Multiple-antennas and isotropically-random unitary inputs: the received signal density in closed-form

Computing the capacity of a multiple antenna wireless link subject to flat Rayleigh block-fading, with no channel-state information available either to the transmitter or to the receiver, is an important open problem. The isotropically-random (i.r.) unitary matrix-having orthonormal columns, and a probability density that is invariant to pre-multiplication by an independent unitary matrix-plays a central role in the calculation of capacity and in some special cases is capacity-achieving. We take an important step towards computing this capacity by obtaining, in closed-form, the probability density of the received signal when transmitting i.r. unitary matrices. This enables us to evaluate the mutual information for any case of interest, something that could previously only be done for single transmit and receive antennas. Simulation results show that at high SNR the mutual information is maximized for M=min(N, T/2) transmit antennas, where N is the number of receive antennas and T is the length of the coherence interval, whereas at low SNR the mutual information is maximized by allocating all transmit power to a single antenna.