Sequential Necessary and Sufficient Conditions for optimal channel input distributions of channels with memory and feedback

Abstract

We derive Sequential Necessary and Sufficient Conditions (SNSC) for any channel input distribution equation to maximize directed information for channel distributions of the form equation, where Xn t {X0., Xn} and Yn t {Y0., Yn} are the channel input and output random variables, and M is nonnegative and finite. The results are obtained using the information structures of the optimal channel input distributions and the corresponding Finite Transmission Feedback Information (FTFI) capacity, convexity properties of directed information, and dynamic programming recursions. The conditions are applied to a finite alphabet channel with M = 1 to derive recursive closed form expressions for the optimal (nonstationary) distributions, which achieve the FTFI capacity. Further, ergodic feedback capacity is obtained in closed form, using the asymptotic properties of the optimal distributions. A numerical example is presented to illustrate the convergence properties of the per unit time limiting version of the FTFI capacity. © 2016 IEEE.