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&#x2019;s location is <italic>unknown and random</italic>, while its distribution information is known <italic>a priori</italic> based on empirical data or target movement pattern. First, we derive the <italic>posterior Cram&#x00E9;r-Rao bound (PCRB)</italic> 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 <italic>closed form</italic>, which indicates that the transmit beamforming should achieve a <italic>&#x201C;probability-dependent power focusing&#x201D;</italic> 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&#x2019;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 <italic>two-stage</italic> 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 <italic>tight</italic>, based on which the <italic>optimal solution</italic> 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.