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

Abstract

Abstract-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 non-linear 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.Index Terms-channel capacity, directed information, dynamic programming, feedback capacity, Gauss-Markov source, information rate, intersymbol interference, Kalman-Bucy filter, linear Gaussian noise channel, noise whitening filter

Feedback Stabilization Over a First Order Moving Average Gaussian Noise Channel

“…We show that the results on feedback capacity in [11] (and parallel results in [15]) and the SNR constrained stabilization results of [8] are linked for the case of an MA1 channel and relative degree one, minimum phase plant with a single unstable pole at z = . In this case stabilization within an SNR constraint is possible precisely when the feedback capacity of the channel, CFB (as in (2)) satisfies C FB log 2 (jj): (3) Moreover, if stabilization is possible, it can be achieved by a linear scheme.…”

“…These results focus primarily on autoregressive noise coloring, and use the linear coding structure of [12] to provide a lower bound on the feedback channel capacity. The authors of [15] discuss Kalman-Bucy filtering in relation to feedback communication over Gaussian channels with memory.…”

Capacity of the (1, ∞)-RLL input-constrained erasure channel with feedback

“…Subsequent works included the formulation of capacity as an infinite-horizon dynamic program for channels where the state is a function of the input [16], Markov channels [17] and Gaussian channels with memory [18]. To apply known algorithms from DP, such as Value and Policy iteration, quantization is required and, therefore, only lower bounds were derived in the above papers.…”

Delayed Feedback Capacity of Stationary Sources over Linear Gaussian Noise Channels

“…For channels with memory, bounds have been developed for the feedback capacity [3], [4], [5], [6], [7]. In [8], the optimal feedback source distribution is derived in terms of a state-space channel representation and Kalman filtering. The maximal information rate for stationary sources is derived in an analytically explicit form in [9].…”

“…The delayedfeedback information rate of the original Gaussian noise channel equals the instantaneous-feedback information rate of the derived state-space channel. By generalizing the methodology and results derived in [9], [8], we show that 1) a feedback-dependent Gauss-Markov source is optimal for achieving the delayed-feedback capacity, and the necessary Markov memory length equals the larger of a) the moving average (MA) noise spectral order, and b) the sum of the feedback delay and the autoregressive (AR) noise spectral order; 2) a state estimator (Kalman-Bucy filter) for the derived state-space channel model is optimal for processing the (delayed) feedback information, and the solution of its steady-state Riccati equation delivers the maximal information rate for stationary sources. Notation: Random variables are denoted by upper-case letters, e.g., X t , and their realizations are denoted using lower case letters, e.g., x t .…”