Asymptotic capacity of space-time coding for arbitrary fading: a closed form expression using Girko's law

Several works addressed the problem of deriving the, asymptotic capacity of a wireless system with space diversity in random fading. However, the theory of random matrices was never used in evaluating the asymptotic optimal performance in closed form. By increasing the number of transmit and receive antennas the resulting capacity tend to be a stable value independent of the fading realization. This surprising result is a consequence of Girko's (1984) law, stating that the asymptotic distribution of the eigenvalues of a random matrix, with independent identically distributed zero mean complex entries, is a circle. The conditions on the probability density function of the matrix entries are satisfied by the majority of random non-line of sight fading models. Using this theory in this paper we derive the close form expression for the asymptotic capacity of a system with transmit and receive diversity, assuming independent flat fading for each transmit-receive antenna link, with equal distribution. Our formula fits the numerical results even if the number of transmit an receive antennas as small as ten.