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

For a stationary additive Gaussian-noise channel with a rational noise power spectrum of a finite-order $L$, 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- $n$ feedback capacity by using a dynamic programming approach that decomposes the computation into $n$ sequentially identical optimization problems where each stage involves optimizing $O(L^{2})$ 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.