conv codes

7/31/2019 Conv Codes

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Channel Coding 1

Dr.-Ing. Dirk Wbben

Institute for Telecommunications and High-Frequency TechniquesDepartment of Communications Engineering

Room: N2300, Phone: 0421/218-62385

[email protected]

www.ant.uni-bremen.de/courses/cc1/

Lecture

Tuesday, 08:30 10:00 in S1270

Exercise

Wednesday, 14:00 16:00 in N2420Dates for exercises will be announced

during lectures.

Tutor

Carsten Bockelmann

Room: N2370

Phone [email protected]


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2

Outline Channel Coding I

1. Introduction

Declarations and definitions, general principle of channel coding

Structure of digital communication systems

2. Introduction to Information Theory Probabilities, measure of information

SHANNONs channel capacity for different channels

3. Linear Block Codes

Properties of block codes and general decoding principles

Bounds on error rate performance

Representation of block codes with generator and parity check matrices

Cyclic block codes (CRC-Code, Reed-Solomon and BCH codes)

4. Convolutional Codes

Structure, algebraic and graphical presentation

Distance properties and error rate performance

Optimal decoding with Viterbi algorithm


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Convolutional Codes

Basics

Implementation of encoder and algebraic description

Graphical presentation in finite state diagram and trellis diagram

Classification of convolutional codes

Non-recursive and recursive convolutional encoders

Catastrophic convolutional codes

Truncated, terminated and tailbiting convolutional codes Optimal decoding

MAP and ML criterion

Viterbi algorithm

Puncturing of convolutional codes

Distance properties of convolutional codes

Error rate performance


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Basics of Convolutional Codes (Faltungscodes)

Shift register(Schieberegister)structure withLckmemory elements memory leads to statistical dependence of successive code words

In each cycle kbits are shifted each bit affects the output wordLc times

Lc is called constraint length(Einflusslnge), memory depthm= Lc-1 Coded symbols are calculated by modulo 2 additions of memory contents

generators Code word contains n bit code rateRc = k/n

Our further investigation is restricted to codes with rate

Rc = 1/

n

, i.e.k=

1!

u21 k 21 k 21 k

2 1n n-1x

Lc


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Example: (n, k,Lc)=(2,1,3)-convolutional code with generators g1= 78 and g2= 58Encoder is non-systematic and non-recursive (NSC-Code)

Rc = 1/2, Lc = 3 m = 2 Information word u = [1 0 0 1 1 | 0 0] tail bits to finish in state 0

Structure and Encoding

g10 =1

g20 =1

g12 =1

g22 =1

g11 =1

g21 =0

u(l) state following state output

1 00 10 11

0 10 01 10

0 01 00 11

1 00 10 11

1 10 11 01

0 11 01 01

0 01 00 11

1

1

1

0

1

1

1

1

0

1

0

1

1

1

u(l-1) u(l-2)u(l)

0 00011001

1 0001100

0 100110

0 00011

1 0001

1 100

0 10

0 0


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Equivalence of block codes and convolutional codes

Convolutional codes:

kinformation bits are mapped onto code word x including n bits

Code words x are interdependent due to the memory

Block codes:

Code words x are independent

Block codes are convolutional codes without memory

Only convolutional encoded sequences of finite length are viewed in practice

Finite convolutionally encoded sequence can be viewed as single code wordgenerated by a block code

Convolutional codes are a special case of block codes


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Properties of Convolutional Codes

Only a small number of simple convolutional codes are of practical interest

Convolutional codes are not constructed by algebraic methods but by computer

search(with the advantage of a simple mathematical description)

Convolutional decoders can easily process soft-decision input and compute

soft-decision output(only hard-decision decoding has been considered for block codes)

Similar to block codes, systematic and non-systematic encoders are

distinguished for convolutional codes

(mostly non-systematic convolutional encoders are of practical interest)


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Algebraic Description (1)

