LQG Control Approach to Gaussian Broadcast Channels with Feedback

TL;DR

This paper introduces a LQG control-based coding scheme for Gaussian broadcast channels with feedback, demonstrating how noise correlation affects capacity and providing bounds on the sum capacity pre-log for various noise covariance structures.

Contribution

It develops a novel LQG control approach for broadcast channels with feedback, extending capacity results to correlated noise scenarios and improving existing codes for two receivers.

Findings

Performance depends on receiver noise correlation.

Achieves sum rate of 1/2 log(1+P*phi) for independent noises.

Pre-log of sum capacity can exceed one with correlated noises.

Abstract

A code for communication over the k-receiver additive white Gaussian noise broadcast channel with feedback is presented and analyzed using tools from the theory of linear quadratic Gaussian optimal control. It is shown that the performance of this code depends on the noise correlation at the receivers and it is related to the solution of a discrete algebraic Riccati equation. For the case of independent noises, the sum rate achieved by the proposed code, satisfying average power constraint P, is characterized as 1/2 log (1+P*phi), where the coefficient "phi" in the interval [1,k] quantifies the power gain due to the presence of feedback. When specialized to the case of two receivers, this includes a previous result by Elia and strictly improves upon the code of Ozarow and Leung. When the noises are correlated, the pre-log of the sum-capacity of the broadcast channel with feedback can be strictly greater than one. It is established that for all noise covariance matrices of rank r the pre-log of the sum capacity is at most k-r+1 and, conversely, there exists a noise covariance matrix of rank r for which the proposed code achieves this upper bound. This generalizes a previous result by Gastpar and Wigger for the two-receiver broadcast channel.