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Optimal signalling over the Gaussian MIMO wire-tap channel is studied underthe total transmit power constraint. A closed-form solution for an optimaltransmit covariance matrix is obtained when the channel is strictly degraded.In combination with the rank-1 solution, this provides the completecharacterization of the optimal covariance for the case of two transmitantennas. The cases of weak eavesdropper and high SNR are considered. It isshown that the optimal covariance does not converge to a scaled identity in thehigh-SNR regime. Necessary optimality conditions and a tight upper bound on therank of an optimal covariance matrix are established for the general case,along with a lower bound to the secrecy capacity, which is tight in a number ofscenarios.

Optimal Signaling for Secure Communications over Gaussian MIMO Wiretap Channels