## 1 Introduction  
Source localization has been fundamental across various applications, including radar, sonar, and navigation [1, 2, 3], and has further expanded its utility into diverse commercial domains such as indoor sensing, robotics, and sensor networks [4, 5]. This trend underscores the growing significance of precise positioning in modern society. Localization traditionally involves two stages. The first stage acquires parameters through signals transmitted from a source to several spatially distributed sensors. Subsequently, the second stage formulates a nonlinear optimization problem to determine the source position. Typical parameters for stationary source localization include Time Difference of Arrival (TDOA) [6, 7, 8, 9], Time of Arrival (TOA) [10, 11, 12], Angle of Arrival (AOA) [13, 14], and Received Signal Strength (RSS) [15, 16]. Among them, TDOA stands out as one of the most popular choices. Distinguished from TOA-based technologies, TDOA localization avoids the requirement of synchronization between the source and sensors, and its performance commonly exceeds that of AOA and RSS-based localization methods. When relative motion exists between the source and sensors, the Frequency Difference of Arrival (FDOA) can additionally be obtained. Combining FDOA with TDOA enhances the estimation accuracy of both the position and velocity of the source. However, moving source localization using TDOA and FDOA measurements presents challenges due to the highly nonlinear and nonconvex nature of the Maximum Likelihood Estimation (MLE) problem. The conventional approach to handling the nonlinear MLE problem typically involves linearizing it through Taylor-series expansion [17]. This results in an iterative algorithm that can reach the optimal estimate. However, this method requires an accurate initial guess to ensure convergence; otherwise, the iteration may be trapped by local minima. This limitation prompted researchers to seek better solutions, one of which is the closed-form solution (CFS). Two-Step Weighted Least Squares (2 SWLS) [18] is perhaps the most popular CFS for solving TDOA-FDOA localization [19; 20]. This approach first linearizes the measurement equations by introducing two redundant parameters after squaring, forming a pseudo-linear Weighted Least Squares (WLS) problem that can straightforwardly find the solution. The second stage leverages constraints between the redundant parameters and the source position and velocity to enhance the location accuracy. Studies such as [21; 22; 23] took account of the sensor uncertainties of sensor position and velocity errors, developing new CFSs to reduce estimation errors. Since sensor uncertainties are a type of common mode error that is identical for different sources, an asymptotically efficient algebraic solution for multiple disjoint sources was proposed to better calibrate the error [24]. Ref. [25] formulated the problem as a quadratic optimization with two quadratic equality constraints and solved it using iterative Constrained Weighted Least Squares (CWLS), significantly improving upon previous methods. Although the aforementioned CFSs achieve the Cramer-Rao Lower Bound (CRLB) of Mean-Square Error (MSE) and avoid possible divergence problems, they are not noise resilient if the noise is relatively large. Therefore, Multidimensional Scaling (MDS) analysis was incorporated to improve robustness against noise [26]. Another approach to enhancing noise robustness is resorting to direct position determination [27], but this comes with demanding computational complexity. Compromising between performance and complexity, convex optimization [28; 29; 30] is a competitive technology superior to the aforementioned approaches. It typically formulates TDOA-FDOA localization as a semidefinite programming (SDP) problem by utilizing Semidefinite Relaxation (SDR) to relax the nonconvex problem to a nearly equivalent convex one. The SDP solution can mitigate the divergence issue that may occur in MLE implemented by iteration.