Optimality of beamforming and closed form secrecy capacity of MIMO wiretap channels with two transmit antennas

A Gaussian multiple-input multiple-output (MIMO) wiretap channel model is considered. The input is a two-antenna transmitter, while the outputs are the legitimate receiver and an eavesdropper, both equipped with multiple antennas. All channels are assumed to be known. The problem of obtaining the optimal input covariance matrix that achieves secrecy capacity subject to a sum power constraint is addressed, and a closed-form expression for the secrecy capacity is obtained. The sufficient and necessary condition for beamforming to be optimal is also given.