  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.
