A list-type reduced-constraint generalization of the Viterbi algorithm

Abstract

The Viterbi algorithm (VA), an optimum decoding rule for a <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</tex> -ary trellis code of constraint length <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</tex> , operates by taking the best survivor from each of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q^{K-1}</tex> lists of candidates at each decoding step. A generalized VA (GVA) is proposed that makes comparisons on the basis of a label of length <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L(L\leq K)</tex> . It selects, incorporating the notion of list decoding, the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</tex> best survivors from each of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q^{L-1}</tex> lists of candidates at each decoding step. Coding theorems for a discrete memoryless channel are proved for GVA decoding and shown to be natural generalizations of those for VA decoding. An example of intersymbol interference removal is given to illustrate the practical benefits that the GVA can provide.