Time-Invariant Feedback Strategies Do Not Increase Capacity of AGN Channels Driven by Stable and Certain Unstable Autoregressive Noise
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The capacity of additive Gaussian noise (AGN) channels with feedback, when the noise is described by, stable and unstable, autoregressive models, is characterized, under the condition that, channel input feedback strategies are time-invariant and induce asymptotic stationarity, and ergodicity of the channel output process. This condition is more relaxed than imposing stationarity or asymptotic stationarity of the joint input and output process, as done in [1]. Based on this characterization, new closed form capacity formulas, and lower bounds are derived, for the stable and unstable AR(c) noise, V_t=cV_t-1+ W_t, V_0=v_0, t=1, ..., n, where c∈ (-∞,∞), W_t, t=1,..., n, is a zero mean, variance K_W, independent Gaussian sequence, independent of V_0. Our capacity formulas are fundamentally different from existing formulas found in the literature. The new formulas illustrate multiple regimes of capacity, as a function of the parameters (c,K_W,κ), where κ is the total average power allocated to the transmitter. In particular, 1) feedback increases capacity for the regime, c^2 ∈ (1, ∞), for κ > K_W/(c^2-1)^2, 2) feedback does not increase capacity for the regime c^2 ∈ (1, ∞), for κ≤K_W/(c^2-1)^2, and 3) feedback does not increase capacity for the regime c ∈ [-1,1], for κ∈ [0,∞). We show that our disagreement with [1] is mainly attributed to the use of necessary and sufficient conditions for convergence of generalized difference Riccati equations (DREs) to analogous generalized algebraic Riccati equations (AREs), of Gaussian estimation problems, known as detectability and stabilizability conditions, to ensure asymptotic stationarity and ergodicity of the channel output process.
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