Description by generators (Generatoren)gj (octal)

Example code withLc=3 andRc = 1/2

Encoding by discrete convolution in GF(2)

generally for=1,,n

1 1,0 1,1 1,2 8

2 2,0 2,1 2,2 8

1 1 1 7

1 0 1 5

g g g

g g g

g

g

,0

mod2m

i i

i

x g u

1 1

2 2

x u g

x u g

Octal description: Left-MSB (not always in literature)

ForLc3 append zeros from the left Example: g = [1 0 0 1 1] 010 011 238Problem: In literature you will sometimes find Right-MSB

Sometimes zeros are appended from the right

g10 =1

g20 =1

g12 =1

g22 =1

g11 =1

g21 =0

u()u(l-2)u(l-1)

x1()

x2()


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z-Transform (delay:z-1) D-Transform (delay:D)

Generator polynomials

Encoding (polynomial multiplication)

Encoded sequence

with generator matrix

Code space

0

i

i

i

z x zX

0

i

i

iDX D x

2 2

1 1,0 1,1 1,2

2 2

2 2,0 2,1 2,2

1

1

G D g g D g D D D

G D g g D g D D

,

0

mi

i

i

G D g D

X D U D G D

1 2( ) ( ) ( )nD X D X D X D U D D X G

Example

1 2( ) ( ) ( )nD G D G D G DG

, GF(2)iU D D U D U G

for=1,,n


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Example: u = [1 0 0 1 1]

Generator polynomials

Encoding

1 2

1 2

3 4 2 3 4 2

2 3 6 2 3 4 5 6

( ) ( )

1 1 1 1

1 1

11 10 11 11 01 01 11

D X D X D

U D G D U D G D

D D D D D D D

D D D D D D D D D

X

3 41U D D D

2 21 21 1G D D D G D D u(-1) u(-2)u()

x1()

x2()


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Interpretation as a Block Code

Encoding can (alternatively) be described by matrix multiplication

The input sequence is not necessarily finite, thus the generator matrix G of

convolutional codes is semi-infinite (here forn = 2)

Example from previous slide

x u G1,0 2,0 1,1 2,1 1,2 2,2 1, 2,

1,0 2,0 1,1 2,1 1, 1 2, 1 1, 2,

1,0 2,0 1, 2 2, 2 1, 1 2, 1 1, 2,

m m

m m m m

m m m m m m

g g g g g g g g

g g g g g g g g

g g g g g g g g

G

11 10 11

11 10 111 0 0 1 1 11 10 11 11 01 01 1111 10 11

11 10 11

11 10 11

x

G is a convolution matrix Toeplitz structure


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Convolutional encoder can be interpreted as Mealy state machine Output signal depends on current state and current input signal x() =fx( u( ), S( ) )

Next state S(+1) depends on current state and current input S(+1) =fS( u( ), S( ) )

Description by state transition diagram (Zustandsdiagramm) with 2m states

Example: (2,1,3)-NSC-code with generators g1 = 78 and g2 = 58

Graphical Presentation in Finite State Diagram

0/00

1/11

0/10

1/00

1/01

1/10

0/01

0/110 0

1 0 0 1

1 1

g10 =1

g20 =1

g12 =1

g22 =1

g11 =1

g21 =0

u()

u(

-2)u(

-1)

x1()

x2()

u/[x1x2]


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Finite state diagram does not contain any information in time

State diagram expanded by temporal component results in Trellis diagram Trellis starts in S0 and is fully developed afterLc times


Graphical Presentation in the Trellis Diagram

0 0

1 0 0 1

1 1

0/00

1/11

0/10

1/00

1/01

1/10

0/01

0/11

00

10

01

11

0/00

0/10

1/01

1/00

0/01

0/111/11

1/10=0 =4 =3 =2 =1


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Classification of Convolutional Codes

Non-recursive, non-systematic Convolutional Encoders (NSC-Encoders)

