Constrained DFEs for Reduced Error Propagation: Theoretical Results and an Adaptive Soft Error Variance Controlled DFE

Abstract

The problem of error propagation in a decision feedback equalizer (DFE) is acknowledged to be an important factor in the functioning of the equalizer. In order to determine conditions which guarantee the non-propagation of errors we formulate the DFE as a dynamical system in the form of a matrix equation. We analyse the propagation of errors in this matrix system and formulate conditions on the system matrices which guarantee: a) Using deterministic analysis, the non-propagation of errors; b) Using a stochastic analysis, the non-propagation of the expected value of the norm of the hard error vector. These conditions translate into constraints on the size of the feedback tap-weights so that there is linear convergence of the errors (or of expected values of norms of errors for the stochastic analysis) with constant C < 1. This is equivalent to exponential decay of the hard errors (or of expected values of norms of errors for the stochastic analysis) over time with decay constant C < 1. This ensures that errors do not propagate, but decay exponentially with time.The stochastic analysis gives weaker constraints which may be of more practical use in filter design, as the deterministic constraint guards against a worst case scenario for error propagation. The constraints derived in the stochastic case also combine the variance of the soft errors. Hence feedback of soft error variance may be used to adapt the constraint used on the feedback filter, and this motivates an algorithm for a soft error variance controlled adaptive DFE.