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ECSE413B ♦ THO LE-NGOC ♦ BAND-LIMITED SIGNALING SCHEMES IN AN AWGN ENVIRONMENT PAGE- 1
BANDLIMITED SIGNALING OVER AWGN CHANNELS
POWER SPECTRA OF LINEARLY MODULATED SIGNALS Bandpass signal { }tj cetvts ω)(Re)( = , v(t): complex baseband signal, ωc: center (carrier) frequency
Power spectral density (psd): [ ])()(21)( cvcvs ffSffSfS −−+−=
Consider a complex baseband signal ∑∞
−∞=
−=n
n nTtgItv )()(
g(t): Tx sprectrum-shaping pulse, G(f): Fourier transform of g(t) In: symbol, real for M-PAM, complex for M-PSK, M-QAM, M-APK
In: wide-sense stationary with mean μi { } ∑∞
−∞=
−=⇒n
i nTtgtvE )()( μ
and, autocorrelation function: { }mnnII IIEm +=Φ *
21)(
PAM: ⎩⎨⎧
=+≠
=Φ0mfor 0mfor
)( 22
2
ii
iII m
σμμ
for {In}: real, mutually uncorrelated
M-ary DEMOD
+ s(t) r(t)=s(t)+n(t)
white Gaussian noise, n(t), with power spectral density
of No/2 (per Hz)
ADDITIVE WHITE GAUSSIAN NOISE (AWGN) CHANNEL
M-ary MODULATOR
No/2
frequency, f [Hz]
Bit-
to-s
ymbo
l CO
NVE
RTE
R
I/P BIT STREAM {bn}
sym
bol-t
o-bi
t CO
NVE
RTE
R
O/P BIT STREAM
{bn’}
BIT-TO-SYMBOL CONVERSION: Input binary sequence: {bn},
⎩⎨⎧
====
=p-10}Pr{bwith0
p1}Pr{bwith1b
i
in
{ai} {ai
’}
CHANNEL WITH CONSTANT
ATTENUATION C(f)=α≠0
Tx AFE
Rx AFE
THE PHYSICAL CHANNEL INCLUDES THE ANALOG Tx & Rx SECTIONS AND THE PHYSICAL MEDIUM, WHICH CAN BE A WIRELESS OR WIRELINE MEDIUM WITH C(f)=α≠0. THERMAL NOISE @ Rx INPUT IS ADDED TO THE ATTENUATED Tx SIGNAL & THE RESULTING Rx SIGNAL CAN BE AMPLIFIED FOR AN APPROPRIATE LEVEL FOR THE DEMOD
ECSE413B ♦ THO LE-NGOC ♦ BAND-LIMITED SIGNALING SCHEMES IN AN AWGN ENVIRONMENT PAGE- 2
autocorrelation function of v(t):
{ } { }
444444 3444444 21
43421
T periodin t with Periodic:)(
)()(*).(
)()(*)(
21)()(*
21),( *
mTt
mTnTtgnTtgm
nTmTtgnTtgm
IIEtvtvEtt
nmII
n mII
nmnvv
−+
⎥⎦
⎤⎢⎣
⎡+−−−Φ=
−−+−
Φ
=+=+Φ
∑∑
∑ ∑
∞
−∞=
∞
−∞=
∞
−∞=
∞
−∞=+
τγ
τ
τττ
Φvv(t+τ;t): periodic in t with period T, E{v(t)}: periodic with period T ⇒ v(t): cyclostationary (periodically stationary in wide sense) averaging Φvv(t+τ;t) over a single period to remove t
44444 344444 21∫∑
∑ ∫∑∫∞+
∞−
∞
−∞=
∞
−∞=
+−
−−
∞
−∞=−
−+Φ=
−+Φ=+Φ=Φ
dumTugugmT
dumTugugT
mdtttT
m
n
TnT
TnTm
T
T vvvv
)(*)(*)(1
)()(*1).(),(1)(
II
2/
2/II
2
2
τ
τττ
