An Innovations Approach to Viterbi Decoding of Convolutional Codes

We introduce the notion of innovations for Viterbi decoding of convolutional codes. First we define a kind of innovation corresponding to the received data, i.e., the input to a Viterbi decoder. Then the structure of a Scarce-State-Transition (SST) Viterbi decoder is derived in a natural manner. It is shown that the newly defined innovation is just the input to the main decoder in an SST Viterbi decoder and generates the same syndrome as the original received data does. A similar result holds for QLI codes as well. In this case, however, the precise innovation is not defined. We see that this innovation-like quantity is related to the linear smoothed estimate of the information. The essence of innovations approach to a linear filtering problem is first to whiten the observed data, and then to treat the resulting simpler white-noise observations problem. In our case, this corresponds to the reduction of decoding complexity in the main decoder in an SST Viterbi decoder. We show that the distributions related to the main decoder in an SST Viterbi decoder (i.e., the input distribution and the state distribution in the code trellis) are much biased under moderately noisy conditions. Note that the state distribution in the corresponding error trellis is also biased under the same channel conditions. These biased distributions enable decoding with remarkably low average complexity. Index Terms Convolutional codes, Viterbi decoding, innovations, linear filtering, linear smoothing, Scarce-State-Transition (SST) Viterbi decoder. M. Tajima was with the graduate School of Science and Engineering, University of Toyama, 3190 Gofuku, Toyama 930-8555, Japan (e-mail: [email protected]). Manuscript received April 19, 2005; revised August 26, 2015. JOURNAL OF LTEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 1 An Innovations Approach to Viterbi Decoding of Convolutional Codes