Secrecy Rate Optimizations for a MISO Secrecy Channel with Multiple Multiantenna Eavesdroppers

This paper investigates secrecy rate optimization problems for a multiple-input-single-output (MISO) secrecy channel in the presence of multiple multiantenna eavesdroppers. Specifically, we consider power minimization and secrecy rate maximization problems for this secrecy network. First, we formulate the power minimization problem based on the assumption that the legitimate transmitter has perfect channel state information (CSI) of the legitimate user and the eavesdroppers, where this problem can be reformulated into a second-order cone program (SOCP). In addition, we provide a closed-form solution of transmit beamforming for the scenario of an eavesdropper. Next, we consider robust secrecy rate optimization problems by incorporating two probabilistic channel uncertainties with CSI feedback. By exploiting the Bernstein-type inequality and S-Procedure to convert the probabilistic secrecy rate constraint into the determined constraint, we formulate this secrecy rate optimization problem into a convex optimization framework. Furthermore, we provide analyses to show the optimal transmit covariance matrix is rank-one for the proposed schemes. Numerical results are provided to validate the performance of these two conservative approximation methods, where it is shown that the Bernstein-type inequality-based approach outperforms the S-Procedure approach in terms of the achievable secrecy rates.