Multiple-Antennas and Isotropically-Random Unitary Inputs: The Received Signal Density in Closed-Form

An important open problem in multiple antenna communications theory is to compute the capacity of a wireless link subject to flat Rayleigh block-fading, with no channel-state information available either to the transmitter or to the receiver. 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 happens to be capacity- achieving. In this paper, 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. The technique is based on analytically computing the expectation of an exponential quadratic function of an i.r. unitary matrix and makes use of a Fourier integral representation of the constituent Dirac delta functions in the underlying density. Our formula for the received signal density enables us to evaluate the mutual information for the case of an i.r. input and multiple anttennas, something that could previously be done only for single transmit and receive antennas. Numerical 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.