Constellation design for the noncoherent MIMO Rayleigh-fading channel at general SNR

Constellation design for the noncoherent multiple-input-multiple-output (MIMO) block Rayleigh-fading channel is considered. For general signal-to-noise ratios (SNRs), starting from a given base unitary constellation of finite cardinality, and using the cutoff rate expression as the design criterion, input probabilities and per-antenna amplitudes for the constellation points are obtained via a difference of convex programming formulation. Using the mutual information as a performance metric, it is shown 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 unitary 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 the optimum constellation design technique 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.