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Decoding of Convolutional Codes

- Let Cm be the set of allowable code sequences

of length m. - Not all sequences in 0,1m are allowable code

sequences! - Each code sequence can be

represented by a unique path - through the trellis diagram
- What is the probability that the code sequence

is sent and the - binary sequence is received?
- where p is the probability of bit error of BSC

from modulation

Decoding Rule for Convolutional Codes

- Maximum Likelihood Decoding Rule
- Choose the code sequence through the trellis

which has the - smallest Hamming distance to the received

sequence!

The Viterbi Algorithm

- The Viterbi Algorithm (Viterbi, 1967) is a

clever way of - implementing Maximum Likelihood Decoding.
- Computer Scientists will recognize the Viterbi

Algorithm as an - example of a CS technique called Dynamic

Programming

- Reference G. D. Forney, The Viterbi

Algorithm, - Proceedings of the IEEE, 1973
- Chips are available from many manufacturers

which - implement the Viterbi Algorithm for K lt 10
- Can be used for either hard or soft decision

decoding - We consider hard decision decoding initially

Basic Idea of Viterbi Algorithm

- There are 2rm code sequences in Cm .
- This number of sequences approaches infinity as

m - becomes large
- Instead of searching through all possible

sequences, - find the best code sequence "one stage at a

time"

The Viterbi Algorithm(Hamming Distance Metric)

- Initialization
- Let time i 0.
- We assign each state j a metric Z j (0) at time

0. - We know that the code must start in the state 0.
- Therefore we assign
- Z j (0) 0
- Z j (0) for all other states

The Viterbi Algorithm (continued)

- Consider decoding of the ith segment
- Let be the segment of n bits received

between times i - and i 1
- There are several code segments of n bits

which lead - into state j at time i1. We wish to

find the most likely one. - Let be the state from which the code

segment emerged - For each state j, we assume that is the

path leading into - state j if
- is the smallest of all the code

segments leading into state j.

The Viterbi Algorithm (continued)

- Iteration
- Let
- Let ii1
- Repeat previous step
- Incorrect paths drop out as i approaches infinity.

Viterbi Algorithm Decoding Example

- r 1/2, K 3 code from previous example
- (0 0 1 1 0 1 0 0 10 10

1 1) is sent - (0 1 1 1 0 1 0 0 10 10

1 1) is received. - What path through the trellis does the Viterbi

Algorithm choose?

Viterbi Algorithm Decoding Example(continued)

Viterbi Decoding Examples

- There is a company Alantro with a example

Viterbi - decoder on the web, made available to

promote their - website
- http//www.alantro.com/viterbi/workshop.html
- Your browser must have JAVA-enabled

Summary of Encoding and Decoding ofConvolutional

Codes

- Convolutional are encoded using a finite state

machine. - Optimal decoder for convolutional codes will

find the path - through the trellis which lies at the shortest

distance to the - received signal.
- Viterbi algorithm reduces the complexity of this

search by - finding the optimal path one stage at a time.
- The complexity of the Viterbi algorithm is

proportional to the number of states - exponential relationship to constraint length

Implementation of Viterbi Decoder

- Complexity is proportional to number of states
- increases exponentially with constrain length

K 2K - Very suited to parallel implementation
- Each state has two transitions into it
- Each state has two transitions out of it
- Each node must compute two path metrics, add

them to previous metric and compare - Much analysis as gone into optimizing

implementation of this - Butterfly calculation

Other Applications of Viterbi Algorithm

- Any problem that can be framed in terms of

sequence detection can be - solved with the Viterbi Algorithm
- MLSE Equalization
- Decoding of continuous phase modulation
- Multiuser detection

Continuous Operation

- When continuous operation is desired, decoder

will automatically synchronize with transmitted

signal without knowing state - Optimal decoding requires waiting until all

bits are received to trace back path. - In practice, it is usually safe to assume that

all paths have merged after approximately 5K time

intervals - diminishing returns after delay of 5K

Frame Operation of Convolutional Codes

- Frequently, we desire to transfer short (e.g.,

192 bit) - frames with convolutional codes.
- When we do this, we must find a way to terminate
- code trellis
- Truncation
- Zero-Padding
- Tail-biting
- Note that the trellis code is serving as a

block code in - this application

Trellis Termination Zero Padding

- Add K-1 0s to the end of the data sequence to

force - the trellis back to the all zeros state
- Performance is good
- Now both start and ending state are known by the

decoder - Wastes bits in short frame

Performance of Convolutional Codes

- When the decoder chooses a path through the

trellis which diverges from the correct path,

this is called an "error event - The probability that an error event begins

during the current time interval is the

"first-event error probablity Pe - The minimum Hamming distance separating any two

distinct path through the trellis is called the

free distance dfree.

Calculation of Error Event Probability

- What's the pairwise probability of choosing a

path at distance d from the correct path?

Calculation of First Event Error Probability

Evaluating Error ProbabilityUsing the Transfer

Function Bound

Finding T(D) from State Diagram

- Break all 0s state in two, creating a starting

state and a terminating state - Re-label every output 1 as a D
- ad is the number of distinct paths leading from

the starting state to the terminating state while

generating the function Dd

Example of State Diagram

Performance Example for Convolutional Code

Performance of r1/2 Convolutional Codeswith

Hard Decisions

Performance of r1/3 Convolutional Codeswith

Hard Decisions

Punctured Convolutional Codes

Practical Examples of Convolutional Codes

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