Nonanticipative Rate Distortion Function and Filtering Theory: A weak Convergence Approach
For information theory and control communities, this work provides a theoretical foundation for real-time joint source-channel coding, though the result is incremental as it extends known concepts to general Polish spaces.
The paper establishes a relation between nonanticipative rate distortion function and Bayesian filtering theory on Polish spaces, proving existence of optimal reproduction distributions via weak convergence. It demonstrates optimality of linear encoders for multidimensional partially observable sources over scalar Gaussian channels, achieving joint source-channel coding in real-time.
In this paper the relation between nonanticipative rate distortion function (RDF) and Bayesian filtering theory is further investigated on general Polish spaces. The relation is established via an optimization on the space of conditional distributions of the so-called directed information subject to fidelity constraints. Existence of the optimal reproduction distribution of the nonanticipative RDF is shown using the topology of weak convergence of probability measures. Subsequently, we use the solution of the nonanticipative RDF to present the realization of a multidimensional partially observable source over a scalar Gaussian channel. We show that linear encoders are optimal, establishing joint source-channel coding in real-time.