Transmitter Optimization for Achieving Secrecy Capacity in Gaussian MIMO Wiretap Channels

We consider a Gaussian multiple-input multiple-output (MIMO) wiretap channel model, where there exists a transmitter, a legitimate receiver and an eavesdropper, each node equipped with multiple antennas. We study the problem of finding the optimal input covariance matrix that achieves secrecy capacity subject to a power constraint, which leads to a non-convex optimization problem that is in general difficult to solve. Existing results for this problem address the case in which the transmitter and the legitimate receiver have two antennas each and the eavesdropper has one antenna. For the general cases, it has been shown that the optimal input covariance matrix has low rank when the difference between the Grams of the eavesdropper and the legitimate receiver channel matrices is indefinite or semi-definite, while it may have low rank or full rank when the difference is positive definite. In this paper, the aforementioned non-convex optimization problem is investigated. In particular, for the multiple-input single-output (MISO) wiretap channel, the optimal input covariance matrix is obtained in closed form. For general cases, we derive the necessary conditions for the optimal input covariance matrix consisting of a set of equations. For the case in which the transmitter has two antennas, the derived necessary conditions can result in a closed form solution; For the case in which the difference between the Grams is indefinite and has all negative eigenvalues except one positive eigenvalue, the optimal input covariance matrix has rank one and can be obtained in closed form; For other cases, the solution is proved to be a fixed point of a mapping from a convex set to itself and an iterative procedure is provided to search for it. Numerical results are presented to illustrate the proposed theoretical findings.