Non-systematic encoders

No separation between information bits and parity bits within the code word

Higher performance than systematic encoders

Usually applied in practice

Systematic Convolutional Encoders Code word explicitly contains information bits

Not relevant for praxis due to lower performance

Exception: recursive systematic convolutional encoders for Turbo-Codes and

Trellis Coded Modulation (TCM) Channel Coding II

Recursive Systematic Convolutional Encoders (RSC-Encoders)


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Recursive Convolutional Encoders (RSC-Encoders) (1)

Consecutive state depends on

current state

encoder input

and feedback structure of the encoder

Recursive encoders of practical interest are mostly systematic and can be

derived from NSC codes

Beginning with a NSC encoder the generator polynomials are converted to get

a systematic but recursive encoder

Generator polynomials of RSC encoder derived from NSC generator polynomial

11

222

1

( ) ( ) 1

( )( ) ( )

( )

G D G D

G DG D G D

G D


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Output of the systematic RSC encoder is given by

with

Using the delay operatorD and g1,0=1 it follows

a() can be regarded as the current content of the register and depends on the

current input u() and the old register contents a(-i)

1

( )

( ) ( )

U D

A D G D 1,0( ) ( )

mi

i

iD g D U D

1,

1( ) ( ) ( )

m

i

i

a g a i u

1,1( ) ( ) ( )m

i

i

a u g a i

11

22 2 2

1

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) (( ) ( ))( )

X D U D G D U D

X D U D G D GU D A D G DG

DD

NSC and RSC generatesame code space X(D) =U(D)G(D) is CW of NSC

same X(D) is generated by

RSC for info word U(D)G1(D)

IIR:

infinite impulse response for finite output weight

wH(u) 2 required


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Example: (2,1,3)-RSC encoder with generators g1 = 78 (recursive) and g2 = 58

Block diagram and state diagram

1

22 2

( ) ( )

( )( ) ( ) (1 ) with ( ) : ( ) ( ) ( 1) ( 2)1

X D U D

U DX D A D D A D a u a aD D

0 0

1 0 0 1

1 1

0/00

1/11

1/10

0/00

0/01

1/10

0/01

1/11

g20 =1

g12 =1

g22 =1

g11 =1

g21 =0

a(-

1)

u()

1x

2x

a(

-2)

a()

u/[x1

x2

]


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Now other polynomial used for feedback

Example: (2,1,3)-RSC encoder with generators g1 = 78 and g2 = 58 (recursive)

g10 =1

g22 =1

g12 =1

g21 =0

g11 =1

a(-1)u()

1x

2x

a(-2)

1

2

2 2

( ) ( )( )

( ) ( ) (1 ) with ( ) : ( ) ( ) ( 2)1

X D U DU D

X D A D D D A D a u aD

0 0

1 0 0 1

1 1

0/00

1/11

0/01

0/00

1/10

0/01

1/10

1/11

a()

u/[x1x2]


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Catastrophic Convolutional Encoder (1)

Catastrophic convolutional encoder can produce sequences of infinite length

with finite weight that do not return to the all-zero-path

Finite number of transmission errors can lead to infinite decoding errors

Example: (2,1,3)-NSC encoder with generators g1 = 58 and g2 = 68 u = [1 1 1 1 1 ] x = [11 10 00 00 00 ] wH(u) = but wH(x)=3

Fory = [00 00 00 00 00 ] ML-decoder decides for all-zero sequence inf. errors

0 0

1 0 0 1

1 1

0/00

1/11

0/01

1/01

1/10

1/00

0/11

0/10

g10 =1

g20 =1

g12 =1

g22 =0

g11 =0

g21 =1

u(-1) u(-2)

u()


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Catastrophic Convolutional Encoder (2)

A convolutional encoder is catastrophic if there exists an information sequenceU(D) such that wH(U(D))= and wH(X(D)) <

