Joint Analog Beamforming and Antenna Position Design for Secure Communication systems With Movable Antennas

Movable antennas (MA) are a novel technology that allows for the flexible adjustment of antenna positions within a specified region, thereby enhancing the performance of wireless communication systems. In this paper, we explore the use of MA to improve physical layer security in an analog beamforming (AB) communication system. Our goal is to maximize the secrecy rate by jointly optimizing the transmit AB and MA position, subject to constant modulus (CM) constraints on the AB and position constraints for the MA. The resulting problem is non-convex, and we propose a penalty product manifold (PPM) method to solve it efficiently. Specifically, we convert the inequality constraints related to MA position into a penalty function using smoothing techniques, thereby reformulating the problem as an unconstrained optimization on the product manifold space (PMS). We then derive a parallel conjugate gradient descent (PCGD) algorithm to update both the AB and MA position on the PMS. This method is efficient, providing an analytical solution at each step and ensuring convergence to a KKT point. Simulation results show that the MA system achieves a higher secrecy rate than systems with fixed-position antennas.