Optimal Zero Delay Coding of Markov Sources: Stationary and Finite Memory Codes

TL;DR

This paper proves that for irreducible, aperiodic finite alphabet Markov sources, deterministic stationary Markov codes are optimal for infinite horizon zero delay coding, with finite horizon performance approaching the infinite horizon optimum at rate O(1/T).

Contribution

It establishes the optimality of deterministic stationary Markov coding policies for infinite horizon zero delay coding of Markov sources, extending previous results.

Findings

Deterministic stationary Markov codes are optimal for infinite horizon.

Finite horizon performance approaches infinite horizon optimum at rate O(1/T).

Results extend to noisy channels with feedback.

Abstract

The optimal zero delay coding of a finite state Markov source is considered. The existence and structure of optimal codes are studied using a stochastic control formulation. Prior results in the literature established the optimality of deterministic Markov (Walrand-Varaiya type) coding policies for finite time horizon problem, and the optimality of both deterministic nonstationary and randomized stationary policies for the infinite time horizon problem. Our main result here shows that for any irreducible and aperiodic Markov source with a finite alphabet, \emph{deterministic and stationary} Markov coding policies are optimal for the infinite horizon problem. In addition, the finite blocklength (time horizon) performance on an optimal (stationary and Markov) coding policy is shown to approach the infinite time horizon optimum at a rate $O(1/T)$. The results are extended to systems where zero delay communication takes place across a noisy channel with noiseless feedback.