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

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(SNR²), 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 per 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 nonconvex 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.