A lower bound on the bit‐error‐rate resulting from mismatched viterbi decoding

Abstract

AbstractA lower bound is derived on the bit error rate that results when a rate l/n convolutionally encoded binary data stream is transmitted over a noisy symmetric channel, and is then decoded using a mismatched Viterbi decoder, i.e., a Viterbi decoder that performs maximum‐likelihood sequence estimation using possibly incorrect branch metrics. The branch metrics are assumed to be symmetric, but are generally differrent from the log‐likelihood function. The proposed bound is expressed in terms of the pairwise error probability for a given error event normalized by the sum of the events length and the code's memory. While the bound can be somewhat loose for medium signal‐to‐noise ratios, it is usually asymptotically tight at high signal‐to‐noise ratios. For the special case where the branch metrics employed are equal to the log‐likelihood function our lower bound does not coincide with the standard bound that was derived by Forney and Mazo. It falls short by a multiplicative constant that depends on the code's memory and the shortest error event of minimum Hamming weight. We apply our lower bound to the study of convolutionally encoded direct‐sequence spread‐spectrum communication with Laplacian noise and show that nearest‐neighbor decoding, which is optimal for Gaussian noise, is sub‐optimal and asymptotically (for high processing gain) results in a 3 dB loss in performance when compared with the optimal maximum‐likelihood decoder. Extensions to rate c/n convolutional encoders, general trellis encoders and to symmetric channels with memory are also presented.