170300504 Bai 9 Ma Hoa Kenh Compatibility Mode

BI 9: M HA KNHng L Khoang L Khoa

Email: [email protected]

Faculty of Electronics & Telecommunications, HCMUS1

Ni dung trnh by:

M ha knh ( Channel coding ) M ha khi (Block codes)

+ M lp (Repetition Code)+ Hamming codes+ Cyclic codes+ Cyclic codes

* Reed-Solomon codes M ha chp (Convolutional codes)

+ Encode+ Decode

iu ch m li (Trellis Coded Modulation)2

S khi DCS

FormatSourceencode

Channelencode

Pulsemodulate

Bandpassmodulate

Digital modulation

FormatSourcedecode

Channeldecode

Demod.SampleDetect

Channel

Digital modulation

Digital demodulation

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Channel coding l g?

Tn hiu truyn qua knh truyn s b nh hng bi nhiu, can nhiu, fading l tn hiu u thu b sai.

M ha knh: dng bo v d liu khng b sai bng cch thm vo cc bit d tha (redundancy).

tng m ha knh l gi mt chui bit c kh nng sa linng sa li

M ha knh khng lm gim li bit truyn m ch lm gim li bit d liu (bng tin)

C hai loi m ha knh c bn l: Block codes v Convolutional codes

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Cc loi m ha sa sai

M lp (Repetition Code) M khi tuyn tnh (Linear Block Code), e.g.

Hamming M vng (Cyclic Code), e.g. CRC BCH v RS Code BCH v RS Code M chp (Convolutional Code)

Truyn thng, gii m Viterbi M Turbo M LDPC

Coded Modulation TCM BICM 5

M lp

Recovered state

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Kim tra chn l (Parity Check)

Thm 1 bit xor cc bit c kt qu l 0 D liu truyn, sa li, khng th sa li

Kim tra hng v ct ng dng: ASCII, truyn d liu qua cng ni

tip7

M khi tuyn tnh (Linear block codes) Chui bit thng tin c chia thnh tng khi k bit. Mi khi c encode thnh tng khi ln hn c

n bit. Cc bit c m ha v gi trn knh truyn. Qu trnh gii m c thc hin pha thu.

Data blockChannelencoder Codeword

k bits n bits

rate Code

bits Redundant

n

kR

n-k

c =

8

Linear block codes contd

Khong cch Hamming gia hai vector U v V, l s cc phn t khc nhau.

Khong cc ti thiu ca m ha khi l)()( VUVU, = wd

)(min),(min wdd UUU == V d: Tnh khong cch Hamming ca C1: 101101

v C2 :001100Gii: V =>d12=W(100001)=2 => Ta c th gii m sa sai bng cch chn

codewords c dmin

)(min),(minmin iijiji wdd UUU ==

100001001100101101 =

9


Kh nng pht hin li c cho bi:

Kh nng sa li t ca m ha c nh ngha l s li ti a c th sa c trn 1 t m (codeword)

1min = de

m (codeword)

=

21mindt

10


Encoding trong b m ha khi (n,k)

mGU =1

2( , , , ) ( , , , )u u u m m m

=

VV

Cc hng ca G th c lp tuyn tnh.

21 2 1 2

1 2 1 1 2 2

( , , , ) ( , , , )

( , , , )

n k

k

n k k

u u u m m m

u u u m m m

=

= + + +

V

VV V V

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Example: Block code (6,3)

=

= 00

10

01

10

11

011

VV

G

000

100

010

100

110

010

000

100

010

Message vector Codeword

=

=

10

01

00

11

01

10

3

2

VVG

11

11

10

00

01

01 1

111

10

110

001

101

111

100

011

110

001

101

010110010

12


M ha khi (n,k) k phn t u tin (hoc cui cng) trong t m l cc

bit thng tin.

