Secure Beamforming for Multiple-Antenna Amplify-and-Forward Relay Networks

In this paper, we investigate the secure relay beamforming problems for the amplify-and-forward (AF) relay network, where the transmitter sends the information to the receiver with the help of a multiple-antenna relay in the presence of an eavesdropper. We first study the secrecy rate maximization (SRM) problem in the case of complete channel state information (CSI). The rank-1 relay beamformer, which has been solved in our previous work in "Secure relay beamforming with imperfect channel side information," IEEE Trans. Veh. Technol., vol. 62, no. 5, pp. 2140-2155, June 2013, is rigorously proved to be globally optimal. Then, we turn to the case of partial eavesdropper's CSI (ECSI), where the uncertainty of ECSI is characterized as a bounded region, and study the worst-case secrecy rate maximization (WCSRM) problem. Different from the common spherical uncertainty region, the uncertainty region adopted in our paper imposes independent constraints on the channel gain and direction. With the proposed uncertainty region, we give an in-depth analysis of the WCSRM problem. We solve the rank-1 relay beamformer in an explicit form and move a step further by establishing the sufficient conditions for the global optimality of the rank-1 solution. Besides, the rank-2 relay beamformer with a special singular value decomposition (SVD) structure is also considered. By reducing the optimization variables into single variable, we show that the proposed rank-2 solution can be solved by simple line search. The asymptotic optimality of the proposed rank-2 relay beamformer in the high relay power region is also proved under mild conditions. Finally, numerical results are presented to verify the analysis in the paper.