decoding of convolutional codes let c m be the set of allowable code sequences of length m. not...

Decoding of Convolutional Codes

Let Cm be the set of allowable code sequences of length m.

Not all sequences in {0,1}m 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

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mCc

c

y

cydHmp

cydHpcy

,)1.(

,|Pr

Decoding Rule for Convolutional Codes

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Maximum Likelihood Decoding Rule:

Choose the code sequence through the trellis which has the

smallest Hamming distance to the received sequence!

cyH

dmCc

cymCc

,min,Prmax

The Viterbi Algorithm

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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 < 10 Can be used for either hard or soft decision decoding

We consider hard decision decoding initially

Basic Idea of Viterbi Algorithm

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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)

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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

Z j 0 for all other states

The Viterbi Algorithm (continued)

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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 i+1. 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.

iy

ic

ics ic

ic

yicH

diicsZ ,

The Viterbi Algorithm (continued)

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Iteration:

• Let

• Let i=i+1

• Repeat previous step

•Incorrect paths drop out as i approaches infinity.

yicH

diicsZi

jZ ,1

Viterbi Algorithm Decoding Example

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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?

c

y

Viterbi Algorithm Decoding Example(continued)

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Viterbi Decoding Examples

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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

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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

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Other Applications of Viterbi Algorithm

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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

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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

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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

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

the trellis back to the all zeros state

Performance is goodNow both start and ending state are known by the decoder

Wastes bits in short frame

Performance of Convolutional Codes

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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

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What's the pairwise probability of choosing a path at distance d from the correct path?

Calculation of First Event Error Probability

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Evaluating Error ProbabilityUsing the Transfer Function Bound

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Finding T(D) from State Diagram

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Break all 0’s 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

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Performance Example for Convolutional Code

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Performance of r=1/2 Convolutional Codeswith Hard Decisions

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Performance of r=1/3 Convolutional Codeswith Hard Decisions

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Punctured Convolutional Codes

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Practical Examples of Convolutional Codes

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