the distance between two identical block R is the block size

loz For exampleNN cp

L = N R = NL = N R = Ncp74 Synchronization2010722

Estimation rangeEstimation rangegg

loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval

f ( ) ( )s s⎣ ⎦

loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)

loz DVB-T (L=N) 802 11 (L N4)

[ ]05 05s sf fminus

[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)

(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus

loz Note that is the subcarrier spacing1sf NT=

75

sNT

Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz

plusmng q y

loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn

76 Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz

plusmng q y

loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for

subcarrier spacing is required

plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn

p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs

77 Synchronization2010722

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )

65 Synchronization2010722

TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )

loz Drawback Plateau effect

Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)

((N2) ((N2) ((N2)

helliphellip

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)

Sliding window (length = N)

66 Synchronization2010722

TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )

loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula

( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus

扣掉相加

( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +

loz The method also called delay correlator

67 Synchronization2010722

IEEE 80211aIEEE 80211a

68 Synchronization2010722

Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )

loz Minnrsquos Method [2][ ]B B －B －B

( )( )

( )

22

2Minn

P dM d =

1 4 1N minus

( )( )( )2

2

MinnR d

( ) ( ) ( ) ( )

1 4 1

2 2 2 40 0

N

d N m k d N m k Nm k

P d r r+ + + + += =

= sum sum

( ) ( ) ( )2

21 4 1

2 40 0

N

d N m k Nk

R d rminus

+ + += sum sum69

0 0m k= =

Synchronization2010722

Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )

loz Parkrsquos Method [3][ ]

where D is the symmetric version of CC D C D

where D is the symmetric version of C

( )( ) 2

3k

P dM d =( )

( )( )23

ParkM dR d

2N

( ) 2

30

N

d k d kk

P d r rminus +=

= sdotsum

( )3

22

0

N

d kk

R d r += sum70

0k=

Synchronization2010722

Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )

loz Renrsquos Method [4][ ]

AS AS

where S is the PN sequence weighted of the original bl 2preamble

( )( )( )( )

24

Re 24

n

P dM d

R d=

( )( )4R d

( ) 2 1

4 2 2

N

k k N d k d k NP d s s r rminus

+ + + += sum0k=sum

( )21

41 N

d kR d rminus

= sum71

( )402 d k

kR d r +

=sum

Synchronization2010722

Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )

72 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

73 Synchronization2010722

Fractional CFO estimation Fractional CFO estimation

loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected

11 Rminus⎛ ⎞sum

h L i th di t b t t id ti l bl k R i

0

12 d m d m L

ms

f r rLTπ + + +

=

⎛ ⎞= ang⎜ ⎟⎝ ⎠sum

where L is the distance between two identical block R is the block size

loz For exampleNN cp

L = N R = NL = N R = Ncp74 Synchronization2010722

Estimation rangeEstimation rangegg

loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval

f ( ) ( )s s⎣ ⎦

loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)

loz DVB-T (L=N) 802 11 (L N4)

[ ]05 05s sf fminus

[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)

(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus

loz Note that is the subcarrier spacing1sf NT=

75

sNT

Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz

plusmng q y

loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn

76 Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz

plusmng q y

loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for

subcarrier spacing is required

plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn

p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs

77 Synchronization2010722

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )

loz Drawback Plateau effect

Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)

((N2) ((N2) ((N2)

helliphellip

r(0) r(1) helliphellipr((N2)-

2)r((N2)-

1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)

Sliding window (length = N)

66 Synchronization2010722

TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )

loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula

( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus

扣掉相加

( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +

loz The method also called delay correlator

67 Synchronization2010722

IEEE 80211aIEEE 80211a

68 Synchronization2010722

Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )

loz Minnrsquos Method [2][ ]B B －B －B

( )( )

( )

22

2Minn

P dM d =

1 4 1N minus

( )( )( )2

2

MinnR d

( ) ( ) ( ) ( )

