Optimal Constellations for the Low SNR Noncoherent MIMO Block Rayleigh Fading Channel

TL;DR

This paper derives optimal signal constellations for low SNR noncoherent MIMO Rayleigh channels, revealing a unified solution called STORM that maximizes mutual information growth and energy efficiency.

Contribution

It introduces a novel, closed-form optimal constellation design called STORM for low SNR noncoherent MIMO channels under power constraints.

Findings

Optimal constellations are identical for both power constraints.

STORM achieves capacity growth up to second order in SNR.

Explicit characterization of first- and second-order optimal constellations.

Abstract

Reliable communication over the discrete-input/continuous-output noncoherent multiple-input multiple-output (MIMO) Rayleigh block fading channel is considered when the signal-to-noise ratio (SNR) per degree of freedom is low. Two key problems are posed and solved to obtain the optimum discrete input. In both problems, the average and peak power per space-time slot of the input constellation are constrained. In the first one, the peak power to average power ratio (PPAPR) of the input constellation is held fixed, while in the second problem, the peak power is fixed independently of the average power. In the first PPAPR-constrained problem, the mutual information, which grows as O(SNR^2), is maximized up to second order in SNR. In the second peak-constrained problem, where the mutual information behaves as O(SNR), the structure of constellations that are optimal up to first order, or equivalently, that minimize energy/bit, are explicitly characterized. Furthermore, among constellations that are first-order optimal, those that maximize the mutual information up to second order, or equivalently, the wideband slope, are characterized. In both PPAPR-constrained and peak-constrained problems, the optimal constellations are obtained in closed-form as solutions to non-convex optimizations, and interestingly, they are found to be identical. Due to its special structure, the common solution is referred to as Space Time Orthogonal Rank one Modulation, or STORM. In both problems, it is seen that STORM provides a sharp characterization of the behavior of noncoherent MIMO capacity.