Exact error probability for convolutional codes with soft decision Viterbi decoding

The Viterbi decoding algorithm was proposed as a scheme of decoding convolutional codes, and realizes maximum likelihood decoding. Although a large number of studies have been made on its applications, little is known about its exact error performance. Actually, most of results of error performance studies are based on a bound of error performance or by computer simulation. Only few studies have so far been made for exact analyses, but these results were discussed for hard decisions. In general, it is difficult to exactly analyze the error performance of convolutional codes for soft decisions. The error performance for a high signal-to-noise power ratio (SNR) can be obtained by the error bound, however, little attention has been devoted to the exact error performance for low SNR. In this paper, we propose a new technique for the exact analysis of state probability of convolutional codes with Viterbi decoding. This technique employs an iterative calculation of the probability density function (pdf). The proposed technique gives the exact state probability for low SNR. It can be applied for soft decisions