On MMSE Properties and I-MMSE Implications in Parallel MIMO Gaussian Channels

TL;DR

This paper generalizes the scalar MMSE crossing property to parallel MIMO Gaussian channels, revealing a single crossing point per eigenvalue, and uses this to derive new insights and proofs in information theory, including the broadcast channel capacity.

Contribution

It extends the single crossing point property of MMSE to parallel MIMO channels and applies this to derive new theoretical results in information theory.

Findings

Matrix Q(t) has at most one eigenvalue crossing point.

The property enables an alternative proof of the broadcast channel capacity.

Establishes a fundamental link between MMSE properties and information-theoretic limits.

Abstract

This paper extends the "single crossing point" property of the scalar MMSE function, derived by Guo, Shamai and Verd\'u (first presented in ISIT 2008), to the parallel degraded MIMO scenario. It is shown that the matrix Q(t), which is the difference between the MMSE assuming a Gaussian input and the MMSE assuming an arbitrary input, has, at most, a single crossing point for each of its eigenvalues. Together with the I-MMSE relationship, a fundamental connection between Information Theory and Estimation Theory, this new property is employed to derive results in Information Theory. As a simple application of this property we provide an alternative converse proof for the broadcast channel (BC) capacity region under covariance constraint in this specific setting.