Computable Bounds for Rate Distortion with Feed-Forward for Stationary and Ergodic Sources

الأصل · EN

In this paper we consider the rate distortion problem of discrete-time, ergodic, and stationary sources with feed forward at the receiver. We derive a sequence of achievable and computable rates that converge to the feed-forward rate distortion. We show that, for ergodic and stationary sources, the rate align Rₙ(D)=1/n I(Xⁿ→ Xⁿ)align is achievable for any n, where the minimization is taken over the transition conditioning probability p(xⁿ|xⁿ) such that d(Xⁿ,Xⁿ)≤ D. The limit of Rₙ(D) exists and is the feed-forward rate distortion. We follow Gallager's proof where there is no feed-forward and, with appropriate modification, obtain our result. We provide an algorithm for calculating Rₙ(D) using the alternating minimization procedure, and present several numerical examples. We also present a dual form for the optimization of Rₙ(D), and transform it into a geometric programming problem.