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

We consider moduloadditive noise channels, where the noise process is a stationary irreducible and aperiodic Markov chain of order . We begin by investigating the capacity-cost function ( ( )) of such additive-noise channels without feedback. We establish a tight upper bound to ( ( )) which holds for general (not necessarily Markovian) stationary -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 ( ) 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 ( ). 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 ( ( )). We show (both analytically and numerically) that for a particular feedback encoding strategy and a class of Markov noise sources, the lower bound to ( ) is strictly greater than ( ). This demonstrates that feedback can increase the capacity-cost function of discrete channels with memory.