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.

中文翻译：

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利用目标位置分布实现安全集成传感和通信的最佳波束赋形


在本文中，我们研究了一种安全的集成传感和通信 （ISAC） 系统，其中一个多天线基站 （BS） 同时与一个单天线用户通信，并通过其反射的回波信号感应作为潜在窃听者的目标的位置参数。特别是，我们考虑了一个具有挑战性的场景，即目标的位置是未知和随机的，而其分布信息是根据经验数据或目标移动模式先验已知的。首先，我们推导出目标位置传感中均方误差 （MSE） 的后验 Cramér-Rao 边界 （PCRB），其表达式很复杂。为了获得更多见解，我们推导出了封闭形式的 PCRB 的紧密近似值，这表明发射波束成形应在可能的目标位置上实现“概率依赖性功率聚焦”效果。接下来，考虑在基站处采用基于人工噪声 （AN） 的波束成形结构，以减轻信息窃听并提高目标的传感反射信号功率，我们制定了发射波束成形优化问题，以在所有可能的目标（窃听者）位置中最大化最坏情况的保密率，在传感 PCRB 的最大阈值下。公式化问题是非凸的，难以求解。为了解决这个问题，我们首先表明这个问题可以通过两阶段方法来解决，首先在窃听者的信干噪比 （SINR） 上获得与任何给定阈值相对应的最佳波束形成，然后通过对阈值的一维搜索获得最佳阈值，从而获得最佳波束形成。 通过应用 Charnes-Cooper 等效变换和半定松弛 （SDR），我们将第一个问题松弛为凸形式，并进一步证明 1 阶松弛是紧密的，在此基础上，可以通过多项式时间复杂度的两阶段方法获得原始波束形成优化问题的最优解。然后，我们通过在可能的窃听者信道和/或用户信道的零空间中设计信息波束和/或AN波束，进一步提出了两种复杂度较低的次优解。数值结果验证了我们的设计在目标未知（窃听者）位置的挑战性场景中实现安全通信和高质量传感的有效性。