1 4 1

2 2 2 40 0

N

d N m k d N m k Nm k

P d r r+ + + + += =

= sum sum

( ) ( ) ( )2

21 4 1

2 40 0

N

d N m k Nk

R d rminus

+ + += sum sum69

0 0m k= =

Synchronization2010722

Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )

loz Parkrsquos Method [3][ ]

where D is the symmetric version of CC D C D

where D is the symmetric version of C

( )( ) 2

3k

P dM d =( )

( )( )23

ParkM dR d

2N

( ) 2

30

N

d k d kk

P d r rminus +=

= sdotsum

( )3

22

0

N

d kk

R d r += sum70

0k=

Synchronization2010722

Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )

loz Renrsquos Method [4][ ]

AS AS

where S is the PN sequence weighted of the original bl 2preamble

( )( )( )( )

24

Re 24

n

P dM d

R d=

( )( )4R d

( ) 2 1

4 2 2

N

k k N d k d k NP d s s r rminus

+ + + += sum0k=sum

( )21

41 N

d kR d rminus

= sum71

( )402 d k

kR d r +

=sum

Synchronization2010722

Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )

72 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

73 Synchronization2010722

Fractional CFO estimation Fractional CFO estimation

loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected

11 Rminus⎛ ⎞sum

h L i th di t b t t id ti l bl k R i

0

12 d m d m L

ms

f r rLTπ + + +

=

⎛ ⎞= ang⎜ ⎟⎝ ⎠sum

where L is the distance between two identical block R is the block size

loz For exampleNN cp

L = N R = NL = N R = Ncp74 Synchronization2010722

Estimation rangeEstimation rangegg

loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval

f ( ) ( )s s⎣ ⎦

loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)

loz DVB-T (L=N) 802 11 (L N4)

[ ]05 05s sf fminus

[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)

(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus

loz Note that is the subcarrier spacing1sf NT=

75

sNT

Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz

plusmng q y

loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn

76 Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz

plusmng q y

loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for

subcarrier spacing is required

plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn

p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs

77 Synchronization2010722

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )

loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula

( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus

扣掉相加

( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +

loz The method also called delay correlator

67 Synchronization2010722

IEEE 80211aIEEE 80211a

68 Synchronization2010722

Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )

loz Minnrsquos Method [2][ ]B B －B －B

( )( )

( )

22

2Minn

P dM d =

1 4 1N minus

( )( )( )2

2

MinnR d

( ) ( ) ( ) ( )

1 4 1

2 2 2 40 0

N

d N m k d N m k Nm k

P d r r+ + + + += =

= sum sum

( ) ( ) ( )2

21 4 1

2 40 0

N

d N m k Nk

R d rminus

+ + += sum sum69

0 0m k= =

Synchronization2010722

Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )

loz Parkrsquos Method [3][ ]

where D is the symmetric version of CC D C D

where D is the symmetric version of C

( )( ) 2

3k

P dM d =( )

( )( )23

ParkM dR d

2N

( ) 2

30

N

d k d kk

P d r rminus +=

= sdotsum

( )3

22

0

N

d kk

R d r += sum70

0k=

Synchronization2010722

Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )

loz Renrsquos Method [4][ ]

AS AS

where S is the PN sequence weighted of the original bl 2preamble

( )( )( )( )

24

Re 24

n

P dM d

R d=

( )( )4R d

( ) 2 1

4 2 2

N

k k N d k d k NP d s s r rminus

+ + + += sum0k=sum

( )21

41 N

d kR d rminus

= sum71

( )402 d k

kR d r +

=sum

Synchronization2010722

Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )

72 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

73 Synchronization2010722

Fractional CFO estimation Fractional CFO estimation

loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected

11 Rminus⎛ ⎞sum

h L i th di t b t t id ti l bl k R i

0

12 d m d m L

ms

f r rLTπ + + +

=

⎛ ⎞= ang⎜ ⎟⎝ ⎠sum

where L is the distance between two identical block R is the block size

loz For exampleNN cp

L = N R = NL = N R = Ncp74 Synchronization2010722

Estimation rangeEstimation rangegg

loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval

f ( ) ( )s s⎣ ⎦

loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)

loz DVB-T (L=N) 802 11 (L N4)

