On the Feedback Capacity of Power-Constrained Gaussian Noise Channels With Memory

Abstract

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> For a stationary additive Gaussian-noise channel with a rational noise power spectrum of a finite-order <emphasis><formula formulatype="inline"> <tex>$L$</tex></formula></emphasis>, we derive two new results for the feedback capacity under an average channel input power constraint. First, we show that a very simple feedback-dependent Gauss–Markov source achieves the feedback capacity, and that Kalman–Bucy filtering is optimal for processing the feedback. Based on these results, we develop a new method for optimizing the channel inputs for achieving the Cover–Pombra block-length- <emphasis><formula formulatype="inline"><tex>$n$</tex></formula></emphasis> feedback capacity by using a dynamic programming approach that decomposes the computation into <emphasis><formula formulatype="inline"><tex>$n$</tex></formula></emphasis> sequentially identical optimization problems where each stage involves optimizing <emphasis><formula formulatype="inline"><tex>$O(L^{2})$</tex></formula></emphasis> variables. Second, we derive the explicit maximal information rate for stationary feedback-dependent sources. In general, evaluating the maximal information rate for stationary sources requires solving only a few equations by simple nonlinear programming. For first-order autoregressive and/or moving average (ARMA) noise channels, this optimization admits a closed-form maximal information rate formula. The maximal information rate for stationary sources is a lower bound on the feedback capacity, and it equals the feedback capacity if the long-standing conjecture, that stationary sources achieve the feedback capacity, holds. </para>