Channel Coding in Communication Networks

8/6/2019 Channel Coding in Communication Networks

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Convolutional Codes

Representation 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


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Modification to Known Codes

1. Puncturing: delete a parity symbol

(n,k) codep (n-1,k) code

2. Shortening: delete a message symbol

(n,k) codep (n-1,k-1) code

3. Expurgating: delete some subset of codewords

(n,k) codep (n,k-1) code

4. Extending: add an additional parity symbol

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



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Modification to Known Codes

5. Lengthening: add an additional message symbol

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

6. Augmenting: add a subset of additional code words

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



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Interleaving We have assumedso far that bit errors are independent from one

bit to the next

In mobile radio,fadingmakes bursts of errorlikely.

Interleaving is used to try to make these errors independent again


Depth

Of

Interleaving

1

16

2

11

3

16

4

21

5

~

31

7

~

26

6

5

2910

30

30

34

35

35

Length

Order

Bits

Transmitted

Order

Bits

Received


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Block Diagram of Concatenated Coding Systems


Data

Bits

Outer

EncoderInterleave

Inner

Encoder

Modulator

Channel

De-ModulatorInner

Decoder

De-

Interleave

Outer

Decoder

Data

Out


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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 dont exploit full power of the error

correction coder



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Practical Application: Galileo Deep Space Probe

Uses concatenated coding

Inner code rate is , 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



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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 PCSProposed 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


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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 beenfound:

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


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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


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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


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Example of Convolutional

Code

k=1, n=2, K=3 convolutionalcode


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Code

k=2, n=3, K=2convolutionalcode


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Representations of Convolutional

Codes

Encoder Block Diagram (shown above)

GeneratorRepresentation

Trellis Representation

State Diagram Representation


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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 Kis 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


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Code

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

code


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Code

k=2, n=3, K=2convolutionalcode


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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


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Example of State Diagram Representation

Of Convolutional Codesk=1, n=2, K=3 convolutional code


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Example of Trellis Diagram

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

code


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Encoding Example Using Trellis

Representation

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

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


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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.


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Best Rate 1/2 Codes


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Best Rate 1/3

Codes


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Best Rate 2/3 Codes


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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

Channel Coding in Communication Networks

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