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 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.