On the Capacity Enlargement of Gaussian Broadcast Channels with Passive Noisy Feedback

TL;DR

This paper characterizes the conditions under which noisy feedback can enlarge the capacity region of a two-user Gaussian Broadcast Channel with independent noises, revealing that unequal forward noise variances limit feedback benefits.

Contribution

It provides a sharp, explicit characterization of the noise variance conditions that determine capacity enlargement in Gaussian BCs with noisy feedback.

Findings

Capacity enlargement depends on the noise variances of feedback and forward channels.

No capacity increase occurs when feedback noise is too high relative to forward noise variances.

Unequal forward noise variances can negate the benefits of noisy feedback.

Abstract

It is well known that the capacity region of an average transmit power constrained Gaussian Broadcast Channel (GBC) with independent noise realizations at the receivers is enlarged by the presence of causal noiseless feedback. Capacity region enlargement is also known to be possible by using only passive noisy feedback, when the GBC has identical noise variances at the receivers. The last fact remains true even when the feedback noise variance is very high, and available only from one of the receivers. While such capacity enlargements are feasible for several other feedback models in the Gaussian BC setting, it is also known that feedback does not change the capacity region for physically degraded broadcast channels. In this paper, we consider a two user GBC with independent noise realizations at the receivers, where the feedback links from the receivers are corrupted by independent additive Gaussian noise processes. We investigate the set of four noise variances, two forward and two feedback, for which no capacity enlargement is possible. A sharp characterization of this region is derived, i.e., any quadruple outside the presented region will lead to a capacity enlargement, whereas quadruples inside will leave the capacity region unchanged. Our results lead to the conclusion that when the forward noise variances are different, too noisy a feedback from one of the receivers alone is not always beneficial for enlarging the capacity region, be it from the stronger user or the weaker one, in sharp contrast to the case of equal forward noise variances.