MIMO Linear Equalization with an H ∞ Criterion

In this paper we study the problem of linearly equalizing the Multiple Input Multiple Output (MIMO) communications channels from an H∞ point of view. H∞ estimation theory has been recently introduced as a method for designing filters that have acceptable performance in the face of model uncertainty and lack of statistical information on the exogenous signals. In this paper we obtain a closed-form solution to the square MIMO linear H∞ equalization problem and parameterize all possible H-optimal equalizers. In particular, we show that, for minimum phase channels, a scaled version of the zero-forcing equalizer is H-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. Our analysis also suggests certain remedies in the non-minimum phase case, namely, allowing for finite delay, oversampling, or using multiple sensors. For example, we show that H∞ equalization of non-minimum phase channels requires a time delay of at least l units, l is the number of non-minimum phase zeros of the channel.