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 $(a,c),a\in(-\infty,\infty),c\in(-\infty,\infty)$ (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 $(a,c),a\in(-{1},{1}), c\in$(−1, 1) and for all transmit powers $\kappa\in(0,\infty)$, feedback does not increase capacity, (ii) for the regime that includes the unstable noise, ARMA $(a,c), |c| \gt 1,a\in$(−1, 1), and for transmit power above a threshold $\kappa \gt \kappa_{{\min}}$, feedback increases capacity, and the higher the $|c|$ 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 $(a,c)$, $a\in$(−1, 1), $c\in$(−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].