Time-Invariant Feedback Strategies Do Not Increase Capacity of AGN Channels Driven by Stable and Certain Unstable Autoregressive Noise

Abstract

The capacity of additive Gaussian noise (AGN) channels, $Y_t=X_t+V_t,t=1,\dots ,n$, $\frac{1}{n}\mathbf{E}\{{\sum }_{t=1}^n|X_t{|}^2\}\le \kappa ,\kappa \in [0,\infty )$, with time-invariant channel input feedback strategies, is characterized and conditions are identified for entropy rates, and limit of average power to exist, when the noise is described by {\it stable and unstable} autoregressive models, AR$(c)$, $V_t=c{V}_{t-1}+W_t,V_0=v_0,t=1,\dots ,n$, where $c\in (-\infty ,\infty )$, $W_t,t=1,\dots ,n$, is a zero mean, variance $K_W$, independent Gaussian sequence, independent of $V_0$. For stable AR$(c),c\in (-1,1)$ the conditions are necessary and sufficient for asymptotic stationarity of the processes $(X_t,Y_t),t=1,2,\dots$. New closed form capacity formulas and lower bounds are derived, for the AR$(c),c\in (-\infty ,\infty )$ noise, which are fundamentally different from existing formulas in the literature, and illustrate multiple regimes of capacity, as a function of the parameters $(c,K_W,\kappa )$, as follows.\\ 1) feedback increases capacity for the regime, $c^2\in (1,\infty ),$ for $\kappa >\frac{K_W(1+\sqrt{4c^2-3})}{2(c^2-1{)}^2}$, \\ 2) feedback does not increase capacity for the regime $c^2\in (1,\infty )$, for $\kappa \le \frac{K_W(1+\sqrt{4c^2-3})}{2(c^2-1{)}^2}$, and \\ 3) feedback does not increase capacity for the regime $c\in [-1,1]$, for $\kappa \in [0,\infty )$.

submitted to IEEE Transactions on Information Theory (IT-19-0520)