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 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,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,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 ∈When compared to [1], our capacity formulas for AGN channels driven a stable AR(c), c ∈ (−1, 1) July 26, 2019 DRAFT noise, state that feedback does not increase capacity, contrary to the main results of the characterizations of feedback capacity derived in [1, Theorem 4.1 and Theorem 6.1], for stationary or asymptotically stationary process. We show that our disagreement with [1] is mainly attributed to the use of necessary and sufficient conditions, known as detectability and stabilizability conditions, for convergence of generalized difference Riccati equations (DREs) to analogous generalized algebraic Riccati equations (AREs), of Gaussian estimation problems, to ensure asymptotic stationarity and ergodicity of the channel output process, which are not accounted for in [1, Theorem 4.1 and Theorem 6.1]. I. INTRODUCTION, MOTIVATION, AND MAIN RESULTS OF THE PAPER The feedback capacity of additive Gaussian noise (AGN) channels driven by nonstationary, and stationary limited memory Gaussian noise, is addressed, since the early 1970's, in an anthology of papers, under various assumptions [1]-[8]. Two of the fundamental problems, which are addressed are related to (Q1): characterizations and computations of feedback capacity of noiseless feedback codes, and (Q2): bound on feedback capacity, based on linear feedback coding schemes of communicating Gaussian random variables (RVs), Θ : Ω → R, and coding schemes of communicating digital messages. . , ⌈M n ⌉}, when the initial state S 0 = s 0 of the noise is known to the encoder and the decoder.This paper is mainly concerned with question (Q1), for AGN channels driven by stable and unstable noise, under the condition of, channel input strategies are time-invariant and induce asymptotic stationarity, and ergodicity of the channel output process. For asymptotically stationary noise, this condition, as it will become apparent in latter parts of this paper, is sufficient and necessary, t...