Non-Coherent Rayleigh Fading Channels: Properties of the Capacity-Achieving Input

The capacity-achieving input distribution of noncoherent Rayleigh fading channels is known to be discrete, and this result is refined in several ways. First, a detailed analysis is provided for bounded peak-power constraints. Second, scenarios for which binary inputs are optimal are characterized by analyzing the interplay between average-and peak-power constraints, including conditions under which the average-power constraint remains active or inactive. Third, cardinality bounds on the support of the capacity-achieving distribution are provided using methods from complex analysis, such as the oscillation theorem. Fourth, bounds on the location of the support points are provided. Fifth, new methods are derived to bound the probability masses of the optimal input distribution, along with techniques to connect the support-point locations and probabilities. Several other results are provided, such as capacity bounds.