Optimal realizable MMSE linear and decision feedback equalizers: time domain results

Motivated by the fact that the Kalman filter is the optimal linear estimator, we apply Kalman filter theory to the study of realizable minimum mean square error (MMSE) linear and decision feedback equalizers. We establish state-space realizations of linear equalizers (LEs) and decision feedback equalizers (DFEs), based on which the Kalman filter can be directly applied to find the optimal estimates of the transmitted symbols. This state space approach yields optimal realizable LE and DFE. More importantly, it provides insights into the properties of the LE and DFE that are not clearly seen otherwise. We show that, for both LE and DFE, increasing the detection delay results in smaller estimation error at the expense of higher complexity. It is shown that the Kalman filter based LE is equivalent to an MMSE LE of infinite filter length. However, to our surprise, the Kalman filter based DFE is shown to be equivalent to a finite length MMSE DFE of orders bounded by the detection delay and the channel length. All the derivations are carried out in the time domain. Thus, the results obtained are applicable to time-invariant channels and to time-varying channels, which are commonly encountered in mobile communications.