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 which serves 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. First, we derive the posterior Cram\'er-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 it in closed form, which indicates that the transmit beamforming should achieve a "probability-dependent power focusing" effect over possible target locations, with more power focused on highly-probable locations. Next, considering an artificial noise based beamforming structure, we formulate the transmit beamforming optimization problem to maximize the worst-case secrecy rate among all possible target (eavesdropper) locations, subject to a threshold on the sensing PCRB. The formulated problem is non-convex and difficult to solve. We 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 via one-dimensional search. By applying the semi-definite relaxation (SDR) technique, we relax the first problem into a convex form and further prove that the relaxation is tight, based on which the optimal solution of the original beamforming optimization problem can be obtained with polynomial-time complexity. Then, we further propose two suboptimal solutions with lower complexity. Numerical results validate the effectiveness of our designs.

翻译：本文研究了一种安全集成感知与通信（ISAC）系统，其中多天线基站（BS）同时与单天线用户通信，并通过目标反射的回波信号感知潜在窃听者（即目标）的位置参数。特别地，我们考虑了一个具有挑战性的场景：目标位置未知且随机，但其分布信息已知。首先，我们推导了目标位置感知均方误差（MSE）的后验克拉美-罗界（PCRB），该表达式较为复杂。为深入分析，我们推导了其闭式紧近似，表明发射波束成形应在可能的目标位置上实现“概率依赖型功率聚焦”效应，即更多功率集中在高概率区域。随后，考虑基于人工噪声的波束成形结构，我们构建了发射波束成形优化问题，在满足感知PCRB阈值约束的条件下，最大化所有可能目标（窃听者）位置中的最差情况保密速率。该问题非凸且求解困难。我们证明该问题可通过两阶段方法求解：首先针对窃听者信干噪比（SINR）的任意给定阈值获取最优波束成形，然后通过一维搜索获得最优阈值。通过应用半定松弛（SDR）技术，我们将第一个问题松弛为凸形式，并进一步证明松弛的紧致性，从而在多项式时间复杂度内获得原波束成形优化问题的最优解。随后，我们提出两种低复杂度的次优解。数值结果验证了所提方案的有效性。