Insufficiency of Linear-Feedback Schemes in Gaussian Broadcast Channels With Common Message

We consider the K ≥ 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 and Kailath´s scheme for point-to-point channels or Ozarow and 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 → ∞, 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 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 Lth order exponential decay of the probability of error if the feedback rate Rfb is at least (L - 1)R for some positive integer L.