Encoder is noncatastrophic if generator polynomials contain no common factor

or

D

corresponds to simple delay (no delay corresponds to 1)

Properties:

The state diagram of a catastrophic encoder contains a circuit in which a nonzeroinput sequence corresponds to an all-zero output sequence

state diagram contains weight-zero loop about the all-one state Systematic encoder are always noncatastrophic

Encoder is noncatastrophic if at least one output stream is formed by thesummation of an odd number of taps

1 2GCD ( ), ( ), , ( ) with integer 0ng D g D g D D l

1 2GCD ( ), ( ), , ( ) 1ng D g D g D


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Truncated Convolutional Codes

Only sequences offinite length are viewed in practice

For information sequence u with arbitrary tail (Ende), the trellis can end in any

state

last state is not known by decoder last bits are decoded with lower reliability worse performance

Interpretation as block code: Description by generator matrix

0 1

0 1

0 1

0 1

0

m

m

m

G G GG G G

G G G G

G G

G

1, 2, ,i i i n ig g g G with

G is a truncated (abgeschnitten) convolutionmatrix

dimension: Kx Kn


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Terminated Convolutional Codes

Appending tail bits to information sequence u

encoder stops in a predefined state (usually state 0) reliable decoding of the last information bits

Number of tail bits equals memory of encoder, tail bits depend on encoder

NSC: adding m times zero

RSC: adding m tail bits, value depends on last state encoded by information bits

Adding tail bits reduces the code rateRc Sequence u with Kinformation bits

and 1/n convolutional code:

Generator matrix:

0 1

0 1

0 1

m

m

m

G G G

G G GG

G G G

Tail

c c

K KR R

n K m K m

G is a convolution matrix

Toeplitz structure

dimension: Kx (K+m)n


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Tailbiting Convolutional Codes

For small sequence lengthNthe addition of tail bits significantly reduces the

code rate

Tailbiting Convolutional Codes: last state corresponds to first state

No tail bits are required

State machine does not start in state 0 decoder more complex

NSC: initialize encoder with last m bits ofu

Generator matrix:

0 1

0 1

0 1

0 1

1 0

m

m

m

m

m

G G GG G G

G G G G

G

G G

G G G

G is a circular

convolution matrix

dimension: Kx Kn


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Optimal Decoding (1)

Information sequence u contains Kinformation bits and is encoded into thecode sequence consisting ofNsymbols x() (each symbol contains n bits)

The received sequence is y, is the estimated coded sequence and a

denotes an arbitrary coded sequence

MAP criterion (Maximum A-posteriori Probability)

Optimum decoder that calculates the sequence which maximizes

1 10 0 1 1n nx x x N x N x

0x 1Nx

x

Pr Pr

Pr Pr

Pr Pr

p pp p

p p

x y a y

x ay x y a

y y

y x x y a a arg max Pr p a

x y a a

x Pr x yA-Posteriori Probability (APP)

Pr{a}: a-priori information

about source


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Maximum Likelihood criterion (ML)

If all sequences are equally likely or if the receiver does not know

the statistics Pr{a} of the source, no a-priori information can be used

Decision criteria becomes

For equally likely input sequences the MAP criteria and ML criteria yield the

identical (optimal) result

If the input sequences are not equally likely but the input statistic Pr{a} is not

known by the receiver the ML criteria is suboptimal

argmax pa

x y a p py x y a

Pr Pr 2 K a x


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ML-Decoding assuming a memory less channel (DMC)

As the logarithm is a strictly monotone increasing function we can write

Incremental metric (yi()|ai()) describes the transition probabilities of the channel

AWGN channel:

