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The problem of channel shortening equalization for optimal detection in ISIchannels is considered. The problem is to choose a linear equalizer and apartial response target filter such that the combination produces the bestdetection performance. Instead of using the traditional approach of MMSEequalization, we directly seek all equalizer and target pairs that yieldoptimal detection performance in terms of the sequence or symbol error rate.This leads to a new notion of a posteriori equivalence between the equalizedand target channels with a simple characterization in terms of their underlyingprobability distributions. Using this characterization we show the surprisingexistence an infinite family of equalizer and target pairs for which anymaximum a posteriori (MAP) based detector designed for the target channel issimultaneously MAP optimal for the equalized channel. For channels whose inputsymbols have equal energy, such as q-PSK, the MMSE equalizer designed with amonic target constraint yields a solution belonging to this optimal family ofdesigns. Although, these designs produce IIR target filters, the ideas areextended to design good FIR targets. For an arbitrary choice of target andequalizer, we derive an expression for the probability of sequence detectionerror. This expression is used to design optimal FIR targets and IIR equalizersand to quantify the FIR approximation penalty.

A Posteriori Equivalence: A New Perspective for Design of Optimal Channel Shortening Equalizers