(time-autocorrelation function of g(t): Tjm
gg efGmT ϖτ −ℑ⎯→⎯−Φ 2)()( )
PAM: ⎩⎨⎧
=+≠
=Φ0mfor 0mfor
)( 22
2
ii
iII m
σμμ ⇒
⎪⎪
⎭
⎪⎪
⎬
⎫
⎪⎪
⎩
⎪⎪
⎨
⎧
+= ∑∞+
−∞=
−
43421
1/Tofperiodwithfinperiodic
222)(1)(m
mTjIIv efG
TfS ωμσ
∑+∞
−∞=
−Φℑ Φ=⎯⎯⎯ →⎯m
mTjIIv emfG
TfSvv ωτ )()(1)( 2))((
22 22
2( ) ( ) ( )
Continuous Discrete spectral linesspectrum
I Iv
m
m mS f G f G fT T TTσ μ
δ+∞
=−∞
⎛ ⎞= + −⎜ ⎟⎝ ⎠∑
14243 1444442444443
∑∑+∞
−∞=
+∞
−∞=
− −=mm
mTj
Tmf
Te )(1 δω
To remove the discrete spectral lines, we need μI=0 zero mean
sequence or:
2
0mGT
⎛ ⎞ =⎜ ⎟⎝ ⎠
for all m
ECSE413B ♦ THO LE-NGOC ♦ BAND-LIMITED SIGNALING SCHEMES IN AN AWGN ENVIRONMENT PAGE- 3
TIME-LIMITED g(t) ⇒ G(f) with INFINITE BW EXAMPLE: PAM signal using rectangular pulse:
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-4 -2 0 2 4
-40
-35
-30
-25
-20
-15
-10
-5
0
-4 -2 0 2 4
2sinc( )fT 2sinc( )fT 2sinc( )fT
1020log sinc( )fT
-T/2 T/2
A g(t)
)(sincsin2
)(2/
2/2/
2/
2
fTATfT
fTATfj
AdtAefGTt
TtT
T
ftj
==
−==
−=
+=+
−
−∫
ππ
ππ
ECSE413B ♦ THO LE-NGOC ♦ BAND-LIMITED SIGNALING SCHEMES IN AN AWGN ENVIRONMENT PAGE- 4
⇒ FOR BANDLIMITED TRANSMISSION, WE WANT G(f) with LIMITED
BANDWIDTH ⇒ g(t) is no longer time-limited
-B/2 +B/2
a G(f)
)(sincsin2
)(2/
2/2/
2/
2
BtaBBt
BtaBtj
adfaetgBt
BtB
B
ftj
==
==−=
+=+
−
+∫
ππ
ππ
ECSE413B ♦ THO LE-NGOC ♦ BAND-LIMITED SIGNALING SCHEMES IN AN AWGN ENVIRONMENT PAGE- 5
Tx AND Rx PARTITIONING:
Tx Pavg=E{|v(t)|2}= ∫∞+
∞−dffG
TI 22
)(σ⇒Tx Es=T Pavg= ∫
+∞
∞−dffGI
22 )(σ
Define h(t)= g(t)∗gR(t), at the Rx filter output:
[ ]
[ ]{noise G.
einteferenclIntersymbo
)(
sampleMain
)0()()(
)()()()()()()(
0
k
mm
mkkkn
n
Rn
nR
nmThIhInTnkhIkTy
tgtnnTthItgtntvty
++=+−=
⊗+−=⊗+=
∑∑
∑∞+
≠−∞=
−
∞
−∞=
∞
−∞=
44 344 21
321
nk: Gaussian noise with zero mean and variance: ∫∞+
∞−= dttgN
Ro
n22 )(
2σ
RECALL: for time-limited g(t), i.e., g(t)=0 for t∉[0,T], the Rx uses a matched filter: gR(t)=g(T-t), ⇒ Tj
RR efGfGtg ω−ℑ =⎯→⎯ )()()( * (THE MATCHED FILTER PROVIDES MAXIMUM output SNR) gR(t)=0 for t∉[0,T] ⇒h(t)= g(t)∗gR(t)=0 for t∉[-T,T] (or [0,2T]) ⇒ the intersymbol interference (ISI) term is 0, i.e., no ISI For bandlimited signaling, g(t)≠0 for t∉[0,T]. However, the ISI term can be eliminated when h(mT)=0 for all m≠0 ⇒ WE WANT BANDLIMITED G(f), GR(f)=G*(f)e-jωT AND H(f)= G(f)GR(f)
