The Viterbi Decoding Algorithm

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Viterbi Algorithm Summary:

Distance Properties of Convolutional Codes

Fact: Convolutional codes are linear. the all-zero codeword is a codeword in any convolutional code There is no loss in generality, therefore, in finding the minimum distance between the all-zero word and each codeword.

与全零码字的汉明距 1

Def: Minimum Free Distance dfree is the minimum Hamming distance of all paths through the trellis to the all-zero path .

From the theory of block codes, we know that the code with the minimum distance dmin can correct all t or less errors, where means the largest integer no greater than y.

In the convolutional code, however, dmin is to be replaced by dfree, which is called the (minimum) free distance, so that the error-correcting capability of a convolutional code is

Transfer Functions of Convolutional Codes We need to know not only the minimum free distance but also all possible distance properties of the convolutional code under consideration to assess the error-correcting capabilities of the code in an average statistical sense. The transfer Function of Convolutional codes can be used to Give the all possible distance properties of Convolutional codes

How to construct Transfer Functions ? State diagram of the convolutional encoder with branch words replaced with Dj, where j is the weight of the branch sequence.

State diagram of Figure split open at the node a = (00).

The state equation of each node:

By solving the simultaneous equations, we obtain the result, The generating function or transfer function, denoted T(D), is defined as By solving the simultaneous equations, we obtain the result, 1条重量（距离）为5，从a e的通路 2条重量（距离）为6，从a e的通路

Result analysis: The first term (the term with the lowest exponent) is the minimum free distance df of the convolutional code.

the modified state transfer chart 1 State diagram with branches labeled with Hamming weight D, length L, and N for branches caused by an input data 1.

the modified state equation

The transfer function T(D,L,N) in this case is found to be

There is one path of free distance 5, which differs in one input bit position from the all-zeros path and has length 3; There are two paths of distance 6, one having length 4 and the other having length 5, and both differ in two input bits from the all-zeros path; If we are interested in the jth node level, we consider all terms up to and including the jth power (Lj) term.

Performance Bounds for Viterbi Decoding of Convolutional Codes

The transfer function T(D,L,N)

The First-Event probability Def: the first-event probability is defined as the probability that the correct path is rejected (not a survivor) for the first time at time t = t j .

The error probability of df=5 m0 m1 the situation is equivalent to that of a binary communications system where m0 and m1 are two equally likely messages to be communicated using codewords C0 = 0 0 0 0 0 0 and C1 = 1 1 1 0 1 1 , respectively. The distance between C0 and C1 is The error-correcting capability of this two-codeword system is given by

that is, as long as the channel induces one or two errors, the all-zeros sequence will be favored. If, however, three or more errors are made in the channel, then the correct all-zeros sequence will be rejected and the sequence 1 1 10 1 1 will be favored . Let us consider a BSC with p as the symbol crossover probability. Then the probability of a path sequence with free distance 5, denoted P2(e; 5), is given by the binomial distribution

where p is the symbol error probability of the BSC, which depends on the modulation used for transmission of the channel symbols. The subscript 2 indicates a two-codeword system that pertains to the convolutional decoder decision at state a for the first time.

考虑一般情况下： Suppose that the path being compared with the all-zeros path at some node j at time t = tj has distance d from the all-zeros path.

If d is odd, the all-zeros path will be correctly chosen if the number of errors in the received sequence is less than (d + 1)/2 ; otherwise, the incorrect path is chosen. Thus, the probability of selecting the incorrect path is:

If d is even, the incorrect path is selected when the number errors exceeds d/2 ; that is, when k > d/2 + 1. If the number of errors equals d/2, we must flip an honest coin and one of the two selected. Thus, in this situation, we have

the probability of incorrectly selecting a path with Hamming distance d is

We have seen from the transfer function T(D) given in series form that there are many paths with different distances that merge with the all-zeros path at a given node, indicating thereby that there is no simple exact expression for the first-event error probability. Thus, we overbound the error probability using the union bound, which is the sum of the pairwise error probabilities P2(e; k) over all possible paths that merge with the all zeros path at the given node.

where ak is the number of nonzero paths with Hamming distance k, with respect to the all-zeros path, which merge with the all-zeros path at node j and diverge once from the all-zeros path at some previous node.

If there are ak paths with weight k, and if we still use P2(e; k) to denote the pairwise error probability of paths with weight k, the union bound on the probability of decoding error is given by

For the encoder characterized by the connection vectors g1 = (1 1 1) and g2 = (1 0 I), we had

Viterbi has also shown that the expression given in (P60 ) for the probability of incorrectly selecting a path with Hamming distance d can be bounded by where p is the symbol crossover probability (symbol error probability) of the BSC.

Bit-Error Probability for the BSC channel for Viterbi Decoding of Convolutional Codes `

In the series expansion for the transfer function T(D, N), the exponents of N indicate the number of nonzero information bits that are in error when an incorrect path is selected over the all-zeros path. Consider the transfer function given

By differentiating T(D, N) with respect to N and setting N = 1, the exponents of N become multiplication factors of the corresponding P2(e; d). From this expression, we obtain the bit-error probability PB(e) as we have obtained for the first-event error probability

Probability of Error for Convolutional Soft-Decision Decoding 1: the system model

b: the branch metrics and path metrics the matched-filter output of mth bit in jth branch the transmitted bit of the mth bit of the jth branch the additive noise of the mth bit of the jth branch the transmitted signal energy for each code word b: the branch metrics and path metrics Branch metrics: the branch metrics for the jth branch of ith path the match-filter output for jth branch

the transmitted sequence for jth branch of ith path for soft-decision decoding Neglecting the term that are common to all branch

c: The Pairwise Error probability at node B the all zero e sequence (path) is assumed to transmit

As Defining

d: the first event error probability denotes the number of paths of distance d from the all-zero path

e: The bit error probability As with the path property of transfer-function e: The bit error probability In the transfer function T(DN), the exponents in the factor N indicate the number of information bit error (number of is) in selecting an incorrect path

Probability of Error for Hard-Decision Decoding a: the system mode 1 in BSC channel, ML Criterion minimum distance Criterion

b: the pair wise Error probability The all-zero path is assumed to be transmitted If d is odd, the all-zero path will be correcting selected of the number of errors in the received sequence is less than

e: the first error probability of mode B is the probability of bit error for BSC channel If d is even e: the first error probability of mode B : denotes the number of paths corresponding to the set of distance

f: a upper bound