Minimizing outage probability for a class of MIMO channels

In this paper we investigate the outage probability of multiple-input multiple-output (MIMO) channels, when the probability distribution of the channel gains is not completely known to the transmitter and receiver. The partial knowledge of a true probability distribution of the channel nu_H(h) is modelled by a relative entropy constraint H(nu_H||mu_H,nom) les R, R ges 0, where R is the distance from the so-called nominal channel distribution mu_H,nom(H). The outage probability of this compound channel is equal to the minimax of the probability of the event, "the actual transmission rate exceeds the instantaneous mutual information", where the maximum is taken over all possible channel distributions in the uncertainty set, while the minimum is over all possible transmit covariance matrices. Explicit expressions are derived for minimax outage probability. In addition, a number of properties of maximizing outage probability over the set of channel distributions are derived, including upper and lower bounds on the minimax outage probability. Finally, an alternative relative entropy model of uncertainty which is described using as nominal distribution the eigenvalues of the channel matrix H, is also discussed. The results illustrate how outage probability degrades as a function of uncertainty.