Optimal data detection of data transmitted over 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 a 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 those used for analyzing the achievable rates of a minimum mean square
error (MMSE) receiver
