Global Optimality of Encoders and MSE Decoders for Communicating Unstable Markov Processes over Unstable Gaussian Recursive Models with Feedback: A Nonanticipative RDF Approach

TL;DR: In this article, the authors derived analogous results for communicating unstable Gaussian Markov processes over unstable MIMO Gaussian recursive models (G-RM) with memory, subject to an average cost of quadratic form.

Abstract: Shannon’s coding capacity of memoryless additive Gaussian noise (AGN) channels with noiseless feedback, is known to be achieved by the Elias [1] coding scheme of communicating the mean square-error (MSE) of a Gaussian RV $X \sim N\left({0,\sigma _X^2}\right)$, from past channel outputs, and decoding it using a MSE decoder. Further, it is known that among all encoders, and decoders that minimize the MSE, then the Elias encoder and decoder are globally optimal.In this paper we derive analogous results, for communicating unstable Gaussian Markov processes over unstable multiple-input multiple-output (MIMO) Gaussian recursive models (G-RM) with memory, often called infinite impulse response (IIR) models, subject to an average cost of quadratic form. However, unlike memoryless AGNs, certain conditions are required, for such generalizations. Further, to show global optimality, we need to invoke Gorbunov and Pinsker [2] nonanticipatory entropy, instead of the classical rate distortion function of the source. Another important observation is that we need a two-parameter coding scheme, instead of the one-parameter coding scheme of memoryless AGN channels.