Joint preprocessing and feedback strategies for perfectly reconstructing equalizers

In this paper we consider the transmission of discrete-valued data via a communication channel that is subject to (additive) noise with a known upper bound on its magnitude but otherwise completely unrestricted and unknown behavior. We consider a discrete-time setup and extend previous equalization strategies for perfect reconstruction by allowing linear 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 (possibly nonzero) 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 necessary and sufficient conditions for perfect reconstruction under either linear power-limited preprocessing or linear power- limited preprocessing along with linear feedback. We prove that in order to improve the conditions for perfect reconstruction, it is necessary that the feedback and preprocessing systems are unstable. We also consider the case when a Decision Feedback Equalizer (DFE) structure is imposed at the receiver and provide necessary conditions for improvements in the perfect reconstruction in terms of l¹ norms of appropriate maps. In addition, a procedure that results in parametric l¹ optimization is developed to design a DFE to improve the maximum tolerable noise bound.