On saving decoder states for some trellis codes and partial response channels

Wolf and Ungerboeck [ 11 considered constructing good trellis codes for transmitting information over a partial response channel where the decoder is matched both to the redundancy of the code and to the intersymbol interference generated by the channel. A method of constructing these trellis codes suggested by Wolf and Ungerboeck was to utilize a binary convolutional code with 2 (I states and then to pass the encoded output sequence through a precoder prior to transmission over a partial response channel. In particular, for a class IV partial response channel [2] with transfer function (1 0'). Wolf and Ungerboeck suggested treating the channel as the interleaving of two (1 D) channels and using a precoder with transfer function (1 d D)-'. In that case, the decoder would in general require 2 " + states. Here, we show that for a class of convolutional codes [with a (1 Q D)-I precoder and a (1 D) channel] the number of states in the decoder matched to the code, precoder, and channel is only 2'Within the class are several well-known codes including the rate 112 convolutional code of constraint length 7 and free distance 10. For this code, we will show how to modify the @-state Viterbi decoder matched to the code so that it matches both the code, precoder, and the intersymbol interference of the (1 D) channel. The codes discussed in this paper have similar properties to the trellis codes found by other authors [3]-[4]. The approach taken here, however, is different and is a natural generalization of the work of Wolf and Ungerboeck [I]. In particular, the results of this paper give further evidence to the remarkable fact that good binary convolutional codes for the binary symmetric channel, when properly applied, yield good binary convolution codes for the 1 k D [or 1 f Dn] partial response channel. An outline of this correspondence follows. In the next section, we summarize the applicable work of Wolf and Ungerboeck and describe the class of codes for which we can save decoder states. This is followed by a section containing examples of codes in the class. Included in this section are details of the decoder design for a rate 1/2 constraint length 7 code of free Hamming distance 10. Finally, we conclude with a brief summary.