It is well known that the problem of computing the feedback capacity of a stationary Gaussian channel can be recast as an infinite-dimensional optimization problem; moreover, necessary and sufficient conditions for the optimality of a solution to this optimization problem have been characterized, and based on this characterization, an explicit formula for the feedback capacity has been given for the case that the noise is a first-order autoregressive moving-average Gaussian process. In this paper, via a simple “change of variables” trick, we further examine the above-mentioned infinite-dimensional optimization problem. We prove that unless the Gaussian noise is white, its optimal solution is unique, and we propose an algorithm to recursively compute the unique optimal solution, which is guaranteed to converge in theory and features an efficient implementation for a suboptimal solution in practice. Furthermore, for the case, that the noise is a $k$ -th order autoregressive moving-average Gaussian process, we give a relatively more explicit formula for the feedback capacity; more specifically, the feedback capacity is expressed as a simple function evaluated at a solution to a system of polynomial equations.

Using Kim's variational formulation [1] (with a slight yet important modification), we derive the ARMA(fc) Gaussian feedback capacity, i.e., the feedback capacity of an additive channel where the noise is a k-th order autoregressive moving average Gaussian process. More specifically, the ARMA(fc) Gaussian feedback capacity is expressed as a simple function evaluated at a solution to a system of polynomial equations, which proves to have only finitely many solutions for the cases k = 1,2 and possibly beyond.