### abstract

- 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 IX̂ n → X n ) is achievable for any n , where the minimization is performed over the transition conditioning probability p(x̂ n |x n ) such that E [d(X n , X̂ 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.