An uplink system with a single antenna transmitter and a single receiver with a large number of antennas is considered. We propose an energy-detection-based single-shot noncoherent communication scheme which does not use the instantaneous channel state information (CSI), but rather only the knowledge of the channel statistics. The suggested system uses a transmitter that modulates information on the power of the symbols, and a receiver which measures only the average energy across the antennas. We propose constellation designs which are asymptotically optimal with respect to symbol error rate (SER) with an increasing number of antennas, for any finite signal to noise ratio (SNR) at the receiver, under different assumptions on the availability of CSI statistics (exact channel fading distribution or the first few moments of the channel fading distribution). We also consider the case of imperfect knowledge of the channel statistics and describe in detail the case when there is a bounded uncertainty on the moments of the fading distribution. We present numerical results on the SER performance achieved by these designs in typical scenarios and find that they may outperform existing noncoherent constellations, e.g., conventional Amplitude Shift Keying (ASK), and pilot-based schemes, e.g., Pulse Amplitude Modulation (PAM). We also observe that an optimized constellation for a specific channel distribution makes it very sensitive to uncertainties in the channel statistics. In particular, constellation designs based on optimistic channel conditions could lead to significant performance degradation in terms of the achieved symbol error rates.