2

00

1exp

/ 2 // 2 /

i i

i i

ss

y ap y a

N TN T

joint probabilities

can be factorized

| /i i s sp y x E T | /i i s sp y x E T

/s s

E T /s sE Ty

1 1

0 0 1

N N n

i i

i

p p p y a

y a y a

1 1 1

0 1 0 10 1

ln ln ln N n N n N n

i i i i i i

i ii

p p y a p y a y a

y a

lni i i iy a p y a


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Squared Euclidian distance

Correlation metric

maximize

minimize

2

0

ln/ 2 /

i i

i i i i

s

y ay a p y a C

N T

2

0

2/

i i

i i

s

y ay a

N T

2 2 2

0 0 0 0 0 0

22 4 2

/ 2 / / 2 / / 2 / / /

constantdoes not depend

on i

i i i i i i i si i

s s s s s

a

y y a a y y a Ey a C C

N T N T N T N T N T N

0 0

44

/

si i i i i

s

Ey a y a y

N T N


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Direct approach for ML decoding:

Sum up the incremental metric (yi()|ai()) for all possible code sequences a

Determine sequence a with minimum cost function

Number of sequences a corresponds to 2K effort increases exponentially with K

Elaboration ofMarkovian property of convolutional codes

The current state depends only on previous state and the current input

Successive calculation of path metric

Viterbi Algorithm

1

0 1

arg max arg max ln arg max lnN n

i i

i

p p p y a

a a a

x y a y a


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ML-Decoding of Convolutional Codes (1)

Example: [5,7]8-NSC-encoder, with termination K=3 bit 23=8 CW

[000 00] [00 00 00 00 00]

[100 00] [11 10 11 00 00]

[010 00] [00 11 10 11 00][110 00] [11 01 01 11 00]

[001 00] [00 00 11 10 11]

[101 00] [11 10 00 01 11]

[011 00] [00 11 01 01 11][111 00] [11 01 10 01 11]

00

10

01

11

=0

=4

=3

=2

=1

=5

0/00 0/00 0/00 0/00 0/00

0/10

1/01

1/11 1/11 1/11

0/10

1/01

0/01

1/10

0/01

0/11

1/00

0/11

0/10

0/11

0/00

1/11

0/10

1/01

0/01

1/10

0/11

1/00


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ML-Decoding of Convolutional Codes (2)

Receive word: y = [11 11 00 01 11]

ML decoding

Incremental path metric

Cumulative state metric

00

10

01

11

20 2 2 4 0 4 1 4 2 5

00

02

26

1

11

1

2

3

30

11

3

1

21

21

3

1

1

0

4

2

10

y a y ,a Hd

1 y aj iM M

Estimated code word

Estimated informationword

x 11 10 00 01 11

x 1 0 1

0 0 0M

Hamming distance

instead of Euclidian

distance!

0/00

1/11

0/10

1/01

0/01

1/10

0/11

1/00

=0

=4

=3

=2

=1

=5


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Viterbi Algorithm

1) Start trellis in state 0

2) Calculate (y()|a ()) fory() and any possible code words a ()3) Add the incremental path metric to the old cumulative state metricMj( -1),

j = 0,,2m-14) Select for each state the path with lowest Euclidian distance (largest correlation

metric) and discard the other paths effort increases only linearly with observation length and not exponential

5) Return to step 2) unless allNreceived words y() have been processed6) End of the Trellis

Terminated code (trellis ends in state 0): select path with best metricM0(N),

Truncated code: select path with the overall best metricMj(N)

7) Trace back the path selected in 6) (survivor) and output the correspondinginformation bits


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00

10

01

11

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

Example: (2,1,3)-NSC-encoder with generators g1 = 78 and g2 = 58

Information sequence: u = [1 0 0 1 | 0 0] x = [-1-1 -1+1 -1-1 -1-1|-1+1 -1-1]

Receive sequence y = [+1+1 -1+1 -1-1 -1-1|-1+1 +1-1]

