Wireless Communication Research Lab. CGU

What is Convolution Code?

指導教授：黃文傑 博士學生：吳濟廷

2004.04.21

Outline• Introduction • Encoding structure• State, tree, and trellis diagrams• Veterbi decoding algorithm• Soft decision decoding• Applications• Summary

Code taxonomy

Today

Introduction

• continuous stream of source bits continuous stream of encoded bits• sequence of source bits is convolved to produce

output ‘symbols’• each encoded bit depends on

Current input bit Previous sequence of input bits

Encoding structure

(2,1,2) convolution code with

generator polynomial

code rate = 1/2

2 21 2( ) 1 , ( ) 1g X X X g X X

General encoding structure

General (n,1,m) convolution code encoder

Encoding example

Encoded Sequence 10 10 10 11 01 11

101011 S1 S2 S3

Input Sequence

Input sequence : 101011

Register number : 3

Generator polynomial : 2 21 2( ) 1 , ( ) 1g X X g X X X

Initial state : 0 0 0

Polynomial representation

21

22

( ) 1

( ) 1

g X X X

g X X

2( ) 1m X X Message polynomial:

Generator polynomial:

State representation and diagram

• simpler representation• two transitions

emanating from each state

• not possible to move to any arbitrary state

Code rate=1/2, m=2

2 21 2( ) 1 , ( ) 1g X X X g X X

Using state diagram

• We could also get the same output sequence by using state diagram

“X” signifies “don’t know“

10 10 01 00 01 01 11

U=

Different time slot

Code rate=1/2, m=2

Tree diagram

• the state diagram completely characterizes the encoder

cannot represent time historytree diagram

Code rate=1/2, m=2

Trellis diagram• Branches increase 2L ( L: number of branch words )

Trellis diagram

Code rate=1/2, m=2

Viterbi decoding algorithm

• Discovered and analyzed by Viterbi in 1967• Advantage

• Maximum likelihood decoding• Not a function of the number of symbols• Reduces the decoding complexity

Example of Viterbi decoding

Label each branch with Hamming distance

error

Decoder trellis diagram (rate=1/2, m=2)

Using the encoder state diagram

Path remerging

Two paths are remerged to state 00 at time t5

cumulative hamming path metric

Viterbi decoding procedure(1/2)

Survivors and metric comparisons

Viterbi decoding procedure(2/2)

Survivors and metric comparisons

Soft decision decoding

• Transform 2-level values to m-level values• Measured by Euclidean distance instead of

Hamming distance• 2~3 dB coding gain better than hard

decision decoding

Soft decision diagrammatic explanation

(a) hard decision (b) soft decision (c) soft code symbols

(d) encoding trellis section (e) decoding trellis section

Applications

• GSM: length= 5, rate= 1/2, free distance= 7• IS-95

• Uplink: length= 9, rate= 1/3, d= 18• Downlink: length= 9, rate= 1/2, d= 12

• UMTS (WCDMA) , CDMA2000: turbo code (further development of convolution code)

Summary

• Convolution codes outperform block codes for the same implementation complexity

• Soft decision decoding decreases the error probability

• Widely used in wireless communication systems nowadays

Thank you ~