A Further Note on an Innovations Approach to Viterbi Decoding of Convolutional Codes

In this paper, we show that the soft-decision input to the main decoder in an SST Viterbi decoder is regarded as the innovation as well from the viewpoint of mutual information and mean-square error. It is assumed that a code sequence is transmitted symbol by symbol over an AWGN channel using BPSK modulation. Then we can consider the signal model, where the signal is composed of the signal-to-noise ratio (SNR) and the equiprobable binary input. By assuming that the soft-decision input to the main decoder is the innovation, we show that the minimum mean-square error (MMSE) in estimating the binary input is expressed in terms of the distribution of the encoded block for the main decoder. It is shown that the obtained MMSE satisfies indirectly the known relation between the mutual information and the MMSE in Gaussian channels. Thus the derived MMSE is justified, which in turn implies that the soft-decision input to the main decoder can be regarded as the innovation. Moreover, we see that the input-output mutual information is connected with the distribution of the encoded block for the main decoder.