victor tomashevich-convolutional codes

8/4/2019 Victor Tomashevich-Convolutional Codes

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Name

Convolutional codes

Tomashevich Victor


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Introduction

Convolutional codes map information to code bits sequentially byconvolving a sequence of information bits with generator sequences

A convolutional encoder encodes Kinformation bits to N>Kcode bitsat one time step

Convolutional codes can be regarded as block codes for which theencoder has a certain structure such that we can express theencoding operation as convolution


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Properties of convolutional codes

Consider a convolutional encoder. Input to the encoder is a

information bit sequence u (partitioned into blocks of length K):),,( 10 uuu

),,,()()2()1( K

iiii uuuu

The encoder output the code bit sequence

x(partitioned into blocks

),,( 10 xxx

),,,()()2()1( N

iiii xxxx

N

KR

of length N)


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Example: Consider a rate convolutional code with K=1 and N=2defined by the circuit:

iu

ix

The sequences ),,(

)1(

1

)1(

0

xx , ),,(

)2(

1

)2(

0

xx are generated as follows:

ii ux )1(

1

)2(

iii uuxand

Multiplexing between)1(

ix and)2(

ix gives the code bit sequence

),,()),(),(( 10)2(

1

)1(

1

)2(

0

)1(

0 xxxxxxx


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The convolutional code is linear

The encoding mapping is bijective

Code bits generated at time step iare affected by information bits upto Mtime steps i 1, i2, , i Mback in time. Mis the maximaldelay of information bits in the encoder

Code memory is the (minimal) number of registers to construct anencoding circuit for the code.

Constraint length is the overall number of information bits affectingcode bits generated at time step i: =code memory + K=MK + K=(M +1)K

A convolutional code is systematic if the Ncode bits generated attime step icontain the Kinformation bits


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Example: The rate code defined by the circuit

iuix

has delay M=1, memory 1, constraint length 2, and it is systematic

Example: the rate 2/3 code defined by the circuit

)1(iu

)2(iu

)1(ix

)2(ix

)3(

ix

has delay M=1, memory 2, constraint length 4, and not systematic


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Tree

A

A

A

B

B

A

0u

1u

2u

1

0

00

01

00

00

01

11

00

11

01

10

10

11

10

11


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Trellis

The tree graph can be contracted to a direct graph called trellis

of the convolutional code having at most Snodes at distancei=0,1, to the root

The contents of the (at most) MKencoder registers are assignedthe variables 1,,1,0),2()( KMjGFs ji

The vector ),,()1()1()0( KMiiii ssss

combibing all register contents at time step i is called state ofthe encoder at time step i.

The code bit blockix is clearly a function of is and iu , only


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Example:

The encoder of the rate convolutional code has 221 Sdifferent states. The state is given by

iss The code bit block

ix at time step i is computed from is and iu by

ii ux )1(

andiii sux

)2(

0iu

1i

u

A AAA

B BB

11 1111

00 0000

0101

1010


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Example: Constructing a trellis section

iuix

ii ux )1(

andiii sux

)2(

Two equations are required:

(1) How doesi

s depend on miu and possibly 0,

msmi

1 ii us(2) How does ix depend on is and iu

ii ux )1(

andiii

sux )2(

The branches are labeled with ii xu | called state transitionleading from a state

is to a new state 1is


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Trellissection:

0 0

1 1

0|00

0|01 1|11

1|10

00000

11111

0|000|000|00

1|101|101|10

0|010|010|01 1|111|111|11

0s 1s 2s 2Ls 1Ls


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00000

1111

0|000|000|00

1|101|10

0|010|01 1|111|111|11

0s 1s 2s 2Ls 1Ls

State diagram


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State diagramExample: Trellis of the rate convolutional code

iiux

)1(and iii sux

)2(

0 0

1 1

00

01 11

10

0i

u

1i

u

is

1is

State diagram:

0

1

00

0111

10

0i

u

1i

u

D i ti ith b t i


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Description with submatrices

