  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.
