Capacity of block Rayleigh fading channels without CSI

A system with a single antenna at the transmitter and receiver and no channel state information at either is considered. The channel experiences block Rayleigh fading with a coherence time of T₀ symbol times and the fading statistics are assumed to be known perfectly. The system operates with a finite average transmit power. It is shown that the capacity optimal input distribution in the T₀-dimensional space is the product of the distribution of an isotropically-distributed unit vector and a distribution on the 2-norm in the T₀-dimensional space which is discrete and has a finite number of points in the support. Numerical evaluations of this distribution and the associated capacity for a channel with fading and Gaussian noise for a coherence time T₀ = 2 are presented for representative SNRs.It is also shown numerically that an implicit channel estimation is done by the capacity-achieving scheme.