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

TL;DR

This paper characterizes the capacity of AGN channels with autoregressive noise under time-invariant feedback, revealing regimes where feedback increases capacity and deriving new formulas for both stable and unstable AR noise models.

Contribution

It provides new closed-form capacity formulas and bounds for AGN channels with AR noise, including unstable cases, and identifies when feedback enhances capacity.

Findings

Feedback increases capacity for unstable AR noise when power exceeds a threshold.

Feedback does not increase capacity for stable AR noise or low power regimes.

New capacity formulas differ from existing literature, showing multiple capacity regimes.

Abstract

The capacity of additive Gaussian noise (AGN) channels, $Y_t=X_t+V_t, t=1, \ldots, n$, $\frac{1}{n} {\bf E}\big\{\sum_{t=1}^n |X_t|^2 \big\}\leq \kappa, \kappa \in [0,\infty)$, 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$(c)$, $V_t=cV_{t-1}+ W_t, V_0=v_0, t=1, \ldots, n$, where $c\in (-\infty,\infty)$, $W_t, t=1,\ldots, n$, 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 $(X_t, Y_t), t=1, 2, \ldots$. 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 $\kappa > \frac{K_W\big(1+\sqrt{4c^2-3}\big)}{2\big(c^2-1\big)^2}$, \\ 2) feedback does not increase capacity for the regime $c^2 \in (1, \infty)$, for $\kappa \leq \frac{K_W\big(1+\sqrt{4c^2-3}\big)}{2\big(c^2-1\big)^2}$, and \\ 3) feedback does not increase capacity for the regime $c \in [-1,1]$, for $ \kappa \in [0,\infty)$.