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

The capacity-achieving input distribution of noncoherent 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.