Filtering with fidelity for time-varying Gauss-Markov processes

In this paper, we revisit the relation between Nonanticipative Rate Distortion (NRD) theory and real-time realizable filtering theory. Specifically, we give the closed form expression for the optimal nonstationary (time-varying) reproduction distribution of the Finite Time Horizon (FTH) Nonanticipative Rate Distortion Function (NRDF) and we establish its connection to real-time realizable filtering theory via a realization scheme utilizing time-varying fully observable multidimensional Gauss-Markov processes. As an application we provide the optimal filter with respect to a mean square error constraint. Unlike classical filtering theory, our filtering approach based on FTH NRDF is performed with waterfilling. We also derive a universal lower bound to the mean square error of any causal estimator to Gaussian processes based on the closed form expression of FTH NRDF. Our theoretical results are demonstrated via an illustrative example