An Alternative Proof for the Capacity Region of the Degraded Gaussian MIMO Broadcast Channel

TL;DR

This paper presents a new proof for the capacity region of the degraded Gaussian MIMO broadcast channel, avoiding the channel enhancement technique by leveraging information-theoretic identities and inequalities.

Contribution

It introduces an alternative proof method based on the single-letter capacity description and Fisher information tools, differing from previous enhancement-based proofs.

Findings

Provides a simpler, direct proof of the capacity region.

Utilizes the generalized de Bruijn identity and Dembo's inequality.

Avoids the channel enhancement technique used in prior proofs.

Abstract

We provide an alternative proof for the capacity region of the degraded Gaussian multiple-input multiple-output (MIMO) broadcast channel. Our proof does not use the channel enhancement technique as opposed to the original proof of Weingertan {\it et. al.} and the alternative proof of Liu {\it et. al}. Our proof starts with the single-letter description of the capacity region of the degraded broadcast channel, and directly evaluates it for the degraded Gaussian MIMO broadcast channel by using two main technical tools. The first one is the generalized de Bruijn identity due to Palomar \emph{et. al.} which provides a connection between the differential entropy and the Fisher information matrix. The second tool we use is an inequality due to Dembo which lower bounds the differential entropy in terms of the Fisher information matrix.