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Expanding the Viterbi Algorithm

Creating the Branch Metrics

Viterbi Decoder Implementation

3-7

Thus, by using a shifted version of the branch metrics, we only need one branch metric

for each state pair update. This metric is the one that gets subtracted from one path and

added into the other. We need only generate and store 16 branches.

Any given branch will, for a specified input, have one of four values. The four values

will represent how well the decoder input compares to the recreated encoder outputs of

00, 01, 10, 11. For our example, we assume some soft decision input. The decoder input

data is shifted and scaled so that 1 is changed to 16, and 0 is changed to -16. We obtain

two partial branch metrics B0 and B1, where B0 is the partial branch metric for the

encoder polynomial 1+D+D3+D5 output, and B1 is the partial branch metric for the

1+D2+D3+D4+D5 polynomial output. We can then compute the four branches as B0+B1,

B0-B1, -B0+B1, and -B0-B1, corresponding to recreated encoder values of

11, 10, 01, and 00.

The butterfly loop needs the branches in a specific order dictated by the encoding

polynomials. As seen in Figure 2-2, we update states 2i and 2i+1 using states i and i+16.

We only need the 16 branches for the 16 state pairs. The state gives us five of the six

values needed to determine the encoder output. The remaining value can be found by

examining the butterfly loop assembly code. In the loop, note that we subtract the

branch metric first (from the upper branch), then add it to the lower branch. The

butterfly loop is written assuming that the single branch metric value has the correct

sign to be added in the lower branch. This is equivalent to assuming the encoder input

bit is a 1 (for the purposes of computing the branch metric table). We account for this in

the code by choosing positive 16 for 1Õs and minus 16 for 0Õs (as noted in the preceding

paragraph). The resulting recreated encoder outputs for the first 16 states are shown in

the following table.

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