Modulation and coding course- lecture 10

Today, we are going to talk about:„ Another class of linear codes, known as Convolutional codes.. Convolutional codescoding substantially different from that of block codes.. „ does not

Digital Communications I:

Modulation and Coding Course

Period 3 - 2007 Catharina Logothetis

Lecture 10

Last time, we talked about:

„ Channel coding

„ Linear block codes

Today, we are going to talk about:

„ Another class of linear codes, known as Convolutional codes.

„ We study the structure of the encoder.

„ We study different ways for representing the encoder.

Convolutional codes

coding substantially different from that of block codes

„ A convolutional encoder:

„ encodes the entire data stream, into a single codeword

„ does not need to segment the data stream into blocks of fixed size ( Convolutional codes are often forced to block structure by periodic

truncation ).

„ is a machine with memory.

different nature to the design and evaluation of the code

„ Block codes are based on algebraic/combinatorial

techniques.

„ Convolutional codes are based on construction techniques.

Convolutional codes-cont’d

„ A Convolutional code is specified by

three parameters or

where

number of data bits per coded bit.

assume that from now on

where the encoder has K-1 memory

elements.

„ There is different definitions in literatures for constraint length

) ,

, ( n k K ( k / n , K )

n k

Block diagram of the DCS

Information

source

Rate 1/n

Information

sink

Rate 1/n

4

4 3 4

4 2

1

sequence Input

2

1 , , , , )

=

m

4

4 3 4

4 2 1

4 4

4 4

1

bits) coded ( rd Branch wo 1

sequence

Codeword

3 2 1

, ) , ,

, , (

n

ni ji

i i

i

, ,u , ,u

u U

U U

U U

=

=

= G(m)

U

, ) ˆ

, , ˆ

, ˆ (

m

4 4

4 4

1

d Branch wor per

outputs

1

d Branch wor for

outputs

r Demodulato

sequence received

3 2 1

, ) , ,

, , (

n

ni ji

i i

i

i

, ,z , ,z

z Z

Z Z

Z Z

=

=

Z

A Rate ½ Convolutional encoder

„ 3 shift-registers where the first one takes the

incoming data bit and the rest, form the memory

of the encoder

m

1

u

2

u

First coded bit

Second coded bit

2

1, u

u

(Branch word)

A Rate ½ Convolutional encoder

1

t

1

u

2

u

1 1

2

u

2

t

1

u

2

u

0 1

2

u

3

t

1

u

2

u

0 0

2

u

4

t

1

u

2

u

0 1

2

u

) 101 (

=

m

Message sequence:

A Rate ½ Convolutional encoder

Encoder

) 101 (

=

5

t

1

u

2

u

1 1

2

u

6

t

1

u

2

u

0 0

2

u

Effective code rate

„ Initialize the memory before encoding the first bit (all-zero)

(all-zero)

„ Hence, a tail of zero-bits is appended to data bits.

„ L is the number of data bits and k=1 is assumed:

data tail Encoder codeword

c

K L

n

L

− +

=

) 1 (

Encoder representation

„ Vector representation:

vector for each modulo-2 adder) The i:th element

in each vector, is “1” if the i:th stage in the shift register is connected to the corresponding

modulo-2 adder, and “0” otherwise

„ Example:

m

1

u

2

u

2

1 u u

) 101 (

) 111 (

2

1

=

=

g g

Encoder representation – cont’d

„ Impulse response representaiton:

goes through it

„ Example:

1 1 001

0 1 010

1 1 100

11 10

11 : sequence Output

0 0

1 :

sequence Input

2

u

Branch word Register

contents

11 10

00 10

11

11 10

11 1

00 00

00 0

11 10

11 1

Output

Input m

Modulo-2 sum:

Encoder representation – cont’d

„ Polynomial representation:

modulo-2 adder Each polynomial is of degree K-1 or

less and describes the connection of the shift

registers to the corresponding modulo-2 adder

„ Example:

The output sequence is found as follows:

2 2

) 2 ( 2

) 2 ( 1

) 2 ( 0 2

2 2

) 1 ( 2

) 1 ( 1

) 1 ( 0 1

1

)

(

1

)

(

X X

g X

g g

X

X X

X g

X g

g X

+

= +

+

=

+ +

= +

+

=

g g

) ( ) ( with interlaced

) ( ) ( )

U =

Encoder representation –cont’d

In more details:

11 10

00 10

11

) 1 , 1 ( )

0 , 1 ( )

0 , 0 ( )

0 , 1 ( ) 1 , 1 ( ) (

0

0

0 1 ) ( )

(

0 1

) ( )

(

1 ) 1

)(

1 ( ) ( )

(

1 ) 1

)(

1 ( ) ( )

(

4 3

2

4 3

2 2

4 3

2 1

4 2

2 2

4 3

2 2

1

=

+ +

+ +

=

+ +

+ +

=

+ +

+ +

=

+

= +

+

=

+ +

+

= +

+ +

=

U U

g

m

g

m

g

m

g

m

X X

X X

X

X X

X X

X X

X X

X X

X X

X X

X X

X

X X

X X

X X

X X

State diagram

„ A finite-state machine only encounters a finite number of states

„ State of a machine: the smallest amount

of information that, together with a

current input to the machine, can predict the output of the machine.

„ In a Convolutional encoder, the state is represented by the content of the

memory.

„ Hence, there are states.1

2K−

State diagram – cont’d

„ A state diagram is a way to represent

the encoder

„ A state diagram contains all the states and all possible transitions between

them.

„ Only two transitions initiating from a

state

„ Only two transitions ending up in a state

State diagram – cont’d

00

11

output

Next state

input

Current state

01 0

01 1

10

0 10

00 1

11

0 01

11 1

00

0

000

S

1

S

2

S

S

0

S

2

S

0

S

2

S

1

S

3

S

1

S

0

S

1

S

2

S

3

S

1/11

1/00

1/01

0/11 0/00

0/01 0/10

Input

Output (Branch word)

Trellis – cont’d

diagram that shows the passage of time.

Time

i

State

00

0 =

S

01

1 =

S

10

2 =

S

11

3 =

S

0/00

1/10

0/11

0/10 0/01

1/11

1/01

1/00

Trellis –cont’d

0/11

0/10 0/01

1/11

1/01

1/00

0/00

0/11

0/10 0/01

1/11

1/01 1/00

0/00

0/11

0/10 0/01

1/11

1/01 1/00

0/00

0/11

0/10 0/01

1/11

1/01 1/00

0/00

0/11

0/10 0/01

1/11

1/01 1/00

0/00

Input bits Output bits

Tail bits

Trellis – cont’d

1/11

0/00

0/10 1/11

1/01

0/00

0/11

0/10 0/01

1/11

1/01 1/00

0/00

0/11

0/10 0/01

0/00

0/11 0/00

6

t

1

Input bits Output bits

Tail bits