# Feedback Capacity of Stationary Gaussian Channels Further Examined

## TL;DR

This paper investigates the feedback capacity of stationary Gaussian channels, proving the uniqueness of optimal solutions for non-white noise and providing algorithms and explicit formulas for calculating feedback capacity in autoregressive moving-average noise models.

## Contribution

It establishes the uniqueness of the optimal solution for the feedback capacity problem when noise is not white and introduces an efficient recursive algorithm for its computation.

## Key findings

- Optimal solution is unique for non-white Gaussian noise.
- Proposed recursive algorithm converges and is computationally efficient.
- Explicit formulas for feedback capacity in ARMA noise models for k=1,2 cases.

## Abstract

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, 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, which is amenable to numerical computation for the cases k=1, 2 and possibly beyond.