Convolutional Coding Viterbi Algorithm

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Helsinki University of TechnologyHelsinki University of TechnologyS-72.333 Postgraduate Seminar on Radio Communications

ConvolutionalConvolutional Coding & Coding & ViterbiViterbi AlgorithmAlgorithm

Er Liu

liuer@cc.hut.filiuer@cc.hut.fiCommunications LaboratoryCommunications Laboratory

16.11.200416.11.2004

ConvolutionalConvolutional Coding & Coding & ViterbiViterbi AlgorithmAlgorithm Er Liu (Er Liu (liuer@cc.hut.filiuer@cc.hut.fi))

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OutlineOutline

Convolutional CodingConvolutional codeGenerator sequenceTrellis and state diagram

Viterbi AlgorithmMaximum-Likelihood decodingViterbi algorithm

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ConvolutionalConvolutional EncodingEncoding

Convolutional codes are applied in applications that requiregood performance with low implementation cost. They operateon data stream, not static block.

Convolutional codes have memory that uses previous bits to encode or decode following bits

It is denoted by (n,k,L), where L is code memory depth

Code rate r is determined by input rate and output rate:

1input

output

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ConvolutionalConvolutional EncoderEncoder

Convolutional encoder is a finite state machine (FSM), processing information bits in a serial mannerThus the generated code is a function of input and the states of the FSMIn this (n,k,L)=(2,1,2) encoder each message bits influences a span of n(L+1)=6 successive output bits

j j j j

x m m m

x m m− −

= ⊕ ⊕

(n,k,L)=(2,1,2) encoder

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AnotherAnother EncoderEncoder exampleexample

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GeneratorGenerator SequenceSequence(n,k,L) convolutional code can be described by generator sequencesg(1) , g(2),..., g(n) that are the impulse responses of each coder output branch

Generator sequences specify convolutional code completely by the associated generator matrixEncoded convolutional code is produced by matrix multiplication of input and the generator matrix

[1 0 1 1][1 1 1 1]

⎧ =⎪⎨

=⎪⎩

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ExampleExample of of UsingUsing GeneratorGenerator MatrixMatrix

It can also use polynomial multiplication

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RepresentationRepresentation –– CodeCode TreeTree

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TrellisTrellis and and StateState DiagramDiagram

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MinimumMinimum HammingHamming DistanceDistance

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MaximumMaximum--LikelihoodLikelihood DecodingDecoding

Maximum likelihood decoding means finding the codebranch in the code trellis that was most likely to transmittedTherefore maximum likelihood decoding is based on calculating the hamming distances for each branch formingencode wordProbability to decode sequence is then

The most likely path through the trellis will maximize thismetric

( , ) ( )j jj

p y x p y x∞

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Example of Maximal Likelihood DetectionExample of Maximal Likelihood DetectionAssume a three bit message is to transmitted. To clear the encoder two zero-bits are appended after message. Thus 5 bits are inserted into encoder and 10 bits produced. Assume channel error probability is p=0.1. After the channel 10,01,10,11,00 is produced. What comes after decoder, e.g. what was most likely the transmitted sequence?

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Example of Maximal Likelihood DetectionExample of Maximal Likelihood Detection

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ViterbiViterbi AlgorithmAlgorithm

ML algorithm is too complex to search all available pathesEnd to end calculation

Viterbi algorithm performs ML decoding by reducing itscomplexity

Eliminate least likely trellis path at each transmission stage

Reduce decoding complexity with early rejection of unlike pathes

Viterbi algorithm gets its efficiency via concentrating on suvival paths of the trellis

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ViterbiViterbi decodingdecoding

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ExampleExample of of ViterbiViterbi decodingdecoding

Input data : m =1 1 0 1 1Codeword : X = 11 01 01 00 01Received code : Z = 11 01 01 10 01

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HomeworkHomework

Please use Viterbi algorithm to decode the receivedsequence:

Z=[11 10 10 10 01]Please draw the trellis and state diagram

Helsinki University of TechnologyHelsinki University of TechnologyS-72.333 Postgraduate Seminar on Radio Communications

AnyAny questionsquestions??

Thanks!

Convolutional Coding Viterbi Algorithm

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