Qualitative Analysis of Feedback Capacity of AGN Channels Driven by Unstable Versus Stable Autoregressive Moving Average Noise

Presented are closed-form feedback capacity formulas, and lower bounds on nonfeedback achievable rates, for additive Gaussian noise (AGN) channels, driven by unstable, i.e., nonstationary, and stable, autoregressive moving average noise, ARMA (a, c), a is an element of (-infinity, infinity), c is an element of (-infinity, infinity) with one pole at c and one zero at a), which are independent of the distributions of the initial random variables, i.e., the initial state of the noise. The feedback capacity exhibits multiple regimes of capacity; (i) for the regime that includes the stable noise, ARMA(a; c); a is an element of (1, 1); c is an element of (1, 1) and for all transmit powers kappa is an element of (0, infinity), feedback does not increase capacity, (ii) for the regime that includes the unstable noise, ARMA(a, c), vertical bar c vertical bar > 1, a is an element of (-1, 1), and for transmit power above a threshold kappa > kappa(min), feedback increases capacity, and the higher the vertical bar c vertical bar the higher the capacity. The unstable regime is verified by the semi-definite program of [1]. Although, our answer feedback does not increase capacity for stable noise, ARMA(a, c); a is an element of (-1, 1); c is an element of (1, 1) contradicts the feedback capacity, C-FB, in [2, Theorem 6.1, C-FB], [1] and the maximal information rate, I-max, in [3, Theorem 7, I-max], this is attributed to the fact that rates in [1]-[3], depend on the initial state of the noise, and these rates are not achievable, for asymptotically stationary noise. This paper uses results from [4].