On the Independence Theory of Equalizer Convergence

Adaptive equalization of telephone channels in order to facilitate high-speed data transmission has been successful ever since its introduction by Lucky in the 1960s. This technique uses a linear filter (configured as a tapped delay line) to ramove the harmful effects of the linear channel distortion. At the start of the equalization procedure, a set of parameters, the tap weights, are adjusted so that the final setting of these taps minimizes the intersymbol interference between pulses in the data train. Many theoretical studies have been made concerning steady-state equalization after the optimum tap weights have been achieved; little analysis has been done concerning the convergence of the equalizer tap weights to their final settings. Even in the best published study on this problem (Ungerboeck, Ref. 1), it is necessary to invoke an assumption stating that a sequence of random vectors which direct the operation of the equalizer are statistically independent.-!" This independence assumption will be explained more fully later; for the moment, we only indicate that it is not even approximately true. In fact, given the nth vector of the sequence, all but one component of the next vector will be exactly known. Yet if this assumption is made, surprising agreement with actual performance is obtained.1 Clearly, because of its importance, this situation begs for clarification. Hopefully, what we learn in equalization can be used for other applications where similar adaptive algorithms are used.