Mutual information of non-coherent Rayleigh fading channels with Gaussian input

In this paper, the mutual information of a discrete time Rayleigh fading channel is considered, where neither the transmitter nor the receiver has the knowledge of the channel state information. We derive a closed form expression for the mutual information of single input single output Rayleigh fading channel when the input distribution is complex Gaussian. We compare the channel capacity and the mutual information attained with Gaussian input using the derived closed form expressions. Furthermore, we argue that the mutual information is bounded by the signal to noise ratio with the simulation results showing the sub optimality of Gaussian signalling in non-coherent Rayleigh fading channels.