Definition: A convolutional code is a set Cof code bit sequences

),,,,,( 10 ixxxx

),,,()()2()1( N

iiii xxxx )2()(

GFxj

i

partitioned into lenth Nblocks

There exist many encoders mapping information bit sequences

),,( 10 uuu

),,,()()2()1( K

iiii uuuu )2()(

GFuj

i

(partitioned into length K


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Example: the following two encoding curcuits generate thesame set of code word sequences

)1(

iu

)2(

iu

)1(

ix

)2(

ix )3(ix

)1(

ix

)2(

ix )3(ix

)1(

iu

)2(

iu

Generator matrix


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Generatormatrix

,Gux where

M

M

M

GGGG

GGGG

GGGG

G

210

210

210

iGuxM

m

mmii

,0

The generated convolutional code has rateR=K/N, memoryK*M, and constraint lengthK*(M+1)

E l


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Example:

The rate code is given by

ii ux )1(

iii sux )2(

and

0G governs how iu affects 11:)( 0)2()1( Gxxx iii

1G governs how 1iu affects 10: 1 Gx i

,),,())(),(),(( 210)2(

2

)1(

2

)2(

1

)1(

1

)2(

0

)1(

0 Guuuxxxxxx

where

110111

0111

G

Description with polynomials


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Description with polynomials

)()()(

)()()(

)()()(

)(

)()2()1(

)(

2

)2(

2

)1(

2

)(

1

)2(

1

)1(

1

DgDgDg

DgDgDg

DgDgDg

DG

NKKK

N

N

)2(,)( )(,)(

,

2)(

2,

1)(

1,

)(

0,

)( GFgDgDgDggDg jmiMj

Mi

j

i

j

i

j

i

j

i

))(),(),(()( )()2()1( DuDuDuDu K

where

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

0

)(KjDuDuuDu

ij

i

jjj

))(,),(),(()( )()2()1( DxDxDxDx N

where

NjDxDxxDx ijijjj ,,2,1,)( )()(1

)(

0

)(

)()()( DGDuDx

,,,0,,,1,,,1),,()(, MmNjKijiGg mjmi

NjKiDgMj

iji

,,1,,,1)),(deg(max )(,

Example:


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Example:

The rate code is given by

ii ux )1(

iii sux )2(

and

)(

)1(

1 Dgand

)(

)2(

1 Dg ))()(()(

)2(

1

)1(

1 DgDgDG

From M=1 follows that 1)))(deg(( )( Dg jiThe polynomial )()1(1 Dg governs how ,1,0, mu ml affects

101)(:

)1(

1

)1( DDgxl

The polynomial )()2(1 Dg governs how ,1,0, mu ml affects

111)(: )2(1)2( DDDgxl

DDuu 1)(),0,1,1( ,Gux yielding ),00,01,10,11( x

)()()( DGDuDx yielding211)( DDDx

Punctured convolutional codes


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Punctured convolutional codes

A sequence of code bits is punctured by deleting some of thebits in the sequence according to a fixed rule

In general, the the puncturing of a rate K/Nconvolutionalcode is defined using Npuncturing tables, one for any codebit ,,,1,)( Njx ji in a block ix

Each table contains pbits, where p is the puncturing period. If

a bit is 1, the corresponding code bit is part of the puncturedcode, if the bit is 0, the corresponding code bit is not part of thepunctured code

For a sequence of code bit blocks ,,1,0, ix i the puncturingtables are applied periodically. Npuncturing tables are combinedin a pN puncturing matrixP

Example:


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Example:

The encoder circuit of rate convolutional code given by

22 11)( DDDDG

)0,0,1,0,0(u )11,01,11,00,00(NPxThe sequence NPx is punctured using two different puncturingmatrices:

,

1001

01111

P

1011

01112P

The puncturing period p is 4. Using1P , 3 out of 4 code bits

)1(

ixand 2 out of 4 code bits

)2(

ix of the mother code bits are used, theothers are discarded

5/4)23/()44(2/1 R

and u is encoded to )11,1,1,0,00()11,1,1,0,00( XXXx

Using ,2P the rate of the punctured code is


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Using ,2P the rate of the punctured code is3/2)33/()44(2/1 R and u is encoded to