Decoding of Convolutional Codes with Viterbi-Algorithm

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

2

-2

2

-2

0 0

0

2

-2

2

2

0

-4

0

0

0

0

2

-2

-2

2

0

0

0

0

2

-2

-2

2

4

4

-2

2

2

-2

0

0

0

0

2

-2

-2

2

0

0

0

0

2

4

2

2

4

4

4

4

6

6

6

8

6

6

0 0 11 00 010 10 110 10 11 01 00 01 0

BPSK{0,1} {+1,-1}


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Decoding with Viterbi Algorithm

Rule of Thumb

For continuous data transmission (or very long data blocks) the decoding delay

would become infinite or very high

It has been found experimentally that a delay of 5LC results in a negligibleperformance degradation

Reason: If the decision depth is large enough, the beginnings of different paths

merge and the decision for this part is reliable

00

10

01

11


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Shortest Distance between Bremen and Stuttgart

Finding the path in the trellis (the Autobahn) with the

shortest path cost (distance in km) from the starting

point (Bremen) to the destination (Stuttgart)

121

Bremen Hannover

Osnabrck

Kassel

Dortmund

Wrzburg

Frankfurt

Stuttgart

161 211 137

111 212

190322212

198226

121 282 493 6300

121

232 444

333554

347 480

634

121


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Puncturing of Convolutional Codes (1)

Variable adjustment of the code rate by puncturing (see block codes)

single binary digits of the encoded sequence are not transmitted Advantage of Puncturing:

Flexible code rate without additional hardware effort

Possibly lower decoding complexity

Although the performance of the original code is decreased, the performance of the

punctured code is in general as good as a not punctured code of the same rate

Puncturing Matrix of periodLP repeated application of columns pi

P

P

P

P

1,0 1,1 1, 1

2,0 2,1 2, 1

0 1 1

,0 ,1 , 1

L

L

L

n n n L

p p p

p p p

p p p

P p p p

Each column pi ofP contains the

puncturing scheme of a code

word and therefore consist ofn

elements pi,j GF(2)(pi,j = 0 j-th bit is not transmitted

pi,j = 1 j-th bit is transmitted)


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Instead of transmitting nLP coded bits, only +LP bits are transmitted due to the

puncturing scheme

Parameter with 1 (n1)LP adjusts code rates in the range

Puncturing affects the distance properties

optimal puncturing depends on the specific convolutional codeAttention: puncturing can produce catastrophic convolutional code!

Decoding of punctured codes Placeholders for the punctured bits have to be inserted into the received sequence

prior to decoding (zeros for antipodal transmission). As the distance properties are

degraded by puncturing, the decision depth should be extended.

1

PC

P

LR

L

1

1PC

P

LR

L n n

1 Pn L


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Example: (2,1,3)-NSC-code of code rateRc = 1/2 is punctured to code rateRc = 3/4 with puncturing periodLP = 3 (=1)

g10 =1

g20 =1

g12 =1

g22 =1

g11 =1

g21 =0

u(-1) u(-2)

u() 1 1 0

1 0 1

P

x1()

x2()

x()

Encoded sequence Transmit sequence

x1(0),x2(0),x1(1),x2(1),x1(2),x2(2),x1(3),x2(3), x1(0),x2(0),x1(1),x2(2),x1(3),x2(3),


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Distance Properties of Convolutional Codes (1)

As for block codes, the distance spectrum affects the performance of

convolutional codes

Free distance dfdescribes smallest Hamming-distance between 2 sequences

Free distance df affects the asymptotic (Eb /N0) performance,for average SNR, larger distances affect the performance as well

Distance spectrum

Distance spectrum: Convolutional codes are linear comparison with all-zero-sequence

(instead of comparison of all possible sequence pairs)

Hamming weight wH of all sequences has to be calculated

Modified state diagram Self-loop in state 0 is eliminated, state 0 becomes first state Sb and last state Se

Placeholder at state transitions: L = sequence length

W = weight of uncoded input sequence

D = weight of coded output sequence


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Example: distance spectrum for (2,1,3)-NSC-code with g1 = 78 and g2 = 58 Modified state diagram (values of interest are given in exponent of placeholders)

Linear Equation System

0 0 1 0 0 1

1 1

WD2L0 0

D2L

DL

DL

WL

WDL

WDL

Sb Se

2

10 01

01 10 11

11 11 10

2

01

b

e

S WD L S WL S

S DL S DL S

S WDL S WDL S

S D L S

5 3

2: , ,

1

e

b

S WD LT W D L

S WDL WDL

Solution (transfer function):

L = sequence length

W = weight of uncoded input seq.

