Computable Bounds for Rate Distortion With Feed Forward for Stationary and Ergodic Sources

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 R_n(D) = 1/n min IX̂ⁿ → Xⁿ) is achievable for any n, where the minimization is performed over the transition conditioning probability p(x̂ⁿ|xⁿ) such that E [d(Xⁿ, X̂ⁿ] ≤ D. We also show that the limit of R_n(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 R_n(D) using the alternating minimization procedure and present several numerical examples. We also present a dual form for the optimization of R_n(D) and transform it into a geometric programming problem.