  The problem of channel shortening equalization for optimal detection in ISI
channels is considered. The problem is to choose a linear equalizer and a
partial response target filter such that the combination produces the best
detection performance. Instead of using the traditional approach of MMSE
equalization, we directly seek all equalizer and target pairs that yield
optimal detection performance in terms of the sequence or symbol error rate.
This leads to a new notion of a posteriori equivalence between the equalized
and target channels with a simple characterization in terms of their underlying
probability distributions. Using this characterization we show the surprising
existence an infinite family of equalizer and target pairs for which any
maximum a posteriori (MAP) based detector designed for the target channel is
simultaneously MAP optimal for the equalized channel. For channels whose input
symbols have equal energy, such as q-PSK, the MMSE equalizer designed with a
monic target constraint yields a solution belonging to this optimal family of
designs. Although, these designs produce IIR target filters, the ideas are
extended to design good FIR targets. For an arbitrary choice of target and
equalizer, we derive an expression for the probability of sequence detection
error. This expression is used to design optimal FIR targets and IIR equalizers
and to quantify the FIR approximation penalty.
