U. Glaeser

The basic problem with the algorithm is that it requires both large storage and great number of
computations. All the values of α l(i) must be stored, which requires almost (L + M)q KM memory locations.
The number of multiplications required for determining the α l(i) and β l(i) for each l is q M+1 , and there
are q M additions of q K numbers as well. The computation of γ l(i, j) is not costly and can be accomplished
by a table look-up. Finally, the computation of all σ l(i, j) requires q (M+1)+1 multiplications for each l, and
q − 1 comparisons in choosing the largest i l. Consequently, this is an algorithm with exponential complexity
and in practice can be applied only when M and L are short. Nevertheless, it is used for iterative decoding
where such requirements can be fulfilled, such as for turbo codes. The main advantage of the algorithm
in such cases is its ability to estimate P[s l +1 = j|s l = i], which for the possible transitions equals q −1 only
in the first iteration.
The SOVA Algorithm
The soft-output Viterbi algorithm (SOVA) [15] is a modification of the VA that was designed with the
aim of estimating the reliability of every detected bit by the VA. It is applicable only when q = 2. The VA
is used here in its sliding window form, which detects infinite sequences with delay of δ branches from
the last received one.
The reliability (or soft value) of a bit i, L(i), is defined as L(i) = ln(P[i = 0]/P[i = 1]). The SOVA further
extends the third step in order to obtain this value, in the following way:
() j
Path selection (extension): Let i , j ∈ {0, 1,…,δ − 1} be the information sequences which merge
[ 0,l−j )
with i′ at depths l − j. Their paths have earlier been discarded due to their lower metrics. Let the
[ 0,l )
() j
corresponding metric differences in the merging states be denoted ∆j, and let J = { j:il−δ≠i′ l−δ } . Then
L( i′ l−δ ) ≈
( 1– 2i′ l−δ )minj∈J∆j. Because VA detecting metric can be modified in a way to take into account a priori knowledge of input
bit probabilities, the SOVA can be used as soft input-soft output (SISO) block in turbo decoding schemes.
References
1. A. J. Viterbi and J. Omura, Principles of Digital Communication and Coding, McGraw-Hill, Tokyo,
1979.
2. A. J. Viterbi, “Error bounds for convolutional codes and asymptotically optimum decoding algorithm,”
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final report, University of California, Electrical Eng. Department, July 1996.
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8. H. Kobayashi and D. T. Tang, “Application of partial response channel coding to magnetic recording
systems,” IBM J. Res. and Dev., vol. 14, pp. 368–375, July 1979.
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Theory, vol. IT-17(5), pp. 586–594, 1971.
10. J. B. Anderson and S. Mohan, “Sequential coding algorithms: a survey and cost analysis,” IEEE Trans.
Comm., vol. COM-32(2), pp. 169–176, Feb. 1984.
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codes,” IEEE Trans. Comm., vol. COM-41, pp. 370–380, Feb. 1993.
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© 2002 by CRC Press LLC