Optimal Beamforming for Secure Integrated Sensing and Communication Exploiting Target Location Distribution

In this paper, we study a secure integrated sensing and communication (ISAC) system where one multi-antenna base station (BS) simultaneously communicates with one single-antenna user and senses the location parameter of a target serving as a potential eavesdropper via its reflected echo signals. In particular, we consider a challenging scenario where the target’s location is unknown and random, while its distribution information is known a priori based on empirical data or target movement pattern. First, we derive the posterior Cramér-Rao bound (PCRB) of the mean-squared error (MSE) in target location sensing, which has a complicated expression. To draw more insights, we derive a tight approximation of the PCRB in closed form, which indicates that the transmit beamforming should achieve a “probability-dependent power focusing” effect over possible target locations. Next, considering an artificial noise (AN) based beamforming structure at the BS to alleviate information eavesdropping and enhance the target’s reflected signal power for sensing, we formulate the transmit beamforming optimization problem to maximize the worst-case secrecy rate among all possible target (eavesdropper) locations, subject to a maximum threshold on the sensing PCRB. The formulated problem is non-convex and difficult to solve. To deal with this problem, we first show that the problem can be solved via a two-stage method, by first obtaining the optimal beamforming corresponding to any given threshold on the signal-to-interference-plus-noise ratio (SINR) at the eavesdropper, and then obtaining the optimal threshold and consequently the optimal beamforming via one-dimensional search of the threshold. By applying the Charnes-Cooper equivalent transformation and semi-definite relaxation (SDR), we relax the first problem into a convex form and further prove that the rank-one relaxation is tight, based on which the optimal solution of the original beamforming optimization problem can be obtained via the two-stage method with polynomial-time complexity. Then, we further propose two suboptimal solutions with lower complexity by designing the information beam and/or AN beams in the null spaces of the possible eavesdropper channels and/or the user channel, respectively. Numerical results validate the effectiveness of our designs in achieving secure communication and high-quality sensing in the challenging scenario with unknown target (eavesdropper) location.