Undermodeled equalization: A characterization of stationary points for a family of blind criteria

We attack specific problems related to equalizer performance in undermodele d cases in which assumptions of perfect equalizability are dismissed in fav or of a more realistic situation in which no equalizer setting may achieve perfect channel equalization. We derive a characterization of candidate con vergent points for a family of blind criteria which appeal, tacitly or witt ingly, to maximizing the ratio of different sequence norms of the combined channel-equalizer impulse response. This may be accomplished in a practical implementation by using equalizer output cumulants of different orders. Th e popular Godard and Shalvi-Weinstein schemes are accommodated at one extre me of the family of criteria. We also show that each maximum at the other e xtreme of the family, involving progressively higher order output cumulants , yields, precisely, a Wiener response, This suggests that blind algorithms using progressively higher order statistics may converge more closely to a Wiener response than those using more modest order statistics. We show, mo reover, that the superexponential family of algorithms is also included and establish a convergence proof for undermodeled cases that appeals to no ap proximation. Finally, some apparently novel bounds on attainable open-eye m easures in undermodeled cases are also derived.