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.

The behavior in terms of information theoretic metrics of the discrete-input, continuous-output noncoherent MIMO Rayleigh fading channel is studied as a function of spatial correlations. In the low SNR regime, the mutual information metric is considered, while at higher SNR regimes the cutoff rate expression is employed. For any fixed input constellation and at sufficiently low SNR, a fully correlated channel matrix is shown to maximize the mutual information. In contrast, at high SNR, a fully uncorrelated channel matrix (with independent identically distributed elements) is shown to be optimal, under a condition on the constellation which ensures full diversity.In the special case of the separable correlation model, it is shown that as a function of the receive correlation eigenvalues, the cutoff rate expression is a Schur-convex function at low SNR and a Schur-concave function at high SNR, and as a function of transmit correlation eigenvalues, the cutoff rate expression is Schur-concave at high SNR for full diversity constellations. Moreover, at sufficiently low SNR, the fully correlated transmit correlation matrix is optimal. Finally, for the general model, it is shown that the optimal correlation matrices at a general SNR can be obtained using a difference of convex programming formulation.

Constellation design for the noncoherent multi-input, multi-output (MIMO) block Rayleigh fading channel is considered. For general SNRs, starting from a given base unitary constellation of finite cardinality, and using the cutoff rate expression as the design criterion, we obtain input probabilities and per-antenna amplitudes for the constellation points via a global optimization formulation. Using the mutual information as a performance metric we obtain numerical results that show that the optimized constellations significantly outperform the base unitary designs from which they are obtained in the low-medium SNR regime, and indeed they also similarly outperform the mutual information achieved by isotropically distributed inputs for the continuous input channel (i.e., the so-called unitary space-time capacity (USTC)). At sufficiently high SNRs, the resulting mutual information coincides with that of the base unitary designs. Thus we have an optimum constellation design technique that works over the entire range of SNRs. The bit-energy/spectral-efficiency tradeoff of the optimized constellations are also obtained, and these provide valuable insights on modulation and coding, which are especially useful for wideband channels where the SNR per degree of freedom is low.

-Code design for the low SNR MIMO noncoherent correlated Rayleigh fading channel is considered. Design rules which exploit the correlations in the transmit antennas in the MIMO case, to provide gains over the corresponding SIMO case are presented. The Chernoff bound on the average pairwise error probability (APEP) is used to study the effect of the receive correlation matrix on system performance at different SNR regimes. Based on a lower bound on the APEP, which is related to the Bhattacharya coefficient, a technique is proposed to design codes for use with transmit beamforming, with codewords having unequal prior probabilities. The motivation for such codes with unequal priors arises from recent information theoretic results on the low SNR channel. Such constellations are shown to perform substantially better than constellations designed assuming equal priors, at low SNRs.