Mutual information optimal constellations for the low SNR noncoherent MIMO Rayleigh fading channel

Reliable communication over the discrete input and continuous output noncoherent multiple-input multiple-output (MIMO) Rayleigh fading channel is considered when the SNR per degree of freedom is low. The input constellations are required to satisfy peak and average power constraints. When the peak-to-average power ratio of the input constellation is limited (PAPR-limited) and in the low SNR regime, the mutual information upto second order in SNR is maximized jointly over input signal matrices and their respective probabilities, over all T + 1 point constellations (where T is the coherence length). Even though the problem considered is a finite dimensional non-convex optimization, it admits an elegant solution in closed form. The constellation obtained is referred to as Space Time Orthogonal Rank one Modulation (STORM), and it provides new insights into noncoherent MIMO comunications in the low SNR regime. By deriving an appropriate upper bound, it is shown that in most cases with even moderate values for PAPR and T, STORM is near-optimal with respect to the maximum mutual information achievable with unconstrained cardinality. For the case when the peak-power constraint is a fixed constant (peak-constrained), STORM achieves the capacity per unit energy, while having a wideband slope T times that of the conventional approach of MIMO ON-OFF signaling. This translates to increased bandwidth efficiency or PAPR reduction by a factor of T in the wideband regime.