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
<sub>nrarr

infin</sub>
sup
_p(x

ⁿ

_||y

^n-1

₎
1/n I(X
ⁿ
rarr Y
ⁿ
) where I(X
ⁿ
rarr Y
ⁿ
) = Sigma
_i=1

ⁿ
I(X
ⁱ
; 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(x
ⁿ
||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.