On the exact bit error probability for Viterbi decoding of convolutional codes

Forty years ago, Viterbi published upper bounds on both the first error event (burst error) and bit error probabilities for Viterbi decoding of convolutional codes. These bounds were derived using a signal flow chart technique for convolutional encoders. In 1995, Best et al. published a formula for the exact bit error probability for Viterbi decoding of the rate R = 1/2, memory m = 1 convolutional encoder with generator matrix G(D) = (1 1 + D) when used to communicate over the binary symmetric channel. Their method was later extended to the rate R = 1/2, memory m = 2 generator matrix G(D) = (1+D² 1 + D + D²) by Lentmaier et al. In this paper, we shall use a different approach to derive the exact bit error probability. We derive and solve a general matrix recurrent equation connecting the average information weights at the current and previous steps of the Viterbi decoding. A closed form expression for the exact bit error probability is given. Our general solution yields the expressions for the exact bit error probability obtained by Best et al. (m = 1) and Lentmaier et al. (m = 2) as special cases.