Insufficiency of Linear-Feedback Schemes In Gaussian Broadcast Channels with Common Message

We consider the $K\geq 2$-user memoryless Gaussian broadcast channel (BC) with feedback and common message only. We show that linear-feedback schemes with a message point, in the spirit of Schalkwijk & Kailath's scheme for point-to-point channels or Ozarow & Leung's scheme for BCs with private messages, are strictly suboptimal for this setup. Even with perfect feedback, the largest rate achieved by these schemes is strictly smaller than capacity $C$ (which is the same with and without feedback). In the extreme case where the number of receivers $K\to \infty$, the largest rate achieved by linear-feedback schemes with a message point tends to 0. To contrast this negative result, we describe a scheme for \emph{rate-limited} feedback that uses the feedback in an intermittent way, i.e., the receivers send feedback signals only in few channel uses. This scheme achieves all rates $R$ up to capacity $C$ with an $L$-th order exponential decay of the probability of error if the feedback rate $R_{\textnormal{fb}}$ is at least $(L-1)R$ for some positive integer $L$.