Joint Preprocessing and Feedback Strategies for Perfectly Reconstructing Equalizers

In this paper we consider the transmission of discrete-valued data via a communication chan-nel that is subject to (additive) noise with a known upper bound on its magnitude but otherwise completely unrestricted and unknown behavior. We extend previous equalization strategies for perfect reconstruction by allowing preprocessing of the data and/or linear feedback from the receiver to the transmitter. We are interested in the characterization of general conditions that allow perfect reconstruction of the discrete data (with any given delay and under all possible realizations of channel noise and a limit on the power of transmission) when linear preprocessing of the data and/or linear feedback from the receiver is employed. In particular, we obtain neces-sary and sufficient conditions for perfect reconstruction under linear power-limited preprocessing and under linear power-limited preprocessing along with linear feedback. We also consider the case when a Decision Feedback Equalizer (DFE) structure is imposed at the receiver and pro-vide necessary conditions for improvements in the perfect reconstruction in terms of!1 norms of appropriate maps. In particular, we prove that a necessary condition for improving the perfect reconstructability margin is instability of the feedback and preprocessing systems. In addition, a procedure that results in parametric!1 optimization is developed to design a DFE to improve maximum tolerable noise bound