On Optimality Test in Low Complexity Maximum Likelihood Decoding of Convolutional Codes

This paper considers the average complexity of maximum likelihood (ML) decoding of convolutional codes. ML decoding can be modeled as finding the most probable path taken through a Markov graph. Integrated with the Viterbi algorithm (VA), complexity reduction methods such as the sphere decoder often use the sum log likelihood (SLL) of a Markov path as a bound to disprove (or test) the optimality of other Markov path sets and to consequently avoid exhaustive path search. In this paper, it is shown that the SLL-based optimality tests are inefficient if one fixes the coding memory and takes the codeword length to infinity. Alternatively, the optimality of a source symbol at a given time index can be verified (or tested) using bounds based on partial log likelihood (PLL) of channel output symbols in a fixed-sized time neighborhood. The paper theoretically demonstrates that PLL-based optimality tests, whose efficiency does not depend on the codeword length, can bring significant complexity reduction to ML decoding of convolutional codes. The results are generalized to ML sequence detection in a class of discrete-time hidden Markov systems.