WITH h(mT)=0 for all m≠0
Symbol sequence {In}
v(t)
WGN n(t), N0/2
y(t) {y(kT)} detectionTx filter
G(f) Rx matched filter GR(f)
22
)( fGT
spectrumTx Iσ=
with μI=0
ECSE413B ♦ THO LE-NGOC ♦ BAND-LIMITED SIGNALING SCHEMES IN AN AWGN ENVIRONMENT PAGE- 6
DERIVATION: MATCHED FILTER & MAXIMUM output SNR: y(t)= v(t)∗gR(t)+n(t)∗gR(t) power of noise part n(t)∗gR(t) :
N= ∫ ∫∫∫∞+
∞−
−∞+
∞−
∞+
∞−
∞+
∞−== dtdfetgfGNdffGfGNdffGN ftj
RRo
RRo
Ro π2**2 )()(
2)()(
2)(
2
N= ∫∫∞+
∞−
∞+
∞−= dttgNdttgtgN
Ro
RRo 2* )(
2)()(
2
signal part: a(t)=v(t)∗gR(t)= ∑∞
−∞=
−n
n nTthI )( where h(t)= g(t)∗gR(t)
a(kT)= ∫∑ ∫∞+
∞−
+∞
−∞=
∞+
∞−−=−− ττττττ dggIdTnkggI Rk
nRn )()()]([)(
for time-limited g(t), i.e., g(t)=0 for t∉[0,T]
Signal power: E{a2(kT)}= { }222
2 ,)()( kIRI IEdttgtg =−∫+∞
∞−σσ
Output signal-to-noise power ratio: SNR=∫
∫∞+
∞−
+∞
∞−−
dttg
dttgtg
NR
R
o
I2
2
2
)(
)()(
2/σ
is maximum when gR(t)=g(T-t) or GR(f)=G*(f)e-jωT
note: The Cauchy-Schwarz inequality: ∫∫∫+∞
∞−
+∞
∞−
+∞
∞−≤ dttydttxdttytx 22
2* )()()()(
and the equality holds when y(t)=Cx(t), C: any constant
BANDLIMITED SIGNALING IN AWGN WITH ZERO ISI
⇒ zero ISI when: ⎩⎨⎧
≠=
=000
)( 0
mmh
mTh in time domain
⇒ Nyquist theorem: H(f) satisfies constant:T0-m
hTmfH =⎟⎠⎞
⎜⎝⎛ +∑
∞
∞=
in frequency domain
ECSE413B ♦ THO LE-NGOC ♦ BAND-LIMITED SIGNALING SCHEMES IN AN AWGN ENVIRONMENT PAGE- 7
DERIVATION:
∑∫
∑∫∑∫∫∞+
−∞=
+
−
+∞
−∞=
+
−
+∞
−∞=
+
−
∞+
∞−
+==
+===
k
T
T
mTj
k
T
T
mTj
k
Tk
Tk
mTfjmTfj
TkHZwithdeZ
deTkHdfefHdfefHmTh
)()(,)(
)()()()(
2/1
2/1
2
2/1
2/1
22/)12(
2/)12(
22
λλλλ
λλ
λπ
λπππ
Z(λ) is periodic function of λ with period 1/T, hence can be represented
in a Fourier series, i.e., ∑+∞
−∞=
=n
nTjnezZ λπλ 2)( with ∫−
−=T
T
nTjn deZTz
2/1
2/1
2)( λλ λπ
⇒ h(mT)=z-m/T, i.e.,
for ⎩⎨⎧
≠=
=000
)( 0
mmh
mTh we need Z(λ)= constant :0-k
ThTkH =⎟⎠⎞
⎜⎝⎛ +∑
∞
∞=
λ
Consider simple, ideal, strictly bandlimited ⎪⎩
⎪⎨
⎧
>
≤=
2for 0
2for
)(0
Bf
BfHfH
what is the narrowest bandwidth (B) such that h(mT)=0 for m≠0?
or H(f) satisfies constant -m
TTmfH =⎟⎠⎞
⎜⎝⎛ +∑
∞
∞=
⇒ time-domain: h(0)=aB, h(m/B)=0 frequency-domain:
⇒B=1/T: minimum BW for zero ISI ⇒ spectral efficiency: 1 symbol/s/Hz = m b/s/Hz for linear modulation M=2m
BUT THE IDEAL “BRICK-WALL” FILTER WITH RECTANGULAR FREQUENCY RESPONSE IS NOT PHYSICALLY REALIZABLE!
