Equalization with an H1 Criterion

In this paper we study the problem of equalizing a communications channel from an H 1 point of view. H 1 estimation theory has been recently introduced as a method for designing lters that have acceptable performance in the face of model uncertainty and lack of statistical information on the exogenous signals. The approach may therefore be attractive for channel equalization problems where rapid changes in the system parameters lead to such model uncertainty, and where conventional statistical methods may not be directly applicable. In this paper we obtain a closed-form solution to the H 1 equalization problem and parametrize all possible H 1-optimal equalizers. In particular, we show that, for minimum phase channels, a scaled version of the zero-forcing equalizer is H 1-optimal. The results also indicate an interesting dichotomy between minimum phase and non-minimum phase channels: for minimum phase channels the best causal equalizer performs as well as the best noncausal equalizer, whereas for non-minimum phase channels, causal equalizers cannot reduce the estimation error bounds from their a priori values. In other words, causal equalization of non-minimum phase channels is not possible.] Our analysis also suggests certain remedies in the non-minimum phase case, namely, allowing for nite delay, oversampling, or using multiple sensors. For example, we show that H 1 equalization of non-minimum phase channels requires a time delay of at least d units, d is the number of non-minimum phase zeros of the channel.