][ k=

= IPG

matrix )(matrixidentity

knkkk

k

k

=

=

PI

),...,,,,...,,(),...,,(bits message

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bitsparity

2121 kknnmmmpppuuu

==U

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i vi bt k m ha khi tuyn tnh, chng ta c mt ma trn . Cc hng ca ma trn ny trc giao vi ma trn :

nkn )(HG

0GH =T H c gi l ma trn kim tra parity v cc

hng ca chng c lp tuyn tnh. i vi m ha khi truyn tnh:

0GH =

][ Tkn PIH =

14


Format Channel encoding Modulation

ChanneldecodingFormat

DemodulationDetection

Data source

Data sink

U

r

m

m

channel

eUr +=

Kim tra c trng: S l c trng ca r, tng ng vi error pattern e.

or vectorpattern error ),....,,(or vector codeword received ),....,,(

21

21

n

n

eee

rrr

=

=

e

r

eUr +=

TT eHrHS ==

15


Mng tiu chun Hng c to thnh bng cch cng U

vi pattern

UUU zero

kni = 2,...,3,2ie

kknknkn

k

k

22222

22222

221

UeUee

UeUeeUUU

zero

codeword

coset

coset leaders

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Mng tiu chun v c trng bng gii m1. Tnh2. Tm coset chnh , tng ng vi .3. Tnh v tng ng vi .

TrHS =iee = S

erU += m

)( e(eUee)UerU ++=++=+= Ch :

Nu , error c sa. Nu , b gii m khng th pht hin li.

)( e(eUee)UerU ++=++=+=ee =

ee

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V d: Mng chun cho m (6,3)

000000 110100 011010 101110 101001 011101 110011 000111000001 110101 011011 101111 101000 011100 110010 000110000010 110110 011000 101100 101011 011111 110001 000101

codewords

000100 110000 011100 101010 101101 011010 110111 000110001000 111100010000 100100100000 010100010001 100101 010110

Coset leaders

coset

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110000100011000010101000001000000000

:computed is of syndrome Thereceived. is (001110)

ted. transmit(101110)

===

=

=

r

r

U

TT

Error pattern Syndrome

111010001100100000010010000001001000

(101110)(100000)(001110)estimated is vector corrected The

(100000)is syndrome this toingcorrespondpattern Error

(100)(001110)

=+=+=

=

===

erU

e

HrHS TT

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Hamming codes L trng hp ring ca linear block codes Din t theo hm ca mt s nguyn .

Hamming codes

2m

n m 12 :length Code =

t

mn-kmk

n

m

m

1 :capability correctionError :bitsparity ofNumber

12 :bitsn informatio ofNumber 12 :length Code

=

=

=

=

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Hamming codes

Example: Systematic Hamming code (7,4)

][101110011010101110001

33TPIH =

=

][

1000111010001100101010001110

44=

= IPG

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M ha Hamming

M ha: H(7,4) Nhiu php kim tra tngMessage=[a b c d]

r= (a+b+d) mod 2s= (a+b+c) mod 2

Message=[1 0 1 0]

r=(1+0+0) mod 2 =1

s=(1+0+1) mod 2=0

t=(0+1+0) =1t= (b+c+d) mod 2Code=[r s a t b c d]

Tc m: 4/7 Cng nh, nhiu redundance bit, c bo v tt hn. Khc bit gia pht hin v sa li

t=(0+1+0) mod 2 =1

Code=[ 1 0 1 1 0 1 0 ]

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Hamming codes

Example: Systematic Hamming code (7,4)

][101110011010101110001

33TPIH =

=

][

1000111010001100101010001110

44=

= IPG

23

V d m ha Hamming

H(7,4) Ma trn sinh G: u tin l ma trn n v 4x4 D liu truyn l vector p Vector truyn x (G=[I/P])

Vector nhn r v vector li e

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Sa li Nu khng c li, vector c trng (syndrome) z=zeros

Nu c 1 li v tr th 2

Vector c trng z l

tng vi ct th 2 ca H. Vy, li uc pht hin v tr th 2 v c th sa li cho ng.

25

li m ha (Coding Gain)

Tc m R=k/n, k: s symbol d liu, n tng symbol

SNR t v SNR ca bit

Vi mt s m ha, li m ha ti mt sc xut li bit c nh ngha l s khc bit gia nng lng cn thit cho 1 bit thng tin m ha t c sc xut li cho trc v truyn dn khng m ha

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V d li m ha

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Example of the block codes

BP8PSK

QPSK

[dB] / 0NEb28

Cyclic code

Cyclic codes c quan tm v quan trng v Da trn cu trc i s v c th ng dng rng

ri. D dng thc hin bng thanh ghi dch (shift register) c ng dng rng ri trong thc nghim

Trong thc nghim, cyclic codes c s dng pht hin li (Cyclic redundancy check, CRC) c s dng trong mng chuyn mch gi Khi c 1 li c pht hin b nhn, chng s c yu cu truyn li.

