An Information-Theoretic Derivation of the MMSE Decision-Feedback Equalizer

We consider the discrete-time Gaussian channel with inter-symbol interference (ISI). Under the assumption of perfect feedback, an information-theoretic derivation of the minimum mean-squared error (MMSE) decision-feedback equalizer (DFE) is presented. Whereas previous works have used information theory to analyze the zero-forcing and MMSE DFE structures, this work derives the MMSE DFE directly by means of information-lossless projections applied to the average mutual information of the Gaussian ISI channel. Specifically, the perfect-feedback MMSE DFE works by performing an information-lossless conversion of the ISI channel, which must be viewed in a sequence-wise manner, into a sequence of inputoutput pairs that must be viewed symbol-wise. With ideal interleaving, the latter can also be viewed as a memoryless Gaussian channel. This equivalence resolves the paradoxical result that perfect post-cursor ISI cancellation is in general an information-increasing operation. We find that it is not perfect cancellation that increases the mutual information, but rather the mathematically inconsistent assumption that the perfect-feedback MMSE DFE can be viewed as a sequence-wise channel with memory.