In this paper, we propose a two-stage weighted projection method (TS-WPM) for time-difference-of-arrival (TDOA)-based localization, providing provable improvements in positioning accuracy, particularly under high geometric dilution of precision (GDOP) and low signal-to-noise ratio (SNR) conditions. TS-WPM employs a two-stage iterative refinement approach that dynamically updates both range and position estimates, effectively mitigating residual errors while maintaining computational efficiency. Additionally, we extend TS-WPM to support cooperative localization by leveraging two-way time-of-arrival (TW-TOA) measurements, which enhances positioning accuracy in scenarios with limited anchor availability. To analyze TS-WPM, we derive its error covariance matrix and mean squared error (MSE), establishing conditions for its optimality and robustness. To facilitate rigorous evaluation, we develop a 3rd Generation Partnership Project (3GPP)-compliant analytical framework, incorporating 5G New Radio (NR) physical layer aspects as well as large-scale and small-scale fading. As part of this, we derive a generalized Cram\'er-Rao lower bound (CRLB) for multipath propagation and introduce a novel non-line-of-sight (NLOS) bias model that accounts for propagation conditions and SNR variations. Our evaluations demonstrate that TS-WPM achieves near-CRLB performance and consistently outperforms state-of-the-art weighted nonlinear least squares (WNLS) in high GDOP and low SNR scenarios. Moreover, cooperative localization with TS-WPM significantly enhances accuracy, especially when an insufficient number of anchors (such as 2) are visible. Finally, we analyze the computational complexity of TS-WPM, showing its balanced trade-off between accuracy and efficiency, making it a scalable solution for real-time localization in next-generation networks.

翻译：本文提出一种基于到达时间差（TDOA）的两阶段加权投影方法（TS-WPM），该方法在定位精度方面可证明地实现提升，尤其在高几何精度衰减因子（GDOP）与低信噪比（SNR）条件下表现显著。TS-WPM采用两阶段迭代优化策略，动态更新距离与位置估计值，在保持计算效率的同时有效抑制残差误差。此外，我们将TS-WPM扩展至协同定位场景，通过利用双向到达时间（TW-TOA）测量值，在锚节点可用性受限的情况下提升定位精度。为分析TS-WPM性能，我们推导了其误差协方差矩阵与均方误差（MSE），确立了其最优性与鲁棒性的适用条件。为建立严谨的评估体系，我们开发了符合第三代合作伙伴计划（3GPP）标准的分析框架，该框架融合了5G新空口（NR）物理层特性以及大尺度与小尺度衰落效应。在此框架下，我们推导了适用于多径传播的广义克拉美-罗下界（CRLB），并提出了一种创新的非视距（NLOS）偏差模型，该模型能够反映传播条件与SNR的动态变化。评估结果表明，TS-WPM在高GDOP与低SNR场景下能够实现接近CRLB的性能，且持续优于当前最先进的加权非线性最小二乘法（WNLS）。此外，基于TS-WPM的协同定位显著提升了定位精度，尤其在可见锚节点数量不足（例如仅2个）时效果尤为明显。最后，我们分析了TS-WPM的计算复杂度，证明其在精度与效率之间取得了良好平衡，使其成为下一代网络中实时定位的可扩展解决方案。