I Tlecture 13a

Convolutional CodesRepresentation and Encoding

Many known codes can be modified by an extra code symbol or by

deleting a symbol

* Can create codes of almost any desired rate

* Can create codes with slightly improved performance

The resulting code can usually be decoded with only a slight

modification to the decoder algorithm.

Sometimes modification process can be applied multiple times in

succession

Error Control Coding , © Brian D. Woerner , reproduced by: Erhan A. INCE

Modification to Known Codes

1. Puncturing: delete a parity symbol (n,k) code (n-1,k) code

2. Shortening: delete a message symbol (n,k) code (n-1,k-1) code

3. Expurgating: delete some subset of codewords (n,k) code (n,k-1) code

4. Extending: add an additional parity symbol (n,k) code (n+1,k) code


Modification to Known Codes…5. Lengthening: add an additional message symbol

(n,k) code (n+1,k+1) code

6. Augmenting: add a subset of additional code words (n,k) code (n,k+1) code


Interleaving We have assumed so far that bit errors are independent from one

bit to the next In mobile radio, fading makes bursts of error likely. Interleaving is used to try to make these errors independent again


Depth Of Interleaving

1 1

6 2

11 3

16 4

21 5

31 7

26 6

5 29

10 30

30 34

35 35

Length

Order Bits

Transmitted

Order Bits

Received

Concatenated Codes Two levels of coding

Achieves performance of very long code rates while maintaining

shorter decoding complexity

Overall rate is product of individual code rates

Codeword error occurs if both codes fail.

Error probability is found by first evaluating the error probability of

“inner” decoder and then evaluating the error probability of “outer”

decoder.

Interleaving is always used with concatenated coding


Block Diagram of Concatenated Coding Systems


DataBits

Outer

Encoder Interleave Inner

Encoder Modulator

Channel

De-Modulator Inner

Decoder De-Interleave

Outer

Decoder

DataOut

Practical Application : Coding for CD

•Each channel is sampled at 44000 samples/second

•Each sample is quantized with 16 bits

Uses a concatenated RS code Both codes constructed over GF(256) (8-bits/symbol) Outer code is a (28,24) shortened RS code Inner code is a (32,28) extended RS code In between coders is a (28,4) cross-interleaver Overall code rate is r = 0.75

Most commercial CD players don’t exploit full power of the error correction coder


Practical Application: Galileo Deep Space Probe

Uses concatenated coding Inner code rate is ½, constraint length 7 convolutinal encoder

Outer Code (255,223) RS code over GF(256) – corrects any burst errors from convolutional codes

Overall Code Rate is r= 0.437

A block interleaver held 2RS Code words

Deep space channel is severely energy limited but not bandwidth limited


IS-95 CDMA The IS-95 standard employs the rate (64,6) orthogonal (Walsh)

code on the reverse link

The inner Walsh Code is concatenated with a rate 1/3, constraint length 9 convolutional code


Data Transmission in a 3rd Generation PCS

Proposed ETSI standard employs RS Codes concatenated with

convolutional codes for data communication

Requirements; Ber of the order of 10-6

Moderate Latency is acceptable

CDMA2000 uses turbo codes for data transmission

ETSI has optional provisions for Turbo Coding


A Common Theme from Coding Theory

The real issue is the complexity of the decoder.

For a binary code, we must match 2n possible received sequences with code words

Only a few practical decoding algorithms have been found:

Berlekamp-Massey algorithm for clock codes

Viterbi algorithm (and similar technique) for

convolutional codes

Code designers have focused on finding new codes that work with known algorithms


Block Versus Convolutional Codes

Block codes take k input bits and produce n output bits, where k and n are large

there is no data dependency between blocks

useful for data communcations

Convolutional codes take a small number of input bits and produce a

small number of output bits each time period

data passes through convolutional codes in a continuous stream

useful for low- latency communications


Convolutional Codes

k bits are input, n bits are output

Now k & n are very small (usually k=1-3, n=2-6)

Input depends not only on current set of k input bits, but also on past

input.

The number of bits which input depends on is called the "constraint

length" K.

Frequently, we will see that k=1


Example of Convolutional Code

k=1, n=2, K=3 convolutional code





Representations of Convolutional Codes

Encoder Block Diagram (shown above)

Generator Representation

Trellis Representation

State Diagram Representation


Convolutional Code Generators One generator vector for each of the n output bits:

The length of the generator vector for a rate r=k/n

code with constraint length K is K

The bits in the generator from left to right represent the

connections in the encoder circuit. A “1” represents a link from

the shift register. A “0” represents no link.

Encoder vectors are often given in octal representation








State Diagram Representation

Contents of shift registers make up "state" of code:

Most recent input is most significant bit of state.

Oldest input is least significant bit of state.

(this convention is sometimes reverse)

Arcs connecting states represent allowable transitions

Arcs are labeled with output bits transmitted during transition


Example of State Diagram Representation

Of Convolutional Codes



Trellis Representation of Convolutional Code

State diagram is “unfolded” a function of time

Time indicated by movement towards right

Contents of shift registers make up "state" of code:

Most recent input is most significant bit of state.

Oldest input is least significant bit of state.

Allowable transitions are denoted by connects between

states

transitions may be labeled with transmitted bits


Example of Trellis Diagram



Encoding Example Using Trellis Representation


We begin in state 00:

Input Data: 0 1 0 1 1 0 0

Output: 0 0 1 1 0 1 0 0 10 10 1 1


Distance Structure of a Convolutional Code

The Hamming Distance between any two distinct code sequences

and is the number of bits in which they differ:

The minimum free Hamming distance dfree of a convolutional code is the smallest Hamming distance separating any two distinct code sequences:

Cc 1_ Cc

2_

i i

ci

cccH

d,2,12,1

jcicH

djifree

d ,min


Search for good codes

We would like convolutional codes with large free distance

must avoid “catastrophic codes”

Generators for best convolutional codes are generally found via computer search

search is constrained to codes with regular structure

search is simplified because any permutation of identical

generators is equivalent

search is simplified because of linearity.


Best Rate 1/2 Codes


Best Rate 1/3 Codes


Best Rate 2/3 Codes


Summary of Convolutional Codes

Convolutional Codes are useful for real-time applications because

they can be continously encoded and decoded

We can represent convolutional codes as generators, block

diagrams, state diagrams, and trellis diagrams

We want to design convolutional codes to maximize free distance

while maintaining non-catastrophic performance

I Tlecture 13a

Technology

resulting code

code expurgating

binary code

shortened rs code inner

concatenated rs code

example of convolutional

concatenated codes

inner walsh code