Optimality of beamforming for secrecy capacity of MIMO wiretap channels

A Gaussian multiple-input multiple-output (MIMO) wiretap channel model is considered, where there exists a transmitter, a legitimate receiver and an eavesdropper each equipped with multiple antennas. The optimality of beamforming for secrecy capacity subject to sum power constraint is studied, and two sufficient conditions for beamforming to be globally optimal are given. The first sufficient condition states that when the difference between the Gram matrices of legitimate and eavesdropper channel matrices has exactly one positive eigenvalue, then beamforming is globally optimal. An alternative sufficient condition, which involves convex optimization, is also provided. For the case in which beamforming is globally optimal, the secrecy capacity is obtained. Otherwise, an upper bound of the difference between the secrecy capacity and beamforming secrecy rate is provided.