D = weight of coded output seq.

0 0

1 0 0 1

1 1

0/00

1/11

0/10

1/00

1/01

1/10

0/01

0/11


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Series expansion ofT(W,D,L) yields

Interpretation:

1 sequence of length = 3 with input weight w = 1 and output weight d= 5

1 sequence with input weight w = 2 and output weight d= 6 of length = 4 and = 5

1 sequence with input weight w = 3 and output weight d= 7 of length = 5 and = 7

and 2 sequences of length = 6

5 3

2

5 3

2 6 4 2 6 5

3 7 5 3 7 6 3 7 7

, ,

, ,1

2

w d l

w d lw d l

WD LT W D L

WDL WDL

WD L

W D L W D L

W D L W D L W D L

T W D L

Coefficient Tw,d,l: number ofsequences with input weight w and

output weight dof length


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Presentation of the code sequences until a maximum weight ofd 7 in theTrellis diagram

Free distance df=5


00

10

01

11

w=1,d=5,=3

0/10

0/111/11

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

1/01 0/01

w=2,d=6,=4

1/00

w=2,d=6, =5

1/10

w=3,d=7, =5

w=3,d=7, =7w=3,d=7, =6

w=3,d=7, =6


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General calculation

a: state transition of state 0 into all other states with parameters W,D andL

S: state transition between states S01 up to S11 (without state 0)

b: state transitions of all states into state 0

For the example

0

, , p

p

T W D L

aS b

20 0WD L

a

0 0

0

0

WL

DL WDL

DL WDL

S

2

0

0

D L

b

S01

S10

S11

S01

S10

S11

Transition from

0 0 1 0 0 1

1 1

WD2L0 0

D2L

DL

DL

WL

WDL

WDL

Sb Se


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For sequence of lengthL the exponent ofS becomesp=L-2

2 2

2 3 2 2 3 2 5 3

3

0 0

, , 0 0 0 0 0 0

0 0 0

WL D L D L

T W D L WD L DL WDL WD L W D L WD L

DL WDL

aSb

2 2

2 3 2 2 3 2 2 4 3 2 3 3 3 4 3 2 6 4

4

0 0

, , 0 0 0 0

0 0 0

WL D L D L

T W D L WD L W D L DL WDL W D L W D L W D L W D L

DL WDL

aS b

2

3 2 4 3 2 3 3 3 4 3 2 6 5 3 7 5

5

0 0

, , 0 0

0 0

WL D L

T W D L W D L W D L W D L DL WDL W D L W D L

DL WDL

aS b

5 3 2 6 4 2 6 5 3 7 55 , ,T W D L WD L W D L W D L W D L


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Number of sequences with Hamming weight d

Total number of nonzero information bits (w) associated with code sequences of

Hamming weight d useful to determine BER

Example

, ,d w d

w

a T

, ,

1

, , 1d dw d d

d w dW

T W D Lw T D c DW

, ,d w dwc w T

, , , ,1, , 1 1 1w d l d

w d l w d l

w d l d w l

T W D L T D T D

5 2 6 2 6 3 7 3 7 3 71

1

5 6 6 2 7 2 7 7

1

5 6 7

, , 12

2 2 3 2 3 3

4 12

W

W

W

T W D LWD W D W D W D W D W D

W W

D WD WD W D W D WD

D D D


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45

Distance Spectra for NSC and RSC Encoders (1)

