On optimal signaling over secure MIMO channels

Optimal signalling over the wire-tap MIMO Gaussian channel is studied under the total transmit power constraint. A direct proof of the necessary condition of optimality (signaling on the positive directions of the difference channel) is given using the necessary KKT conditions. Based on it, an explicit, closed-form solution for the optimal transmit covariance matrix is given when the latter is of the full rank. The cases of weak eavesdropper and high SNR are considered. It is shown that the optimal covariance does not converge to a scaled identity in the latter regime. A refined estimate of the rank of an optimal covariance matrix is given for the general case.