⎟⎠⎞
⎜⎝⎛ +
21 BTT
1⎟⎠⎞
⎜⎝⎛ −
21 BT
0 +B/2 -B/2
21
2B
TB
−= want
-B/2 +B/2
a H(f)
)(sincsin2
)(2/
2/2/
2/
2
BtaBBt
BtaBtj
adfaethBt
BtB
B
ftj
==
==−=
+=+
−
+∫
ππ
ππ
ECSE413B ♦ THO LE-NGOC ♦ BAND-LIMITED SIGNALING SCHEMES IN AN AWGN ENVIRONMENT PAGE- 8
raised-cosine filter TRANSFER FUNCTION:
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
+>
+≤≤
−
⎭⎬⎫
⎩⎨⎧
⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛ −
−+
−≤≤
=
Tf
Tf
TTfTT
TfT
fH
210
21
21
21cos1
2
)1(210
)(
α
ααααπ
α
or:
[ ]{ }⎪⎩
⎪⎨
⎧+≤≤−−−
−≤≤=
elseweherexx
xTxR
02/)1(2/)1(2/}2/)1{(cos
2/)1(0),( 2 ααααπ
αα
where x=(f-fc)T
IMPULSE RESPONSE:
⇒ )/(sinc21
cos),( 2 S
S
S Tt
Tt
Tt
tr πα
πα
α
⎟⎟⎠
⎞⎜⎜⎝
⎛−
⎟⎟⎠
⎞⎜⎜⎝
⎛
=
Responses of a raised-cosine filter to an input sequence of:
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
-10 -5 0 5 10
with linear phase response
T
( ), ( 2 ), ( 3 ), ( 8 )t t T t T t T∂ ∂ − ∂ − ∂ −
ECSE413B ♦ THO LE-NGOC ♦ BAND-LIMITED SIGNALING SCHEMES IN AN AWGN ENVIRONMENT PAGE- 9
SPECTRA OF Tx SIGNALS USING RRC FILTERS:
PEAK INSTANTANEOUS POWER RELATIVE TO POWER OF A QPSK SIGNAL USING TIME-LIMITED RECTANGULAR PULSE SHAPE
(from T. Noguchi, Y. Daido, J.A. Nossek, “Modulation Techniques for Microwave Digital Radio”, IEEE Communications Magazine, October 1986, pp. 21-30)
ECSE413B ♦ THO LE-NGOC ♦ BAND-LIMITED SIGNALING SCHEMES IN AN AWGN ENVIRONMENT PAGE- 10
EYE DIAGRAMS:
PAM 4PAM 2
Examples of eye patterns for
2-PAM 4PAM
ECSE413B ♦ THO LE-NGOC ♦ BAND-LIMITED SIGNALING SCHEMES IN AN AWGN ENVIRONMENT PAGE- 11
EYE DIAGRAMS FOR RAISED-COSINE FILTER:
RAISED-COSINE (RC) FILTER
ECSE413B ♦ THO LE-NGOC ♦ BAND-LIMITED SIGNALING SCHEMES IN AN AWGN ENVIRONMENT PAGE- 12
EYE DIAGRAMS FOR RAISED-COSINE FILTER:
ECSE413B ♦ THO LE-NGOC ♦ BAND-LIMITED SIGNALING SCHEMES IN AN AWGN ENVIRONMENT PAGE- 13
DESIGN OF Tx AND Rx FILTERS FOR BANDLIMITED TRANSMISSION IN AWGN CHANNELS WITH ZERO ISI
For maximum output SNR, select: GR(f)=G*(f)e-jωT For zero ISI in y(kT), select: H(f)=GR(f)G
*(f)=R(f,α): raised-cosine (RC) ⇒ optimum partition: linear-phase Tx and Rx filters with root raised-
cosine (RRC) amplitude response, i.e., G(f)=[ R(f,α)]1/2ej2πfτ and GR(f)=[ R(f,α)]1/2ej2πfd where τ and d: constant group delay Tx signal: ∑
∞
−∞=
−=n
n nTtgItv )()( , avg. Tx power: Pavg=T
dffRT
dffGT
III22
22
),()( σα
σσ== ∫∫
+∞
∞−
+∞
∞−
Avg energy per symbol: Es=TPavg= σI2.
