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

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.

The Capacity of Fading Vector Gaussian Channels Under Amplitude Constraints on Antenna Subsets

“…We evaluate the bounds for a random realization of H. If we consider the compound upper bound given by min C 2 , C PA,2 we see that, as predicted by Lemma 1 and Lemma 2, the capacity gap between upper and lower bounds is indeed vanishing both at high SNR, thanks to C PA,2 , and at low SNR, thanks to C 2 . Moreover, we compare the proposed bounds to the previous best in the existing literature, which we proposed in [14]. Specifically, let us denote by C SP the upper bound based on a sphere packing argument [14, Eq.…”

“…Remark 1. To obtain a numerical result, each C i can be further upper-bounded with a suitable technique, like those presented in [8], [11], [14].…”

“…In [12], interesting insights on the capacity-achieving input distribution for low signal-to-noise ratio (SNR) levels are presented. Finally, in [13], [14] the present authors derive an upper bound for arbitrary convex constraint regions that, together with the entropy power inequality (EPI) lower bound, provides a vanishing capacity gap at high SNR.…”