An information theoretic charachterization of channel shortening receivers

Optimal data detection of a linear channel can always be implemented through the Viterbi algorithm (VA). However, in many cases of interest the memory of the channel prohibits application of the VA. A popular and conceptually simple method in this case, studied since the early 70s, is to first filter the received signal in order to shorten the memory of the channel, and then to apply a VA that operates with the shorter memory. We shall refer to this as a channel shortening (CS) receiver. Although studied for almost four decades, an information theoretic understanding of what such simple receiver solution is actually doing is not available. In this paper we will show that an optimized CS receiver is implementing the chain rule of mutual information, but only up to the shortened memory that the receiver is operating with. Further, we will show that the tools for analyzing the ensuing achievable rates from an optimized CS receiver are precisely the same as the tools that are used for analyzing the achievable rates of an minimum mean square error (MMSE) receiver.