This paper studies a multiple intelligent reflecting surfaces (IRSs) collaborative localization system where multiple semi-passive IRSs are deployed in the network to locate one or more targets based on time-of-arrival. It is assumed that each semi-passive IRS is equipped with reflective elements and sensors, which are used to establish the line-of-sight links from the base station (BS) to multiple targets and process echo signals, respectively. Based on the above model, we derive the Fisher information matrix of the echo signal with respect to the time delay. By employing the chain rule and exploiting the geometric relationship between time delay and position, the Cramer-Rao bound (CRB) for estimating the target's Cartesian coordinate position is derived. Then, we propose a two-stage algorithmic framework to minimize CRB in single- and multi-target localization systems by joint optimizing active beamforming at BS, passive beamforming at multiple IRSs and IRS selection. For the single-target case, we derive the optimal closed-form solution for multiple IRSs coefficients design and propose a lowcomplexity algorithm based on alternating direction method of multipliers to obtain the optimal solution for active beaming design. For the multi-target case, alternating optimization is used to transform the original problem into two subproblems where semi-definite relaxation and successive convex approximation are applied to tackle the quadraticity and indefiniteness in the CRB expression, respectively. Finally, numerical simulation results validate the effectiveness of the proposed algorithm for multiple IRSs collaborative localization system compared to other benchmark schemes as well as the significant performance gains.

翻译：本文研究了一种多智能反射面协同定位系统，该系统在网络中部署多个半无源智能反射面，基于到达时间对一个或多个目标进行定位。假设每个半无源智能反射面配备反射元件和传感器，分别用于建立从基站到多个目标的视距链路以及处理回波信号。基于上述模型，我们推导了回波信号关于时延的费舍尔信息矩阵。通过运用链式法则并利用时延与位置之间的几何关系，推导出估计目标笛卡尔坐标位置的克拉美-罗下界。随后，我们提出了一种两阶段算法框架，通过联合优化基站的有源波束成形、多个智能反射面的无源波束成形以及智能反射面选择，以最小化单目标与多目标定位系统中的克拉美-罗下界。针对单目标情况，我们推导了多智能反射面系数设计的最优闭式解，并提出了一种基于交替方向乘子法的低复杂度算法以获得有源波束成形设计的最优解。针对多目标情况，采用交替优化将原问题转化为两个子问题，其中分别应用半定松弛和逐次凸逼近来处理克拉美-罗下界表达式中的二次性和不定性。最后，数值仿真结果验证了所提算法在多智能反射面协同定位系统中相较于其他基准方案的有效性以及显著的性能增益。