1A Closed Form Expression for the Exact Bit Error Probability for Viterbi Decoding of Convolutional Codes

Abstract—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 (2-state) convolutional encoder with generator matrix G(D) = (1 1 + D) when used to communicate over the binary symmetric channel. Their formula was later extended to the rate R = 1/2, memory m = 2 (4-state) convolutional encoder with generator matrix G(D) = (1 +D2 1 +D +D2) by Lentmaier et al. In this paper, a different approach to derive the exact bit error probability is described. A general recurrent matrix equation, connecting the average information weight at the current and previous states of a trellis section of the Viterbi decoder, is derived and solved. The general solution of this matrix equation yields a closed form expression for the exact bit error probability. As special cases, the expressions obtained by Best et al. for the 2-state encoder and by Lentmaier et al. for a 4-state encoder are obtained. The closed form expression derived in this paper is evaluated for various realizations of encoders, including rate R = 1/2 and R = 2/3 encoders, of as many as 16 states. Moreover, it is shown that it is straightforward to extend the approach to communication over the quantized additive white Gaussian noise channel. Index Terms—additive white Gaussian noise channel, binary symmetric channel, bit error probability, convolutional code, con-volutional encoder, exact bit error probability, Viterbi decoding I