The Capacity Gains of Gaussian Channels with Unstable Versus Stable Autoregressive Noise

In this paper, we consider Cover’s and Pombra’s formulation of feedback capacity of additive Gaussian noise (AGN) channels, with jointly Gaussian nonstationary and nonergodic noise. We derive closed-form feedback capacity formulas, using Karush–Kuhn–Tucker (KKT) conditions and convergence properties of difference Riccati equations to limiting algebraic Riccati equations of filtering theory, for unstable and stable autoregressive (AR) noise. Surprisingly, the capacity formulas depend on the parameters of the AR noise, its pole $c\in (-\infty ,\infty )$ and noise variance $K_W\in (0,\infty )$, and the total transmit power $\kappa \in [0,\infty )$, indicating substantial gains for the unstable noise region $\forall c^2\in (1,\infty ),\forall \kappa >{\kappa }_{min}\stackrel{▵}{=}{\displaystyle \frac{K_W\left(1+\sqrt{4c^2-3}\right)}{2{\left(c^2-1\right)}^2}}$ compared to its complement region. In particular, feedback capacity is distinguished by three regimes, as follows. Regime 1, $\forall c^2\in (1,\infty ),\forall \kappa >{\kappa }_{min}$: the optimal channel input includes an innovations part, the capacity increases as $\left|c\right|>1$ increases, while ${\kappa }_{min}$ and the allocated transmit power decrease. Regime 2, $\forall c^2\in (1,\infty ),\forall \kappa \le {\kappa }_{min}$, Regime 3, $\forall c\in [-1,1],\forall \kappa \in [0,\infty )$ (complement of Regime 1): the innovations part of the optimal channel is asymptotically zero and the capacity is fundamentally different compared to Regime 1. The differences of capacity formulas for Regimes 1, 2 and 3 are directly related to their operational meaning: (i) Regime 1 is an ergodic capacity while Regimes 2 and 3 are nonergodic capacities; (ii) Regime 1 is achieved by an asymptotically stationary channel input with a non-zero innovations part, while Regimes 2 and 3 are achieved by an asymptotically zero innovations part. The gains of capacity for Regime 1 are attributed to the high correlation of noise samples compared to stable noise and the use of an informative innovations part by the optimal channel input, which make possible the prediction of future noise samples from past samples, unlike memoryless noise. Our results provide answers to certain open questions regarding the validity of capacity formulas of stable noise that appeared in the literature.