Rx sample y(kT)=Ik+nk since h0=h(0)=1, Ik: signal sample nk: Gaussian noise sample with zero mean, variance:
2)(
22 o
Ro NdffGN
=∫∞+
∞−
{ }
)]([2
)]([2
)]([)(2
)()(2
)()()(2
)()()()(
TklNTklhNdttTklgtgNdxxlTgxkTgN
dxdyylTgxkTgyxNdxdyylTgxkTgywxwEnnE
ooRR
oRR
o
RRo
RRlk
−∂=−=−−=−−=
−−−∂=⎭⎬⎫
⎩⎨⎧
−−=
∫∫
∫ ∫∫ ∫∞+
∞−
∞+
∞−
+∞
∞−
+∞
∞−
+∞
∞−
+∞
∞−
Example: bandlimited M-ASK Ik∈ {A(2n-1-M), n=1,2,…,M} Es=TPavg= σI
2= A2(M2-1) /3 Rx sample y(kT)=Ik+nk Ik: signal sample with dmin=2A or [dmin]
2=12Es/( M2-1)
Performance: same as the time-limited case: ⎥⎦
⎤⎢⎣
⎡
−⎥⎦⎤
⎢⎣⎡ −=
02 1311
NE
Merfc
MP s
e
Symbol sequence {In}
v(t)
WGN n(t), N0/2
y(t) {y(kT)} detectionTx filter
G(f)Rx matched filter GR(f)
22
)( fGT
spectrumTx Iσ= with μI=0
ECSE413B ♦ THO LE-NGOC ♦ BAND-LIMITED SIGNALING SCHEMES IN AN AWGN ENVIRONMENT PAGE- 14
OPTIMUM PARTITION OF Tx/Rx FILTERS FOR BANDLIMITED TRANSMISSION IN AWGN CHANNELS WITH ZERO ISI:
CASE OF INPUT RECTANGULAR PULSES
GENERAL CASE: p(t)↔P(f): Fourier transform of p(t)
)(/),()()()(function transfer channeloverall
),()(filter
)(/),()(filter
xPxRxHxHxH
xRxHRcv
xPxRxHTx
RT
R
T
α
α
α
==
=
=
Signal generated using rectangular pulses:
∑+∞
−∞=
−=n
sn nTtpIts )()(
v(t)
WGN n(t), N0/2
y(t)
{y(kT)} detectionTx filter
G(f)Rx matched filter GR(f)
symbol sequence {In}
v(t)
WGN n(t), N0/2
y(t)
{y(kT)} detectionTx filter
HT(f)Rx matched filter HR(f)
)(sinc)(
,00,1
)(
2SS
s
fTCfW
elsewhereTt
tp
π=⎩⎨⎧ <≤
=
)(sinc/),()()()(function transfer channeloverall
),()(filter
)(sinc/),()(filter
xxRxHxHxH
xRxHRcv
xxRxHTx
RT
R
T
πα
α
πα
==
=
=
x=(f-fc)Ts, fc: carrier frequency Ts: symbol interval
H(x) is normalized for H(0)=1 in this plot
ECSE413B ♦ THO LE-NGOC ♦ BAND-LIMITED SIGNALING SCHEMES IN AN AWGN ENVIRONMENT PAGE- 15
EXAMPLE OF FILTERING REQUIREMENTS: INTELSAT V TDMA/QPSK Carrier frequency: fC=140 MHz Tx bit rate: fb=120.832 MHz Nyquist frequency: fN = fS/2=30.208 MHz Raised cosine filter with α=0.4
ECSE413B ♦ THO LE-NGOC ♦ BAND-LIMITED SIGNALING SCHEMES IN AN AWGN ENVIRONMENT PAGE- 16
EFFECTS OF AMPLITUDE AND PHASE DISTORTIONS ON RECEIVER PERFORMANCE (from T. Noguchi, Y. Daido, J.A. Nossek, “Modulation Techniques for Microwave Digital Radio”, IEEE Communications Magazine, October 1986, pp. 21-30)