A Sphere Packing Bound for Vector Gaussian Fading Channels under Peak Amplitude Constraints

TL;DR

This paper derives a new upper bound on the capacity of MIMO Gaussian fading channels with peak amplitude constraints using geometric sphere packing concepts, extending scalar channel results to MIMO systems.

Contribution

It introduces a novel geometric sphere packing bound for MIMO channels under amplitude constraints, utilizing convex geometry and Steiner's formula, with variants for different SNR regimes.

Findings

Bound is tight at high SNR for various constraints and channel realizations.

Two variants of the bound improve feasibility and low SNR performance.

Specialized bounds for practical transmitter configurations are provided.

Abstract

An upper bound on the capacity of multiple-input multiple-output (MIMO) Gaussian fading channels is derived under peak amplitude constraints. The upper bound is obtained borrowing concepts from convex geometry and it extends to MIMO channels notable results from the geometric analysis on the capacity of scalar Gaussian channels. Relying on a sphere packing argument and on the renowned Steiner's formula, the proposed upper bound depends on the intrinsic volumes of the constraint region, i.e., functionals defining a measure of the geometric features of a convex body. The tightness of the bound is investigated at high signal-to-noise ratio (SNR) for any arbitrary convex amplitude constraint region, for any channel matrix realization, and any dimension of the MIMO system. In addition, two variants of the upper bound are proposed: one is useful to ensure the feasibility in the evaluation of the bound and the other to improve the bound's performance in the low SNR regime. Finally, the upper bound is specialized for two practical transmitter configurations, either employing a single power amplifier for all transmitting antennas or a power amplifier for each antenna.