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

Abstract

We consider the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="TeX">\(K\geq 2\) </tex-math></inline-formula> -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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="TeX">\(C\) </tex-math></inline-formula> (which is the same with and without feedback). In the extreme case where the number of receivers <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="TeX">\(K\to \infty \) </tex-math></inline-formula> , 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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="TeX">\(R\) </tex-math></inline-formula> up to capacity <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="TeX">\(C\) </tex-math></inline-formula> with an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="TeX">\(L\) </tex-math></inline-formula> th order exponential decay of the probability of error if the feedback rate <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="TeX">\(R_{\text {fb}}\) </tex-math></inline-formula> is at least <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="TeX">\((L-1)R\) </tex-math></inline-formula> for some positive integer <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="TeX">\(L\) </tex-math></inline-formula> .