[ ]05 05s sf fminus

[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)

(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus

loz Note that is the subcarrier spacing1sf NT=

75

sNT

Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz

plusmng q y

loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn

76 Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz

plusmng q y

loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for

subcarrier spacing is required

plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn

p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs

77 Synchronization2010722

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

IEEE 80211aIEEE 80211a

68 Synchronization2010722

Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )

loz Minnrsquos Method [2][ ]B B －B －B

( )( )

( )

22

2Minn

P dM d =

1 4 1N minus

( )( )( )2

2

MinnR d

( ) ( ) ( ) ( )

1 4 1

2 2 2 40 0

N

d N m k d N m k Nm k

P d r r+ + + + += =

= sum sum

( ) ( ) ( )2

21 4 1

2 40 0

N

d N m k Nk

R d rminus

+ + += sum sum69

0 0m k= =

Synchronization2010722

Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )

loz Parkrsquos Method [3][ ]

where D is the symmetric version of CC D C D

where D is the symmetric version of C

( )( ) 2

3k

P dM d =( )

( )( )23

ParkM dR d

2N

( ) 2

30

N

d k d kk

P d r rminus +=

= sdotsum

( )3

22

0

N

d kk

R d r += sum70

0k=

Synchronization2010722

Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )

loz Renrsquos Method [4][ ]

AS AS

where S is the PN sequence weighted of the original bl 2preamble

( )( )( )( )

24

Re 24

n

P dM d

R d=

( )( )4R d

( ) 2 1

4 2 2

N

k k N d k d k NP d s s r rminus

+ + + += sum0k=sum

( )21

41 N

d kR d rminus

= sum71

( )402 d k

kR d r +

=sum

Synchronization2010722

Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )

72 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

73 Synchronization2010722

Fractional CFO estimation Fractional CFO estimation

loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected

11 Rminus⎛ ⎞sum

h L i th di t b t t id ti l bl k R i

0

12 d m d m L

ms

f r rLTπ + + +

=

⎛ ⎞= ang⎜ ⎟⎝ ⎠sum

where L is the distance between two identical block R is the block size

loz For exampleNN cp

L = N R = NL = N R = Ncp74 Synchronization2010722

Estimation rangeEstimation rangegg

loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval

f ( ) ( )s s⎣ ⎦

loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)

loz DVB-T (L=N) 802 11 (L N4)

[ ]05 05s sf fminus

[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)

(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus

loz Note that is the subcarrier spacing1sf NT=

75

sNT

Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz

plusmng q y

loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn

76 Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz

plusmng q y

loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for

subcarrier spacing is required

plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn

p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs

77 Synchronization2010722

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )

loz Minnrsquos Method [2][ ]B B －B －B

( )( )

( )

22

2Minn

P dM d =

1 4 1N minus

( )( )( )2

2

MinnR d

( ) ( ) ( ) ( )

1 4 1

2 2 2 40 0

N

d N m k d N m k Nm k

P d r r+ + + + += =

= sum sum

( ) ( ) ( )2

21 4 1

2 40 0

N

d N m k Nk

R d rminus

+ + += sum sum69

0 0m k= =

Synchronization2010722

Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )

loz Parkrsquos Method [3][ ]

where D is the symmetric version of CC D C D

where D is the symmetric version of C

( )( ) 2

3k

P dM d =( )

( )( )23

ParkM dR d

2N

( ) 2

30

N

d k d kk

P d r rminus +=

= sdotsum

( )3

22

0

N

d kk

R d r += sum70

0k=

Synchronization2010722

Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )

loz Renrsquos Method [4][ ]

AS AS

where S is the PN sequence weighted of the original bl 2preamble

( )( )( )( )

