Transmit power minimization beamforming via amplify-and-forward relays in wireless networks with multiple eavesdroppers

In this paper, we consider the collaborative use of amplify-and-forward relays to form a beamforming system and provide physical layer security for a wireless network with multiple eavesdroppers. In this paper, we investigate the relay transmit power minimization under a secrecy rate constraint via secure beamforming. To minimize the relay transmit power, we design an approximate relay power minimization (RPM) beam-forming scheme, in which an iterative algorithm combining the semidefinite relaxation (SDR) technology and the gradient-based method is devised by studying the convexity of the RPM problem. By relaxing the constraints of the RPM problem, we propose a virtual eavesdropper based RPM (VE-RPM) beamforming scheme, which transforms the multivariate RPM problem into a problem of a single variable, and thus obtain an analytical solution. Our proposed beamforming designs can work well even if the number of eavesdroppers is larger than that of relays, while the existing schemes, e.g., the null-space beamforming approaches, can not work under this condition. Simulation results are presented to demonstrate the performance improvement with the RPM beamforming schemes. It is also showed that the virtual eavesdropper approaches significantly reduce the complexity with acceptable performance degradation.