Capacity of non-coherent Rayleigh fading MIMO channels

The capacity of discrete time uncorrelated Rayleigh fading multiple input multiple output (MIMO) channels was investigated without channel state information (CSI) at either the transmitter or the receiver. To achieve the capacity, the amplitude of the multiple input needs to have a discrete distribution with a finite number of mass points with one of them located at the origin. It is shown how to compute the capacity numerically in multi-antenna configuration at any signal-to-noise ratio (SNR), with the discrete input using the Kuhn-Tucker condition for optimality. Furthermore, it is shown that at low SNR, the capacity with two mass points is optimal. As the number of receiver antennas increases, the maximum SNR at which two mass points are optimal decreases. Using this result it is argued that on-off keying is optimal in non-coherent Rayleigh fading MIMO channels at low SNR.