24

Re 24

n

P dM d

R d=

( )( )4R d

( ) 2 1

4 2 2

N

k k N d k d k NP d s s r rminus

+ + + += sum0k=sum

( )21

41 N

d kR d rminus

= sum71

( )402 d k

kR d r +

=sum

Synchronization2010722

Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )

72 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

73 Synchronization2010722

Fractional CFO estimation Fractional CFO estimation

loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected

11 Rminus⎛ ⎞sum

h L i th di t b t t id ti l bl k R i

0

12 d m d m L

ms

f r rLTπ + + +

=

⎛ ⎞= ang⎜ ⎟⎝ ⎠sum

where L is the distance between two identical block R is the block size

loz For exampleNN cp

L = N R = NL = N R = Ncp74 Synchronization2010722

Estimation rangeEstimation rangegg

loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval

f ( ) ( )s s⎣ ⎦

loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)

loz DVB-T (L=N) 802 11 (L N4)

[ ]05 05s sf fminus

[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)

(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus

loz Note that is the subcarrier spacing1sf NT=

75

sNT

Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz

plusmng q y

loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn

76 Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz

plusmng q y

loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for

subcarrier spacing is required

plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn

p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs

77 Synchronization2010722

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )

loz Parkrsquos Method [3][ ]

where D is the symmetric version of CC D C D

where D is the symmetric version of C

( )( ) 2

3k

P dM d =( )

( )( )23

ParkM dR d

2N

( ) 2

30

N

d k d kk

P d r rminus +=

= sdotsum

( )3

22

0

N

d kk

R d r += sum70

0k=

Synchronization2010722

Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )

loz Renrsquos Method [4][ ]

AS AS

where S is the PN sequence weighted of the original bl 2preamble

( )( )( )( )

24

Re 24

n

P dM d

R d=

( )( )4R d

( ) 2 1

4 2 2

N

k k N d k d k NP d s s r rminus

+ + + += sum0k=sum

( )21

41 N

d kR d rminus

= sum71

( )402 d k

kR d r +

=sum

Synchronization2010722

Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )

72 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

73 Synchronization2010722

Fractional CFO estimation Fractional CFO estimation

loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected

11 Rminus⎛ ⎞sum

h L i th di t b t t id ti l bl k R i

0

12 d m d m L

ms

f r rLTπ + + +

=

⎛ ⎞= ang⎜ ⎟⎝ ⎠sum

where L is the distance between two identical block R is the block size

loz For exampleNN cp

L = N R = NL = N R = Ncp74 Synchronization2010722

Estimation rangeEstimation rangegg

loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval

f ( ) ( )s s⎣ ⎦

loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)

loz DVB-T (L=N) 802 11 (L N4)

[ ]05 05s sf fminus

[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)

(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus

loz Note that is the subcarrier spacing1sf NT=

75

sNT

Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz

plusmng q y

loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn

76 Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz

plusmng q y

loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for

subcarrier spacing is required

plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn

p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs

77 Synchronization2010722

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )

loz Renrsquos Method [4][ ]

AS AS

where S is the PN sequence weighted of the original bl 2preamble

( )( )( )( )

24

Re 24

n

P dM d

R d=

( )( )4R d

( ) 2 1

4 2 2

N

k k N d k d k NP d s s r rminus

+ + + += sum0k=sum

( )21

41 N

d kR d rminus

= sum71

( )402 d k

kR d r +

=sum

Synchronization2010722

Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )

72 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

73 Synchronization2010722

Fractional CFO estimation Fractional CFO estimation

loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected

11 Rminus⎛ ⎞sum

h L i th di t b t t id ti l bl k R i

0

12 d m d m L

ms

f r rLTπ + + +

=

⎛ ⎞= ang⎜ ⎟⎝ ⎠sum

where L is the distance between two identical block R is the block size

loz For exampleNN cp

L = N R = NL = N R = Ncp74 Synchronization2010722

Estimation rangeEstimation rangegg

loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval

f ( ) ( )s s⎣ ⎦

loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)

loz DVB-T (L=N) 802 11 (L N4)

