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 =
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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 =
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Linear block codes contd
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
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Linear block codes contd
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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Linear block codes contd
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
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Linear block codes contd
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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Linear block codes contd
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 =
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Linear block codes contd
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 ),....,,(
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21
n
n
eee
rrr
=
=
e
r
eUr +=
TT eHrHS ==
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Linear block codes contd
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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Linear block codes contd
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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Linear block codes contd
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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Linear block codes contd
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
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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.
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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
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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
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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
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V d CRC
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Checking for errors
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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
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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
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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
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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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