Principles of Communications pps

 Constraint length K: the number of shifts over which a single message bit influence the output  M-stage shift register: needs M+1 shifts for a message to enter the shift register and

Principles of Communications

By: Dang Quang Vinh

Faculty of Electronics and Telecommunications

Ho Chi Minh University of Natural Sciences

Convolutional codes

09/2008

 In block coding, the encoder accepts k-bit message block and generates n-bit codeword⇒Block-by-block basis

 Encoder must buffer an entire message block before

generating the codeword

 When the message bits come in serially rather than in large blocks, using buffer is undesirable

 Convolutional coding

Definitions

consists of an M-stage shift register, n modulo-2 adders

with n(L+M) bits

ol) (bits/symb

) (L M n

L r

+

=

ol)(bits/symb

1

n

 Constraint length (K): the number of shifts over which a

single message bit influence the output

 M-stage shift register: needs M+1 shifts for a message to

enter the shift register and come out

 K=M+1

Output Input

Generations

characterized by impulse response or generator

polynomial

cyclic codes

polynomials { }

) ,

, , ,

0

) ( 1

) ( 2

) ( 1

2 ) ( 2

) ( )

( ),

) 1

 Consider the case of (2,1,2)

 Impulse response of path 1 is (1,1,1)

 The corresponding generator polynomial is

 Impulse response of path 2 is (1,0,1)

 The corresponding generator polynomial is

 Message sequence (11001)

 Polynomial representation:

1 )

) 1 ( D = D + D+

g

1 )

) 2

g

1 )

(D = D4 + D3 +

m

Example(2/8)

1

1

) 1 )(

1 (

) ( )

( )

(

2 3

6

2 3

4 5

4 5

6

2 3

4

) 1 ( )

1 (

+ + +

+

=

+ + +

+ +

+ +

+

=

+ + +

+

=

=

D D

D D

D D

D D

D D

D D

D D

D D

D g

D m D

c

1

) 1 )(

1 (

) ( )

( )

(

2 3

5 4

6

2 3

4

) 2 ( )

2 (

+ +

+ +

+

=

+ +

+

=

=

D D

D D

D

D D

D

D g

D m D

c

 Message length L=5bits

 Output length n(L+K-1)=14bits

 A terminating sequence of K-1=2 zeros is

appended to the last input bit for the shift

register to be restored to its zero initial state

00

1011

010

0

10011

001

11

001001

0011

0010

011

00100

11

001001

output 111

m

c (1) =(1001111)

Example(6/8)

 denoting the impulse

input

Output Input

) ,

, , ,

0 , ) ( 1 , )

( 1 , ) ( , )

i

j i

j M i

j M i

j

Output Input

D D

g g

D D

g g

D D

g g

D g

g

D g

g

D D

g g

10

(

1 )

( )

11

(

) ( )

10

(

1 ) ( )

01 (

1 ) ( )

01

(

1 )

( )

11

(

) 1 ( )

3

(

) 1 ( 1 )

3

(

1

) 2 ( 2 )

2

(

2

) 2 ( 1 )

2

(

1

) 1 ( 1 )

1

(

2

) 1 ( 1 )

1

(

1

of input 1 is represented by dashed line

 Output is labeled over the transition line

state Binary description

d

c

1/11

0/10 1/01

c

0/00 1/11

0/10 1/01

1

11

01 c

0

01

00 a

0

11 d 01

1

Trellis(1/2)

 The trellis contains (L+K) levels

 Labeled as j=0,1,…,L,…,L+K-1

 The first (K-1) levels correspond to the

encoder’s departure from the initial state a

 The last (K-1) levels correspond to the

encoder’s return to state a

 For the level j lies in the range K-1 ≤ j ≤ L, all

the states are reachable

Maximum Likelihood Decoding

of Convolutional codes

 m denotes a message vector

 c denotes the corresponding code vector

 r denotes the received vector

 With a given r , decoder is required to make estimate

of message vector, equivalently produce an estimate

of the code vector

 otherwise, a decoding error

happens

 Decoding rule is said to be optimum when the

propability of decoding error is minimized

 The maximum likelihood decoder or decision rule is

described as follows:

