This paper researches the cooperative localization in wireless sensor networks (WSNs) with $2\pi /\pi $ -periodic angle-of-arrival (AOA) measurements. Two types of localizers are developed from the perspectives of Bayesian inference and convex optimization. When the orientation angles are known, the positioning problem is resolved by a phase-only generalized approximate message passing (POG-AMP) algorithm with importance sampling mechanism. From the perspective of convex optimization, the positioning problem under $2\pi /\pi $ -periodic AOAs is converted as a least square (LS) problem and then resolved by the gradient-descent/projected gradient-descent method named as Type-I LS localizer. When the orientations are unknown, expectation-maximization (EM) mechanism is introduced into the POG-AMP localizer, where node positions and orientations are alternatively updated through exchanging their statistical confidences. Type-II LS localizer is constructed by alternatively executing Type-I LS and a maximum-likelihood (ML) estimator of orientation. Cramér-Rao lower bounds (CRLBs) are derived for the proposed localizers. Simulation results validate that the proposed AMP-type and LS-type localizers outperform existing localizers, AMP-type localizers successfully handle nonlinear quantization losses, and EM-framework and ML estimator handle unknown orientation problem. AMP-type localizers outperform LS-type ones, and can approach to the CRLBs even under high noise contaminations.

中文翻译：

---

具有 AOA 测量的无线传感器网络中的协作定位

本文研究了无线传感器网络（WSNs）中的协同定位。 $2\pi /\pi $- 周期性到达角 (AOA) 测量。从贝叶斯推理和凸优化的角度开发了两种类型的定位器。当方向角已知时，定位问题通过具有重要性采样机制的仅相位广义近似消息传递（POG-AMP）算法来解决。从凸优化的角度来看，定位问题 $2\pi /\pi $-周期性AOAs被转换为最小二乘（LS）问题，然后通过称为Type-I LS定位器的梯度下降/投影梯度下降方法解决。当方向未知时，期望最大化（EM）机制被引入到 POG-AMP 定位器中，其中节点位置和方向通过交换它们的统计置信度来交替更新。Type-II LS 定位器是通过交替执行 Type-I LS 和方向的最大似然 (ML) 估计器来构建的。Cramer-Rao 下界 (CRLBs) 是针对建议的定位器得出的。仿真结果验证了所提出的 AMP 型和 LS 型定位器优于现有定位器，AMP 型定位器成功处理非线性量化损失，EM 框架和 ML 估计器处理未知方向问题。