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 I ((X) over cap (n) -> X-n) is achievable for any n, where the minimization is performed over the transition conditioning probability p((x) over cap (n)vertical bar x(n)) such that E [d(X-n, (X) over cap (n))] <= 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.