The capacity of discrete-time memoryless Rayleigh-fading channels

We consider transmission over a discrete-time Rayleigh fading channel, in which successive symbols face independent fading, and where neither the transmitter nor the receiver has channel state information. Subject to an average power constraint, we study the capacity-achieving distribution of this channel and prove it to be discrete with a finite number of mass points, one of them located at the origin. We numerically compute the capacity and the corresponding optimal distribution as a function of the signal-to-noise ratio (SNR). The behavior of the channel at low SNR is studied and finally a comparison is drawn with the ideal additive white Gaussian noise channel.