)11,1,1,00,00()11,1,1,00,00( XXx

)1(

ix

)2(

ix

1 1 1 0

1 0 0 1

1 1 1 0

1 1 0 11P

2P

Puncturing tables

Pucturing period p=41 2 3 4

10|

1|

1|

0|

00

)2(

4

)1(

4

)2(

3

)1(

2

)1(

1

)2(

0

)1(

0 xxxxxxx

00100

43210 uuuuu

Encoder of a rate code punctured to a rate 4/5 (top puncturingtables) or a rate 2/3 code (bottom puncturing tables)


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The rate Rof a punctured code obtained from a rate NKR /0 mother code using the pN puncturing matrix P is given as

Pinof

pK

Pinof

pN

RR 1#1#0

With puncturing we can easily construct convolutional codeswith arbitrary rational rate. However, punctured codes of rateR=K/Nobtained from an optimized good mother code ofmemory musually perform worse than unpunctured rate K/N,memory moptimized codes given by a NK generator matrix

)(DG

This performance gap increases with the number of punctured

bits. The advantage of puncturing is that the decodingcomplexity is not altered, since the original trellis of the mothercode can be used

Consider a rate 1/3 memory 4 mother code given by the submatrices


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Consider a rate 1/3, memory 4 mother code given by the submatrices

)101(),011(),010(),111( 3210 GGGG and

),111(4 G

code# rate punc. tablefd fdc

91/38/24

8

7

6

5

4/11

8/124/108/20

4/98/18

1/28/16

11

9

8

7

7

8

10

2

2

32

1111 11111111 11111111 1111

1111 11111111 11111110 11101111 11111111 11111100 1100

1111 11111111 1111

1000 10001111 11111111 11110000 0000

code# ratepunc. table fd fdc

44/78/14

3

2

1

2/38/12

4/58/10

8/9

8/9

5

4

3

2

8

4

42

2

1111 11111110 11100000 0000

1111 1111

1010 10100000 00001111 11111000 10000000 0000

1111 01111000 1000

0000 0000

Decoding of convolutional codes


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ecod g o co o ut o a codesThe Viterbi algorithm

jujx1

jx2

js1 js2

)1,1,1,1,1,1,1( u

)1,1(),( 2010 ss

)11,11,11,11,11,11,11( x

1/ 1 1


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+1+1 +1+1

-1-1

-1+1

+1-1 +1-1

-1+1

-1-1

jj ss 21 , jj xx 21 , 1211 , jj ss

+1/+1+1

-1/-1-1

+1/+1+1

-1/-1+1

-1/-1+1

+1/+1-1

+1/+1-1

-1/-1-1

Hard decisions


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Hard decisions

y

j

m

jj

m

j

m

j yxyx 2)(

21

)(

1

)(

+1+1 +1+1+1+1+1+1+1+1+1+1-1+1

+1+1

0 -2

+2

0

0

-2

+2

0

-2

0

0

0

+2

-2

+2

j=0 j=6j=5j=4j=3j=2j=1 j=7

+2

-4

+8+6+4

+12+10+8+6

+20+2

+4+2

-2

0 +6

+6

+4+20+2

+4+2+4+2

(+1+1,-1+1,+1+1,+1+1,+1+1,+1+1,+1+1)


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+1+1 +1+1+1+1+1+1+1+1+1+1-1+1

+1+1

0 -2

+2

0

0

-2

+2

0

-2

0

0

0

+2

-2

+2

j=0 j=6j=5j=4j=3j=2j=1 j=7

+2

-4

+8+6+4

+12+10+8+6

+20+2

+4+2

-2

0 +6

+6

+4+20+2

+4+2+4+2

ju +1 +1 +1 +1 +1 +1 +1 - No error

Hard decisions


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y

j

m

jj

m

j

m

jyxyx

2

)(

21

)(

1

)(

+1+1 +1+1+1+1+1+1+1+1+1-1-1+1

+1+1

0 -2

+2

0

0

0

0

-2

+2

+2

0

-2

+2

-2

+2

-2

+2

0

0

00

-2+20

j=0 j=6j=5j=4j=3j=2j=1 j=7

+2

-4

+6+4+2

+10+8+6+4

+4+2+2

+2+2

-2

0 +4

+4

+2+4+20

+6+4+2+4

(+1+1,-1+1,+1-1,+1+1,+1+1,+1+1,+1+1)


