  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.
