Feedback capacity of Gaussian channels with memory

TL;DR

This paper derives a computable expression for the feedback capacity of Gaussian MIMO channels with memory, using a state-space model, and explores how feedback delays affect capacity.

Contribution

It introduces a general state-space model for Gaussian channels with memory and provides a convex optimization-based feedback capacity formula applicable to various models.

Findings

Single delay reduces feedback capacity significantly in stationary regimes.

Large regression parameters allow approaching capacity with delayed feedback.

Detectability condition holds for scalar models and likely for MIMO models.

Abstract

We consider the feedback capacity of a MIMO channel whose channel output is given by a linear state-space model driven by the channel inputs and a Gaussian process. The generality of our state-space model subsumes all previous studied models such as additive channels with colored Gaussian noise, and channels with an arbitrary dependence on previous channel inputs or outputs. The main result is a computable feedback capacity expression that is given as a convex optimization problem subject to a detectability condition. We demonstrate the capacity result on the auto-regressive Gaussian noise channel, where we show that even a single time-instance delay in the feedback reduces the feedback capacity significantly in the stationary regime. On the other hand, for large regression parameters (in the non-stationary regime), the feedback capacity can be approached with delayed feedback. Finally, we show that the detectability condition is satisfied for scalar models and conjecture that it is true for MIMO models.