[ ]05 05s sf fminus

[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)

(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus

loz Note that is the subcarrier spacing1sf NT=

75

sNT

Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz

plusmng q y

loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn

76 Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz

plusmng q y

loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for

subcarrier spacing is required

plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn

p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs

77 Synchronization2010722

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )

72 Synchronization2010722

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

73 Synchronization2010722

Fractional CFO estimation Fractional CFO estimation

loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected

11 Rminus⎛ ⎞sum

h L i th di t b t t id ti l bl k R i

0

12 d m d m L

ms

f r rLTπ + + +

=

⎛ ⎞= ang⎜ ⎟⎝ ⎠sum

where L is the distance between two identical block R is the block size

loz For exampleNN cp

L = N R = NL = N R = Ncp74 Synchronization2010722

Estimation rangeEstimation rangegg

loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval

f ( ) ( )s s⎣ ⎦

loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)

loz DVB-T (L=N) 802 11 (L N4)

[ ]05 05s sf fminus

[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)

(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus

loz Note that is the subcarrier spacing1sf NT=

75

sNT

Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz

plusmng q y

loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn

76 Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz

plusmng q y

loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for

subcarrier spacing is required

plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn

p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs

77 Synchronization2010722

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

OutlineOutline

loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg

73 Synchronization2010722

Fractional CFO estimation Fractional CFO estimation

loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected

11 Rminus⎛ ⎞sum

h L i th di t b t t id ti l bl k R i

0

12 d m d m L

ms

f r rLTπ + + +

=

⎛ ⎞= ang⎜ ⎟⎝ ⎠sum

where L is the distance between two identical block R is the block size

loz For exampleNN cp

L = N R = NL = N R = Ncp74 Synchronization2010722

Estimation rangeEstimation rangegg

loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval

f ( ) ( )s s⎣ ⎦

loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)

loz DVB-T (L=N) 802 11 (L N4)

[ ]05 05s sf fminus

[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)

(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus

loz Note that is the subcarrier spacing1sf NT=

75

sNT

Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz

plusmng q y

loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn

76 Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz

plusmng q y

loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for

subcarrier spacing is required

plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn

p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs

77 Synchronization2010722

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

Fractional CFO estimation Fractional CFO estimation

loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected

11 Rminus⎛ ⎞sum

h L i th di t b t t id ti l bl k R i

0

12 d m d m L

ms

f r rLTπ + + +

=

⎛ ⎞= ang⎜ ⎟⎝ ⎠sum

where L is the distance between two identical block R is the block size

loz For exampleNN cp

L = N R = NL = N R = Ncp74 Synchronization2010722

Estimation rangeEstimation rangegg

loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval

f ( ) ( )s s⎣ ⎦

loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)

loz DVB-T (L=N) 802 11 (L N4)

[ ]05 05s sf fminus

[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)

(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus

loz Note that is the subcarrier spacing1sf NT=

75

sNT

Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz

plusmng q y

loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn

76 Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz

plusmng q y

loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for

subcarrier spacing is required

plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn

p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs

77 Synchronization2010722

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

Estimation rangeEstimation rangegg

loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval

f ( ) ( )s s⎣ ⎦

loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)

loz DVB-T (L=N) 802 11 (L N4)

[ ]05 05s sf fminus

[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)

(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus

loz Note that is the subcarrier spacing1sf NT=

75

sNT

Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz

plusmng q y

loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn

76 Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz

plusmng q y

loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for

subcarrier spacing is required

plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn

p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs

77 Synchronization2010722

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz

plusmng q y

loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn

76 Synchronization2010722

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz

plusmng q y

loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for

subcarrier spacing is required

plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn

p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs

77 Synchronization2010722

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

Maximum CFOMaximum CFO

loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz

plusmng q y

loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for

subcarrier spacing is required

plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn

p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs

77 Synchronization2010722

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )

