Solved and Unsolved Problems in Adaptive Blind Equalization

This paper presents some results and conjectures dealing with channel equalization. For several possible equalizers, the following question is considered: if the equalizer output sequence has identical statistics to the input signal sequence, has the equalizer correctly converged? 1. THE PROBLEM DOMAIN Many communication signal processing problems envisage a transmitted sequence a0 a1, ... drawn from a finite signal alphabet, with passage of that sequence through a linear channel which produces intersymbol interference in the channel output sequence, call it {bk}. There is then possible addition of noise, followed by processing of bk to try to undo the effects of the channel, through a device termed an equalizer. The equalizer output {âk} is supposed to reconstruct the signal alphabet, eg âk=ak-δ for some fixed integer δ and all k. Many equalizers need to be adaptive, and often also blind, ie not even a training sequence of known {ak} is provided to allow equalization. In the blind case, adaptive equalizers rely either on being close to correct to begin with (eg open eye diagram) or computing certain statistics which are used to guide changes in equalizer parameters, such as in Godard blind equalizers [1] and decision directed equalizers [2]. When the parameters or taps in a blind adaptive equalizer settle down, ie cease making gross movements and remain in the neighbourhood of some point, it is reasonable to ask the question: has convergence occurred to a correct point. Such questions can be subtle. For example, for the Godard equalizers, with an infinite number of taps, the existence of incorrect equilibria was demonstrated in [1], but they were shown to be unstable (and thus irrelevant) in [3]. However, with finite equalizer length, stable undesirable equilibria were shown to exist in [4] (and with an 11 year interval from [1] to [4]). In this paper we study the issue of establishing whether convergence for certain equalizers has occurred, given that the equalizer output sequence has the same statistics as the signal sequence. The equalizers are decision directed and decision feedback equalizers, embodying linear system processing together with quantization. 2. DECISION DIRECTED EQUALIZERS The behaviour of decision directed equalizers after adaptation can often be described in the following way. The channel input sequence {ak} is a sequence of independent random variables taking values in an M-ary alphabet, with equal probability. We shall, for convenience, assume M=2, and ak ∈{±1}. The channel impulse response is {d0, d1.......}. In the absence of noise, the channel output is b d a k l l k l = = ∞