The capacity-cost function of discrete additive noise channels with and without feedback

We consider modulo-q additive noise channels, where the noise process is a stationary irreducible and aperiodic Markov chain of order k. We begin by investigating the capacity-cost function (C(β)) of such additive-noise channels without feedback. We establish a tight upper bound to (C(β)) which holds for general (not necessarily Markovian) stationary q-ary noise processes. This bound constitutes the counterpart of the Wyner-Ziv lower bound to the rate-distortion function of stationary sources with memory. We also provide two simple lower bounds to C(β) which along with the upper bound can be easily calculated using the Blahut algorithm for the computation of channel capacity. Numerical results indicate that these bounds form a tight envelope on C(β). We next examine the effect of output feedback on the capacity-cost function of these channels and establish a lower bound to the capacity-cost function with feedback (C_FB(β)). We show (both analytically and numerically) that for a particular feedback encoding strategy and a class of Markov noise sources, the lower bound to C_FB(β) is strictly greater than C(β). This demonstrates that feedback can increase the capacity-cost function of discrete channels with memory