loz Time domain correlationloz Match filters with

coefficient of conjugated preamble waveform modulated by different integer CFOCFO

loz Example CFO is 42 ffs

78 Synchronization2010722

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )

loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )

consecutive symbols

( )1

0 1 2J

g Z Z gminus

Φ = = plusmn plusmnsum( ) 10

0 1 2j ji g i g

jg Z Z gα α+ minus +

=

Φ = = plusmn plusmnsum hellip

( )ˆ arg maxg g= Φ( )g

79 Synchronization2010722

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

IEEE 80211aIEEE 80211a

80 Synchronization2010722

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

IEEE 80211aIEEE 80211a

2010722 Synchronization81

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

Short OFDM training symbol Short OFDM training symbol g yg y

loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS－2626= 0 0 1+j 0 0 0 －1－j 0 0 0 1+j

0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 0 0 0 0 －1－j 0 0 0 －1－j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0

2010722 Synchronization82

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

Long OFDM training symbolLong OFDM training symbolg g yg g y

loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL－2626=1 1 －1 －1 1 1 －1 1 －1 1 1 1 1 1 1

－1 －1 1 1 －1 1 －1 1 1 1 1 0 1 －1 －11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 －1 1 －1 1 －1 －1 －1 －1 －1 －111 1 1 1 1 1 1 1 1 1 11 －1 －1 1 －1 1 －1 1 1 1 1

2010722 Synchronization83

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

Pilot subcarriersPilot subcarriers

loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise

loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21

loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines

2010722 Synchronization84

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

Pilot subcarriersPilot subcarriers

d0 DC d24d29 d30 d42 d43 d47d18 d23P－7 P7 P21d17d5P－21d4

－26 －21 －7 7 21 260

2010722 Synchronization85

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

ReferenceReference

[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997

[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003

[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003

[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005

[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000

[6] J－J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800－1805 Jul 1997

[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428－1429 Oct 20041429 Oct 2004

86 Synchronization2010722

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering

Channel EstimationChannel Estimation

g gg gNational Sun National Sun YatYat－－sensen UniversityUniversity

2010072220100722王森弘

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

882010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

Small scale fadingSmall scale fadinggg

loz Multi-path channelp

892010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

Channel impulse responseChannel impulse responsep pp p

In the absence of noise the received signal can be expressed asloz g p

( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin

= otimes = minusintwhere

( ) is the channel impulse resonseh t

minusinfin

( ) is the channel impulse resonse ( ) is the transmitted signal

( ) i

h tx ty t s the received signal ( ) iy t

1

s the received signalAfter sampling the discrete received signal is given by

L 1

0

[ ] [ ] [ ]L

k

y n x k h n kminus

=

= minussum

902010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths

( ) ( ) ( )[ 0 1 1 ]Th h h L= minush

( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian

d i bl ith d i

( )h l0 1l Lle le minus

2random variables with zero-mean and variance ( )2h lσ

0 5

1

gnitu

de

1 2 3 4 5 6 7 80

05

the

mag

91

the lth tap of the channel

2010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

The multiThe multi--path channel effectpath channel effectpp

loz The multi-path channel effectp

922010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

The transmitted OFDM signal The transmitted OFDM signal

loz After the inverse discrete Fourier transform (IDFT) operation the

gg

ith transmitted OFDM symbol in time domain can be expressed by

H

Hi i=x F X

where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively

1N timesiX HF N Ntimes

932010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix G is constructed as follows

( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )

01 0

hh L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )

( ) ( )0 1 00

h L h= ⎢ ⎥

minus⎢ ⎥⎢ ⎥⎢ ⎥

G

( ) ( )0

0 0 1 0h L h⎢ ⎥

minus⎢ ⎥⎣ ⎦

942010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz The matrix is constructed as follows tailG

( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥

hellip

( )il

01h L

⎢ ⎥⎢ ⎥⎢ ⎥minus

= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

G

0 0⎢ ⎥⎢ ⎥⎣ ⎦

952010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

The matrix form of channelThe matrix form of channel

loz Furthermore a circular convolution matrix can be circularGobtained

circular

circ lar tail=G G + G

( ) ( ) ( )( ) ( )

circular tail

0 0 0 1 11 0 0

h h L hh h

⎡ ⎤minus⎢ ⎥⎢ ⎥

G G G

( ) ( )( )

( ) ( )

1 0 01

1 0 0

h hh L

h L h

⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )

( ) ( )1 0 0

0 1 0h L h

h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥

( ) ( )0

0 0 1 0h L h

⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦

962010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

The received OFDM signalThe received OFDM signalgg

loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as

tail 1i i i i= + +r Gx G x wtail 1

circular tail tail 1 i i i i

i i i i

minus

minus= minus + +

= otimes + +

G G wG x G x G x wx h z w

where denotes the circular convolution denotes an white

i N i i= otimes + +x h z w

Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel

2wσ iz

in the ith OFDM symbol caused by multi-path channel

tail tail 1i i iminus= minus +z G x G x

972010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbolyy

loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as

+ +r Gx G x wtail

circular i i i i

i i

= + += +

r Gx G x wG x w

where denotes the circular convolution We note that there is i N i= otimes +x h w

Notimesno ICI and ISI in each OFDM symbol

982010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Assuming the synchronization is perfect and CP is adopted the

yy

received ith OFDM symbol in the frequency domain can be expressed as

=R Fr

( )i i

i N i

=

= otimes +

R FrF x h w

where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f

N Ntimesi i= +HX W

diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem

Li l bloz Linear algebra

992010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

The received CPThe received CP--OFDM symbolOFDM symbol

loz Any circular matrix can be diagonalized by DFT matrix

yy

( )circular

0 0 0

H

H

FG F H=

⎡ ⎤( )( )

0 0 00 1

0

HH

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥

( )0

0 0 1H N

⎢ ⎥⎢ ⎥

minus⎢ ⎥⎣ ⎦

loz The received signal can be expressed as

( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular

circular

i i i i i

i i i i

+ +

= + = +

R Fr F G x w F G FX wFG FX Fw HX W

1002010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

The frequency response of the channelThe frequency response of the channelq y pq y p

0

1

art

-2

-1real

pa

0 10 20 30 40 50 60 70-2

sub-carrier index

2

0

1

imag

par

t

0 10 20 30 40 50 60 70-1

sub-carrier index

1012010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y

1

2

art

-2

-1

0re

al p

a

0 10 20 30 40 50 60 70-2

sub-carrier index

2

-1

0

1

imag

par

t

0 10 20 30 40 50 60 70-2

sub-carrier index

1022010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

ReferenceReference

[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000

1032010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1042010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

System modelSystem modelyy

loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain

| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise

kk

k

P k qT q QX

Sτ⎧ isin + = minus

= ⎨⎩

loz In addition the power allocation of data and pilot symbols are given by

β⎧

( )

2

2

| 01 1

1

k

k

P k qT q QQ

X

βρ τ

β

⎧ = isin + = minus⎪⎪= ⎨ ( )2 1

otherwisek

kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩

1052010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

System modelSystem modelyy

loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as

where denotes the Hermitian transpose and F stands for

Hi i=x F X

( )Hi p the normalized discrete Fourier transform (DFT) matrix

( )N Ntimes

1062010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

System modelSystem modelyy

loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as

i i i= +R ΛX W

where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier

i i i

Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2

W W Nσ=C IW W NσC I

1072010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

The LS channel estimator The LS channel estimator

loz Define where denotes the received pilot signaliRM

iequivsumR R

loz The channel estimator based on the LS method is given by

1i

i=sum

11

LS1

M M

minusminus= = +

Γ RH H Γ W

where denotes a diagonal matrix whose diagonal

M M

Γ Q Qtimes g gelements are given by

01 1q q kP k qT q QτΓ = isin + = minus

1082010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz The mean square error (MSE) of the LS channel estimator can be expressed as

