A Proof of the Strong Converse Theorem for Gaussian Broadcast Channels via the Gaussian Poincar\'e Inequality

TL;DR

This paper proves the strong converse property for Gaussian broadcast channels using the Gaussian Poincaré inequality, confirming that rates outside the capacity region lead to error probabilities approaching one.

Contribution

It introduces a novel proof of the strong converse for Gaussian broadcast channels leveraging the Gaussian Poincaré inequality, a new mathematical approach in information theory.

Findings

Strong converse holds for Gaussian broadcast channels.

Rates outside the capacity region lead to error probability approaching one.

Uses Gaussian Poincaré inequality as a key mathematical tool.

Abstract

We prove that the Gaussian broadcast channel with two destinations admits the strong converse property. This implies that for every sequence of block codes operated at a common rate pair with an asymptotic average error probability $<1$, the rate pair must lie within the capacity region derived by Cover and Bergmans. The main mathematical tool required for our analysis is a logarithmic Sobolev inequality known as the Gaussian Poincar\'e inequality.