 Choose the estimate for which the log-likelihood

function logp(r/c) is maximum

mˆ

cˆ

c c

m

m ˆ = only if ˆ =

cˆ

Maximum Likelihood Decoding

of Convolutional codes

 Binary symmetric channel: both c and r are binary

sequences of length N

 r differs from c in d positions, or d is the Hamming

distance between r and c

| (

| ( log

i i

i i

c r

c

r p

p c

r

p

if

if 1

)

| ( with

) 1 log(

1 log

) 1 log(

) (

log )

| ( log

p N

p

p d

p d

N p d

c r p

− +

=

⇒

Maximum Likelihood Decoding

of Convolutional codes

 Decoding rule is restated as follows:

 Choose the estimate that minimizes the

Hamming distance between the received vector r

and the transmitted vector c

 The received vector r is compared with each

possible code vector c, and the one closest

to r is chosen as the correct transmitted

code vector

cˆ

The Viterbi algorithm

 Choose a path in the trellis whose coded sequence differs from the received sequence in the fewest number of

positions

The Viterbi algorithm

 The algorithm operates by computing a

metric for every possible path in the trellis

 Metric is Hamming distance between coded sequence represented by that path and

4 3

3

2 5

2 5

Code

Free distance of a conv code

 Performance of a conv code depends on

decoding algorithm and distance properties of

the code.

 Free distance, denoted by dfree, is a measure of code’ s ability to combat channel noise

 Free distance: minimum Hamming distance

between any two codewords in the code

 dfree>2t

 Since a convolutional code doesn't use blocks, processing instead a continuous bitstream, the value of t applies to a quantity of errors located relatively near to each other

Free distance of a conv code

 Conv code has linear property

 So, free distance also defined:

 Calculate dfree by a generating function

 Generating function viewed the transfer

function of encoder

 Relating input and output by convolution

 Generation func relating initial and final state

Free distance of a conv code

 Modify state diagram

00

11

a b

d

c

0/00 1/11

0/10 1/01

Free distance of a conv code

 State equations:

c D a

DLd DLb

d

Dd Db

c

Lc La

D b

2 1

0 2

0

2 1

) ,

(

i

i DL L

D DL

L

D a

a L

D T

4 5

3 7 2

6

5 2 4 2 )

, (D L = D L+ D L + D L + = ∑∞ d− D d L d−T

Free distance of a conv code

 T (D,L) represents all possible transmitted

sequences that terminate with c-e

transition

 For any d ≥ 5, there are 2d -5 paths with

weight w(X)=d that terminate with c-e

transition, those paths are generated by

messages containing d -4 nonzero bits

 The free distance is the smallest of w(X),

so dfree=5

4 5

5 3

7 2

6

),

++

d

d

d D L L

D L

D L

D L

D T

Systematic conv code

 The message elements appear explicitly in the output sequence together with the redundant

elements

Path 1

Path 2

Output Input

Systematic conv code

 Impulse response of path 1 is (1,0,0)

 The corresponding generator polynomial is

 Impulse response of path 2 is (1,0,1)

 The corresponding generator polynomial is

 Message sequence (11001)

2 )

1 ( (D) D

1)

) 2 ( D = D +

g

Systematic conv code

 Output sequence of path 1 (1100100)

 Output sequence of path 2 (1111101)

Systematic conv code

 Another example of systemmatic conv code

Path 1

Path 2

Output Input

Systematic vs nonsystematic

of transmission errors can cause an infinite

number of decoding errors

of systematic code is smaller than that of

nonsystematic code

4 5

5 3

7 2

6

) ,

D L

D L

D L

D T

Systematic vs nonsystematic

 Maximum free distance with systematic and

nonsystematic conv codes of rate 1/2