ARQ (Automatic Repeat-reQuest)

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Cyclic block codes

Mt m tuyn tnh (n,k) c gi l Cyclic code nu khi dch vng 1codeword th cng l codeword.

),...,,,( 1210 = nuuuuU i cyclic shifts of U

V d:

),...,,,,,...,,( 121011)( += inninini uuuuuuuU

UUUUUU

=====

=

)1101( )1011( )0111( )1110()1101(

)4()3()2()1(

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Cyclic block codes

Cu trc i s ca Cyclic codes, suy ra codewords c sinh ra t

Mi quan h gia codeword v thanh ghi dch:

)1( degree ...)( 112210 n-XuXuXuuX nn ++++=U

dch:

Vy:

)1()(

...

...,)(

1)1(

)1(11

)(

12

2101

11

22

10

1)1(

++=

++++++=

+++=

+

n

n

Xu

n

n

n

X

n

nn

n

n

n

n

XuX

uXuXuXuXuu

XuXuXuXuXX

nn

U

U

U

)1( modulo )()()( += nii XXXX UUBy extension)1( modulo )()()1( += nXXXX UU

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Cyclic block codes

Thut ton m ha Cyclic code (n,k):1. Nhn thng tin vi chui bng2. Chia kt qu bc 1 vi a thc sinh .

Ly l phn d3. Thm vo to thnh

)(Xm knX )(Xg

)(Xp)(Xp )(XX kn m3. Thm vo to thnh

codeword )(Xp

)(XU

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Cyclic block codes

Example: For the systematic (7,4) Cyclic code with generator polynomial

1. Find the codeword for the message

)1()()(1)()1011(3 ,4 ,7

6533233

32

++=++==

++==

===

mm

mm

XXXXXXXXXXXXX

knkn

kn

)1011(=m31)( XXX ++=g

)1 1 0 1 0 0 1(1)()()(

:polynomial codeword theForm

1)1()1(:(by )( Divide

)1()()(

bits messagebitsparity

6533

)(remainder generator

3

quotient

32653

6533233

=

+++=+=

++++++=++

++=++==

UmpU

gmmm

pgq

XXXXXXX

XXXXXXXXX)XX

XXXXXXXXXX

X(X)(X)

kn

kn

33

Cyclic block codes

Find the generator and parity check matrices, G and H, respectively.

=

=+++=

010110000101100001011

)1101(),,,(1011)( 321032

G

g ggggXXXX

Not in systematic form.We do the following:

row(3)row(3)row(1) +

1011000 row(4)row(4)row(2)row(1)

row(3)row(3)row(1)++

+

=

1000101010011100101100001011

G

=

111010001110101101001

H

44I33I TPP

34

Cyclic block codes

Gii m Cyclic code: T m b thu c cho bi

c trng phn d c c bng cch chia chui nhn cho a thc sinh:

)()()( XXX eUr +=Received codewor

d

Error pattern

cho a thc sinh:

Vi c trng v mng tiu chun, li s c c lng.

)()()()( XXXX Sgqr += Syndrome

35

V d CRC

36

Checking for errors

37

Cch tnh CRCB1: Nhn M vi 2r (r l s bit CRC),

r = chiu di G 1B2: Chia M.2r cho G. Ly phn d: D ( r bit)B3: Ghp M vi D [M|D]B3: Ghp M vi D [M|D]

Kim traChia d liu nhn cho G Nu phn d=0 => khng c li Nu phn d 0 => c li

38

Kh nng ca CRC Mt li E(X) khng th pht hin khi chng chia

ht cho G(x). Ngc li, th c th pht hin li. C kh nng mnh m trong pht hin li

39

BCH Code

Bose, Ray-Chaudhuri, Hocquenghem C kh nng sa c nhiu li D dng thc hin m ha v gii m

Cc chun trong cng nghip- (511, 493) m ha BCH trong ITU-T. Chun - (511, 493) m ha BCH trong ITU-T. Chun H.261- mt chun m ha video c s dng cho video conferencing v video phone. (40, 32) m ha BCH trong ATM (Asynchronous

Transfer Mode)

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BCH Performance

41

Reed-Solomon Codes

Mt trng hp ring ca non-binary BCH c ng dng rng ri

Storage devices (tape, CD, DVD) Wireless or mobile communication Satellite communication Satellite communication Digital television/Digital Video Broadcast(DVB) High-speed modems (ADSL, xDSL)

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Documents

170300504 Bai 9 Ma Hoa Kenh Compatibility Mode