convolutional codes and its characteristics, Study Guides, Projects, Research of Digital Communication Systems

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Convolutional Codes

• Overview
  • Parameters
  • Encoding
  • Impulse Response/Generator Sequences
  • State Diagram
  • Trellis Diagram
  • Decoding
    • Viterbi Algorithm

Convolutional Codes

• Convert any length message to a single ‘codeword’
• Encoder has memory and has n outputs that at any time depend on k inputs and m previous input blocks
• Typically described by 3 parameters:
  • n = no. of bits produced at encoder output at each time unit
  • k= no. of bits input to encoder at each time unit
  • And one of:
    • m = memory of encoder = no. of prev. input blocks used to generate each output (or)
• K = constraint length = max. no. of bits in a single output stream that can be affected by any input bit = 1+ max i mi

0 *^ 1 1 0 1 0 0

1

0

1

0 00

0

0

Input: 10110

Encoding Example

0 *^ 1 1 0 1 0

0

0

0

0 00

1

0

Input: 10110

0 *^1 1 0

1

0

1

0 01

1

1

Input: 10110

0 *^1 1

0

0

0

1 00

1

0

Input: 10110

0 0 *^1

0

0

0

0 01

0

1

Input: 10110

0

0

0

0 00

0

0

Input: 10110 Output: 11, 01, 01, 01, 11, 01, 11, 00

Impulse Response/Generator Seq.

• These sequences also specify the connections from the shift registers to the outputs: g(0)^ = (1 0 1 1) g(1)^ = (1 1 0 1)
• A 1 means connection, 0 means no connection

1 1 1

1 1 1

0

0

Impulse Response/Generator Seq.

• The generator sequences are often specified in octal form
• The binary representation is left-shifted so that the most significant bit (MSB) of the binary representation is also the MSB of the first digit
• Example: g(0)^ = (1 0 1, 1) = 54 g(1)^ = (1 1 0, 1) = 64

Trellis Diagram

•^

A trellis diagram is a state diagram that has been expanded to show thepassage of time:

Trellis Diagram

• Every possible codeword is represented by a single unique path through the trellis
• Every codeword starts and stops in S 0 , the all-zeros state

Decoding:Viterbi Algorithm

• Usually, the log-likelihood function is used

to simplify implementation

• The log-probabilities, *, known as bit metrics, are usually converted into small integers
• The ML algorithm chooses y’ that maximizes p ( r | y’ )

∑ ∑^ (^ )

• − =

− = 

 

 

 



1 0

1 (^0) *

log ( | ') L^ m log (^ )| '( ) i

n j

j i

j p r y 1 4 2 p 4 ri (^) 4 3 y 4

Decoding: Viterbi Algorithm

• The Viterbi algorithm finds the ML codeword by using the code trellis:

Viterbi Algorithm (Wicker, p. 296-7)

• Let the node corresponding to state Sj at time t be denoted by Sj,t
• Each node in the trellis is assigned a metric value V( Sj,t )
• The metric values are computed by

1. Let V( S0,0 ) =0 and t = 1
2. At time t , compute the partial path metrics for all paths entering each node.
3. Set V( Sk,t ) equal to the best partial path metric entering the node corresponding to state Sk at time t. Ties can be broken by flipping a coin. Nonsurviving branches are deleted from the trellis.
4. If _t