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Abstract: We consider a Gaussian multiple-input multiple-output MIMO wiretap channelmodel, where there exists a transmitter, a legitimate receiver and aneavesdropper, each node equipped with multiple antennas. We study the problemof finding the optimal input covariance matrix that achieves secrecy capacitysubject to a power constraint, which leads to a non-convex optimization problemthat is in general difficult to solve. Existing results for this problemaddress the case in which the transmitter and the legitimate receiver have twoantennas each and the eavesdropper has one antenna. For the general cases, ithas been shown that the optimal input covariance matrix has low rank when thedifference between the Grams of the eavesdropper and the legitimate receiverchannel matrices is indefinite or semi-definite, while it may have low rank orfull rank when the difference is positive definite. In this paper, theaforementioned non-convex optimization problem is investigated. In particular,for the multiple-input single-output MISO wiretap channel, the optimal inputcovariance matrix is obtained in closed form. For general cases, we derive thenecessary conditions for the optimal input covariance matrix consisting of aset of equations. For the case in which the transmitter has two antennas, thederived necessary conditions can result in a closed form solution; For the casein which the difference between the Grams is indefinite and has all negativeeigenvalues except one positive eigenvalue, the optimal input covariance matrixhas rank one and can be obtained in closed form; For other cases, the solutionis proved to be a fixed point of a mapping from a convex set to itself and aniterative procedure is provided to search for it. Numerical results arepresented to illustrate the proposed theoretical findings.



Author: Jiangyuan Li, Athina Petropulu

Source: https://arxiv.org/







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