Capacity-Achieving Input of Non-Coherent Rayleigh Fading Channels: Bounds on the Number of Mass Points

The capacity-achieving input distribution of non-coherent Rayleigh fading channels with average- and peak-power constraints is known to be discrete with a finite number of points. We sharpen this result by deriving upper and lower bounds on the number of amplitude levels. The upper bounds are based on two techniques from complex analysis: counting the number of maxima of a function that characterizes the Karush-Kuhn-Tucker conditions and an oscillation theorem. The latter provides a stronger bound but applies only if the average power constraint is inactive.