LQG Control Approach to Gaussian Broadcast Channels With Feedback

A code for communication over the k-receiver complex additive white Gaussian noise broadcast channel (BC) 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, is characterized as 1/2 log (1 + P phi) where the coefficient phi is an element of [1, k] quantifies the power gain due to the presence of feedback. This includes a previous result by Elia and strictly improves upon the codes by Ozarow and Leung and by Kramer. When the noises are correlated, the prelog of the sum capacity of the BC with feedback can be strictly greater than 1. It is established that for all noise covariance matrices of rank r the prelog 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 et al. for the two-receiver BC.