A Coding Theorem for a Class of Stationary Channels with Feedback

A coding theorem is proved for a class of stationary channels with ´feedback in which the output Y_n = f(X_n-m ⁿ, Z_n-m ⁿ) is the function of the current and past m symbols from the channel input X_n and the stationary ergodic channel noise Z_n. In particular, it is shown that the feedback capacity is equal to where I(Xⁿ rarr Yⁿ) = Sigma_i=1 ⁿ I(Xⁱ;Y_i|Y^i-1) denotes the Massey directed information from the channel input to the output, and the supremum is taken over all causally conditioned distributions p(xⁿpary^n-1) = Pi_i=1 ⁿp(x_i|x^i-1,y^i-1)- The main ideas of the proof are the Shannon strategy for coding with side information and a new elementary coding technique for the given channel model without feedback, which is in a sense dual to Gallager´s lossy coding of stationary ergodic sources. A similar approach gives a simple alternative proof of coding theorems for finite state channels by Yang-Kavcic-Tatikonda, Chen-Berger, and Permuter-Weissman-Goldsmith.