We characterize the feedback capacity$C_{FB}$of general nonlinear decision models (N-DM) through the$n$-finite trans-mission or block length feedback information$(n$- FTFI) capacity,$c_{FB,n}$. For an application we consider the multiple-input multiple-output (MIMO) Gaussian DM with memory on past outputs and inputs driven by nonstationary Gaussian noise, and finite-dimensional Gaussian noise in state space form, subject to an average cost constraint of quadratic form. The main theorems show that optimal randomized control strategies that achieve$c_{FB,n}$, consist of multiple parts, that include control, estimation, and information transmission/signalling strategies. These strategies are determined using decentralized optimization techniques, and involve filtering Riccati equations and a control Riccati equation.

We show that the nonfeedback capacity of multiple-input multiple-output (MIMO) additive Gaussian noise (AGN) channels, when the noise is nonstationary and unstable, is characterized by an asymptotic optimization problem–the per unit time limit of the characterization of a finite block or transmission without feedback information (FTwFI) capacity, that involves two generalized matrix difference Riccati equations (DREs) of filtering theory, and a matrix difference Lyapunov equation of stability theory, of Gaussian systems. Further, we identify conditions and prove, that the characterization of nonfeedback capacity is the uniform asymptotic per unit time limit, over all initial distributions. The asymptotic characterization of capacity involves two generalized matrix algebraic Riccati equations (AREs) and a matrix algebraic Lyapunov equation. We also present an example to illustrate that our characterization of capacity produces a known closed-form expression of the water-filling solution of capacity (for power levels above a minimum power).

The multiple-input multiple-output (MIMO) generalization of Cover’s and Pombra’s Gaussian feedback capacity [1] is considered. A sequential characterization of the finite block or transmission feedback information (FTFI) capacity is derived, with the optimal channel input process expressed as functional of a sufficient statistic and a Gaussian orthogonal innovations process. From the new representations follows that the MIMO version of the Cover and Pombra characterization of the FTFI capacity is expressed as a functional of two generalized matrix difference Riccati equations (DRE) of filtering theory of Gaussian systems. Analogous expressions for nonfeedback capacity are also derived, which involve a Lyapunov equation. The derivations follow directly from [2]; application examples to autoregressive moving average noise are found in [3], and to autoregressive noise in [4], [5]. Asymptotic formulas of feedback, nonfeedback capacity, and achievable lower bound incurred by asymptotically stationary channel inputs follow from the analysis of [2, Section III].

Presented are closed-form feedback capacity formulas, and lower bounds on nonfeedback achievable rates, for additive Gaussian noise (AGN) channels, driven by unstable, i.e., nonstationary, and stable, autoregressive moving average noise, ARMA $(a,c),a\in(-\infty,\infty),c\in(-\infty,\infty)$ (with one pole at c and one zero at a), which are independent of the distributions of the initial random variables, i.e., the initial state of the noise. The feedback capacity exhibits multiple regimes of capacity; (i) for the regime that includes the stable noise, ARMA $(a,c),a\in(-{1},{1}), c\in$(−1, 1) and for all transmit powers $\kappa\in(0,\infty)$, feedback does not increase capacity, (ii) for the regime that includes the unstable noise, ARMA $(a,c), |c| \gt 1,a\in$(−1, 1), and for transmit power above a threshold $\kappa \gt \kappa_{{\min}}$, feedback increases capacity, and the higher the $|c|$ the higher the capacity. The unstable regime is verified by the semi-definite program of [1]. Although, our answer} feedback does not increase capacity for stable noise, ARMA $(a,c)$, $a\in$(−1, 1), $c\in$(−1, 1) contradicts the feedback capacity, $C_{FB}$, in [2, Theorem 6.1,] $C_{FB}]$, [1] and the maximal information rate, $I_{{\max}}$ in [3, Theorem 7,] $I_{{\max}}$, this is attributed to the fact that rates in [1] –[3], depend on the initial state of the noise, and these rates are not achievable, for asymptotically stationary noise. This paper uses results from [4].