On Real-Time Communication Systems with Noisy Feedback

Abstract — We consider a real--time communication system with noisy feedback consisting of a Markov source, a forward and a backward discrete memoryless channels, and a receiver with finite memory. The objective is to design an optimal communication strategy (that is, encoding, decoding, and memory update strategies) to minimize the total expected distortion over a finite horizon. We present a sequential decomposition for the problem, which results in a set of nested optimality equations to determine optimal communication strategies. This provides a systematic methodology to determine globally optimal joint source--channel encoding and decoding strategies for real--time communication systems with noisy feedback. I. Problem Formulation Consider a real--time communication system with noisy feedback as shown in Figure 1. This system consists of a source, a real--time encoder, a noisy forward channel, a noisy backward channel, and a real--time decoder with finite memory. The communication system operates in discrete time for a time horizon T