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+1+1 +1+1+1+1+1+1+1+1+1-1-1+1

+1+1

0 -2

+2

0

0

0

0

-2

+2

+2

0

-2

+2

-2

+2

-2

+2+2

0

0

00

-2+20

j=0 j=6j=5j=4j=3j=2j=1 j=7

+2

-4

+6

+10+8+6+4

+4+2+2

+2+2

-2

0 +4

+4

+20

+2+4

ju +1 +1 +1 +1 +1 +1 +1 - No error

Hard decisions


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y

j

m

jj

m

j

m

j yxyx 2)(

21

)(

1

)(

+1+1 +1+1+1+1+1+1+1+1-1+1-1-1

+1+1

0 -2

+2

0

0

+2

-2

-2

+2

-2

0

+2

-2

+2

0

0

0

0

00

-2+20

j=0 j=6j=5j=4j=3j=2j=1 j=7

+2

-2

+8+6+8

+12+10+8+2

+20+4

00

-2

-2 +6

+10

+8+4+6

+8+6+8+2

(+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+1-1-1


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+1+1 +1+1+1+1+1+1+1+11+11 1

+1+1

0

+2

-2

+2

0

0

+2

-2

-2

+2

-2

0

+2

-2

+2

0

0

+2

-2

0

0

00

0

-2-2+20 +2

j=0 j=6j=5j=4j=3j=2j=1 j=7

+2

-2

+8+6

+12+10+8+2

+20+4

00

-2

-2 +6

+10

+8+4+6

+8+8+2

ju +1 -1 -1 +1 +1 +1 +1- 2 decoding errors

S ft d i i


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Soft decisions

,,2

,,2/1,,2/1

,2

)(

)(

)(

,

)(

ij

m

ij

ij

m

ij

ij

m

ij

ij

m

ij

ij

yx

yxyx

yx

l

GOOD channel

BAD channel

GOOD channel

BAD channel

jjjjjjmj ylxylx 222111)(

CSI values ((G,B),(B,B),(G,G),(G,B),(B,B),(G,G),(G,G))

y (+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+1-1-1


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+1+1

0

+1

-2.5

+2.5

-4

+4

-1

+1

-2.5

+2.5

0

-4

0

0

0

0

0

-4

+4

-4

+4

-1

0

0

0

0

0

0

0

0

00

00

00

00

-4-4-1-2.5+1

+2.5+4 +4 +4

j=0 j=6j=5j=4j=3j=2j=1 j=7

+2.5

-2.5

+10+7+9

+18+14+11+10

+5-0.5+1.5

+7.5+3.5

-2.5

-2.5 +7

+13

+9+5-0.5+1.5

+9+5+4+1.5

G B G GG GB BG G G BB B

+2 +0.5 +2 +2+2 +2+0.5 +0.5+2 +0.5-2 +2-0.5-0.5

+1+1 +1+1+1+1+1+1+1+1-1+1-1-1


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+1+1

0

+1

-2.5

+2.5

-4

+4

-1

+1

-2.5

+2.5

0

-4

0

0

0

0

0

+4+4

0

0

00

-2.5

+2.5+4

j=0 j=6j=5j=4j=3j=2j=1 j=7

+2.5

-2.5

+10

+18+14+11+10

-0.5+1.5

+7.5+3.5

-2.5

-2.5 +7

+13

+1.5

+1.5

G B G GG GB BG G G BB B

+2 +0.5 +2 +2+2 +2+0.5 +0.5+2 +0.5-2 +2-0.5-0.5

ju +1 +1 +1 +1 +1 +1 +1 - No error

victor tomashevich-convolutional codes

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