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 lim_{nrarr infin} sup_p(x ⁿ _||y ^n-1 ₎ 1/n I(Xⁿ rarr Yⁿ) 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(xⁿ||y^n-1) = Pi_i=1 ⁿ p(x_i|x^i-1,y^i-1). The main ideas of the proof are a classical application of 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.