Feedback Capacity of Gaussian Channels Revisited

TL;DR

This paper revisits the feedback capacity of Gaussian channels, introducing a new approach for arbitrary spectral densities and providing computationally tractable solutions for stationary Gaussian noise with finite memory.

Contribution

It presents a novel method to compute feedback capacity over finite transmissions and bridges the gap between stationary and non-stationary cases, showing linear strategies are optimal.

Findings

Capacity can be computed via semi-definite programming for stationary noise.

Linear feedback strategies are proven to be optimal.

The approach generalizes to arbitrary spectral densities and non-stationary noise.

Abstract

In this paper, we revisit the problem of finding the average capacity of the Gaussian feedback channel. First, we consider the problem of finding the average capacity of the analog Gaussian noise channel where the noise has an arbitrary spectral density. We introduce a new approach to the problem where we solve the problem over a finite number of transmissions and then consider the limit of an infinite number of transmissions. Further, we consider the important special case of stationary Gaussian noise with finite memory. We show that the channel capacity at stationarity can be found by solving a semi-definite program, and hence computationally tractable. We also give new proofs and results of the non stationary solution which bridges the gap between results in the literature for the stationary and non stationary feedback channel capacities. It's shown that a linear communication feedback strategy is optimal. Similar to the solution of the stationary problem, it's shown that the optimal linear strategy is to transmit a linear combination of the information symbols to be communicated and the innovations for the estimation error of the state of the noise process.