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, Yt=Xt+Vt, t=1, …, n, 1n E\Σt=1n |Xt|2 \≤ , ∈ [0,∞), 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 stable and unstable autoregressive models, AR(c), Vt=cVt-1+ Wt, V0=v0, t=1, …, n, where c∈ (-∞,∞), Wt, t=1,…, n, is a zero mean, variance KW, independent Gaussian sequence, independent of V0. For stable AR(c), c∈ (-1,1) the conditions are necessary and sufficient for asymptotic stationarity of the processes (Xt, Yt), t=1, 2, …. New closed form capacity formulas and lower bounds are derived, for the AR(c), c∈ (-∞,∞) noise, which are fundamentally different from existing formulas in the literature, and illustrate multiple regimes of capacity, as a function of the parameters (c,KW,), as follows.\\ 1) feedback increases capacity for the regime, c2 ∈ (1, ∞), for > KW(1+4c2-3)2(c2-1)2, \\ 2) feedback does not increase capacity for the regime c2 ∈ (1, ∞), for ≤ KW(1+4c2-3)2(c2-1)2, and \\ 3) feedback does not increase capacity for the regime c ∈ [-1,1], for ∈ [0,∞).
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