22 W Qσ2LS

W QM

σρ

= sdot

1092010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

The LS channel estimatorThe LS channel estimator

loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-

carriers but also by the number of pilot sub-carriers

loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain

1102010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

loz The modified LS (MLS) channel estimator is given by

MLS LSH

L= sdotH FΔ F H

where is a diagonal matrix The entries of are LΔ tδ LΔ

1 01 1t L= minus⎧1 01 10 1t

t Lt L N

δ⎧

= ⎨ = minus⎩

loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme

1112010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )

012

012

012

012

FHL

L+1FL－1 L－1

LSH MLSH

Ｎ－1 Ｎ－1 Ｎ－1

1122010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

The MLS channel estimatorThe MLS channel estimator

loz The MSE of the MLS channel estimator can be easily yobtained by definition

( )( )2 1 t EH⎧ ⎫⎡ ⎤

⎨ ⎬H H H H( )( )2MLS MLSMLS tr E

1NL

σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞

H H H H

1=SINR

LN

⎛ ⎞sdot⎜ ⎟⎝ ⎠

1132010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

ReferencesReferences

[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001

[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002

[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004

1142010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1152010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

IntroductionIntroduction

loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)

1162010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

System architectureSystem architectureyy

1172010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

System architectureSystem architectureyy

1 Input to Time Domain

( ) ( ) 1210 minus=

=Nn

kXIDFTnx

( ) ( )( )⎩

⎨⎧

minus=minus+minusminus=+

=110

11NnnxNNnnNx

nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=

32 Guard Interval Channel

( )⎩ 0 Nnnx

( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY

54 Guard Removal Output to Frequency Domain

( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk

76 Output Channel EstimationAWGNChannel Estimated Ch nn l

( ) ( ) ( ) ( )01 1

Y k X k H k W kk N

= +

= minus( ) ( )

( ) 110 minus== NkkH

kYkXe

e

Channel

1182010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

Pilot arrangementPilot arrangementgg

loz Comb Typeyploz Some sub-carriers are

reserved for pilots for each symbol

loz Block Typeloz All sub-carriers reserved

for pilots with a specific period

1192010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

DecisionDecision--feedback channel estimationfeedback channel estimation

loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier

Decision Feedback EqualizerDecision Feedback Equalizer

110)(

)()( minus== NkkH

kYkX e )(kHe

( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr

( )( ) 01 1( )e

Y kH k k NX k

= = minus

loz For fast fading the comb-type estimation performs much better

( )pureX k

better

1202010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

Linear interpolationLinear interpolation

loz Linear Interpolation

pp

p

( ) ( )e eH k H mL l= +

( ) ( )( ) ( )1p p plH m H m H mL

= + minus +

0 l Lle lt

1212010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

Second order interpolationSecond order interpolationpp

loz Second Order Interpolationp( ) ( )

( 1) ( ) ( 1)e eH k H mL l

c H m c H m c H m= minus

+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +

( 1)α αwhere 1( 1)

2( 1)( 1)

c

c

α α

α α

minus=

=minus minus +0

1

( 1)( 1)( 1)

2

c

c

α α

α αminus

=minus minus +

+=

2lN

α=

122

N

2010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

BlockBlock--type pilot insertiontype pilot insertionyp pyp p

Block type

1232010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

CombComb--type pilot insertiontype pilot insertionyp pyp p

d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4

-26 -21 -7 7 21 260

1242010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

OutlineOutline

loz Effect of multi-path channelloz Frequency domain channel estimation

loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation

loz Pilot arrangementloz Block typeloz Comb type

loz Conclusions

1252010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

ConclusionsConclusions

loz OFDM systemloz Block type

loz Direct or Decision Feedbackloz Comb type

loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques

loz Linearloz Second Orderloz DFT-based

1262010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation

ReferencesReferences

[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993

[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983

[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997

[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002

1272010722 Channel Estimation