Non-recursive convolutional encoder (NSC)

Generator

Each input sequence of weight w

results in output sequences ofweight d(w)

(special for this NSC code)

dmin =df=5 is achieved forw = 1

Number of paths increases

exponentially with distance

2

12

2

1

1

G D D D

G D D

10log10(ad

)


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Distance Spectra for NSC and RSC Encoders (2)

Recursive convolutional encoder (RSC)

Generator

Only forw 2 output sequences

of finite length exist Output sequence of finite weight

only for even input weights

10log10(ad

)

10log10(ad

)

1

2 2

2

1

1 / 1

G D

G D D D D


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Error bounds

An error occurs if the conditional probability of the correct code sequence x is

lower as the probability for another sequence a x

Probability of an error

4 /s s i iE T a x

for all

0 else

1 1

0 0

1 1 1 1

0 0 0 1 0 1

1

0 1

2

Pr ln ln

Pr

Pr Pr

Pr 0

w

N N

N N N n N n

i i i i

i i

N n

i

i

i ia x

P p p

y x y a

y

y x y a

y x y a

y x y a

E b d


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Pairwise error probability Pd of sequences a and x

with distance d = dH(a,x)

The sum overdreceived symbolsyi() is a Gaussian distributed random

variable with

Mean Variance

Probability for mixing up two sequences with pair wise Hamming distance d

becomes

1

0 1

Pr 0N n

d i

i

P y

Error bounds

i ix a

/s SY d E T 2

0 / 2 /Y sd N T

0 0

1 1erfc erfc2 2

s bd c

E EP d dRN N

ii

Y y

E ti ti f th b bilit


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Estimation of the sequence error probability

Probability for error event in a sequence

Estimation of the bit error probability

0

1erfc

2

bw d d d c

d d

EP a P a d R

N

0

1

erfc2

b

b d cd

E

P c d R N

E l (1)


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Example (1)

Example: half-rate code with generators g1 = 78 and g2 = 58

Estimation of the sequence error probability

54 25

, , 1 dd d dd

T W D L W D L L

50

14 2 erfc

2

d bb c

d

EP d dR

N

55

1, , 1 2d d

d

T W D L D

52dda

5

51

1, , 14 2d d

dW

T W D Ld D

W

54 2ddc d

Number of sequences with

Hamming weight d

Number of information bits equal to one (w)

for all sequences with Hamming weight d

E l (2)


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Example (2)

Asymptotic bit error rate

(BER) is determined by free

distance df

For estimating the BER at

moderate SNR, the whole

distance spectrum is required

For large error rates or small

signal-to-noise ratios, the

union bound is very loose and

may diverge

0/ in dBbE N

Comparison of simulation and analytical estimation

0 1 2 3 4 5 610-5

10-4

10-3

10-2

10-1

100

BER

Simulationd=5

d=6d=7d=20

Performance of Convolutional Codes: Quantization


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Performance of Convolutional Codes: Quantization

By quantizing the received

sequence before decoding,

information is lost.

Hard-Decision (q = 2):Strong performance

degradation

3-bit quantization (q = 8):only a small performance

degradation compared to no

quantization

0/ in dBbE N

Influence of quantization

none

0 2 4 6 810

-5

10-4

10

-3

10-2

10-1

100

BER

q=q=2q=8

N i l R l C l i l C d


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Numerical Results to Convolutional Codes

Influence of code rate,Lc=9Influence of constraint length,Rc=0.5

0 1 2 3 4 5 610-5

10-4

10-3

10-2

10-1

100

BER

Lc=3L

c=5

Lc=7Lc=9

0/ in dBbE N 0/ in dBbE N

0 1 2 3 4 5 610

-5

10-4

10-3

10-2

10-1

100

BER

Rc =1/4R

c=1/3

Rc =1/2Rc =2/3Rc =3/4

conv codes

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