Constellation Design for the Noncoherent MIMO Rayleigh Fading Channel at General SNR

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