U. Glaeser

Now the task of detecting i [0,∞) is to find x [0,∞) that is closest to y [0,∞) in the Euclidean sense. Recall that
we stated as an assumption that channel noise is AWGN, while in magnetic recording systems after
equalization the noise is colored so that the minimum-distance detector is not an optimal one, and
additional post-processing is necessary, which will be addressed later in this chapter.
The Viterbi algorithm is a classical application of dynamic programming. Structurally, the algorithm
contains q M lists, one for each state, where the paths whose states correspond to the label indices are stored,
compared, and the best one of them retained. The algorithm can be described recursively as follows:
1. Initial condition: Initialize the starting list with the root node (the known initial state) and set its
metric to zero, l = 0.
2. Path extension: Extend all the paths (nodes) by one branch to yield new candidates, l = l + 1, and
find the sum of the metric of the predecessor node and the branch metric of the connecting branch
(ADD). Classify these candidates into corresponding q M lists (or less for l < M). Each list (except
in the head of the trellis) contains q paths.
3. Path selection: For each end-node of extended paths determine the maximum/minimum* of these
sums (COMPARE) and assign it to the node. Label the node with the best path metric to it, selecting
(SELECT) that path for the next step of the algorithm (discard others). If two or more paths have
the same metric, i.e., if they are equally likely, choose the best one at random. Find the best of all
the survivor paths, x′ , and its corresponding information sequence i ′ and release the bit i′ .
[0,l )
[0,l )
l−δ
Go to step 2.
In the description of the algorithm we emphasized three Viterbi-characteristic operations—add, compare,
select (ADC)—that are performed in every recursion of the algorithm. So today’s specialized signal
processors have this operation embedded optimizing its execution time. Consider now the amount of
“processing” done at each depth l, where all of the q M states of the trellis code are present. For each state
it is necessary to compare q paths that merge in that state, discard all but the best path, and then compute
and send the metrics of q of its successors to the depth l + 1.
Consequently, the computational complexity of the VA exponentially increases with M. These operations
can be easily parallelized, but then the number of parallel processors rises as the number of node
computations decreases. The total time-space complexity of the algorithm is fixed and increases exponentially
with the memory length.
The sliding window VA decodes infinite sequences with delay of δ branches from the last received one.
In order to minimize its memory requirements (δ + 1 trellis levels), and achieve bit error rate only
insignificantly higher than with finite sequence VA, δ is chosen as δ ≈ 4M. In this way, the Viterbi detector
introduces a fixed decision delay.
Example
Assume that a recorded channel input sequence x, consisting of L equally likely binary symbols from the
alphabet {0, 1}, is “transmitted” over PR4 channel. The channel is characterized by the trellis of Fig. 34.57,
i.e., all admissible symbol sequences correspond to the paths traversing the trellis from l = 0 to l = L,
with one symbol labeling each branch, Fig. 34.58. Suppose that the noisy sequence of samples at the
channel output is y = 0.9, 0.2, –0.6, –0.3, 0.6, 0.9, 1.2, 0.3,… If we apply a simple symbol-by-symbol
detector to this sequence, the fifth symbol will be erroneous due to the hard quantization rule for noiseless
channel output estimate
*It depends on whether the metric or the distance is accumulated.
© 2002 by CRC Press LLC
yˆ k
=
⎧
⎪
⎨
⎪
⎩
– 1 yk < – 0.5
1 yk > 0.5
0 otherwise