#### On Low Complexity Maximum Likelihood Decoding of Convolutional Codes

##### Jie Luo

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 the optimality of other Markov path sets and to consequently avoid exhaustive path search. In this paper, it is shown that SLL-based optimality tests are inefficient if one fixes the coding memory and takes the codeword length to infinity. Alternatively, optimality of a source symbol at a given time index can be testified using bounds derived from log likelihoods of the neighboring symbols. It is demonstrated that such neighboring log likelihood (NLL)-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.

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