Analytic expressions for the bit error probabilities of rate-1/2 memory 2 convolutional encoders

Analytic expressions for the exact bit error probabilities of rate R = 1/2, memory m = 2 convolutional encoders are derived for a maximum-likelihood (ML) decoder and transmission over the binary-symmetric channel (BSC). The resulting expressions are rational functions of the crossover probability of the BSC. In addition to classical nonsystematic encoders without feedback, we consider also recursive systematic encoders which became especially important as component encoders in concatenated coding schemes. To attest the validity of the results, they are compared to computer simulations. Based on the presented technique also the bit error probability and the probability distribution of the output log-likelihood ratios of the Max-Log-MAP algorithm are derived in analytic form.