Full Rank Solutions for the MIMO Gaussian Wiretap Channel With an Average Power Constraint

This paper considers a multiple-input multiple-output (MIMO) Gaussian wiretap channel with a transmitter, a legitimate receiver and an eavesdropper, each equipped with multiple antennas. We first study the rank of the optimal input covariance matrix that achieves the secrecy capacity of the MIMO Gaussian wiretap channel under an average power constraint. The rank and other properties of the optimal solution are derived based on certain relationships between the channel matrices for the legitimate receiver and eavesdropper. Next, by obtaining necessary and sufficient conditions on the MIMO wiretap channel parameters, we determine the conditions under which the optimal input covariance matrix is full-rank or rank-deficient. For the case that the optimal input covariance is full-rank, we fully characterize the solution. When the optimal input covariance is rank-deficient, we show that the given MIMO wiretap channel can be modeled by an equivalent wiretap channel whose optimal input covariance is full rank and achieves the same secrecy capacity as the original system. Numerical results are presented to illustrate the proposed theoretical findings.