## 1 Introduction
While SDR sometimes relaxes the problem with insufficient tightness, resulting in a suboptimal estimate, it is sufficiently close to the true value and can serve as an appropriate initialization to guarantee the convergence of iterative MLE [31; 32]. The other strategy to improve the accuracy of SDP is error correction [33], which is capable of achieving the CRB level of performance. The aforementioned studies focus on developing feasible and optimal solutions for estimating the source position, but they did not take into account the influence of sensor placement [34]. The optimal localization performance is primarily determined by the Geometric Dilution of Precision (GDOP). Optimizing the sensor placement can improve the GDOP so a better performance is achievable. In general, the minimum number of sensors required for unambiguously localizing a moving source in an \(N\)-dimensional space using TDOA and FDOA is more than \(N+1\), as the TDOA and FDOA equations are nonlinear with respect to the source position and velocity. Some studies [19; 26] introduce additional instrumental parameters for pseudo-linearization, thus necessitating more sensors to ensure the proposed closed-form algorithms are solvable. Other researchers have explored localization with the minimum number of sensors. For instance, \(N+1\) sensors are required for TDOA localization [35], while elliptic localization in Multiple-Input Multiple-Output (MIMO) systems requires at least one transmitter and three receivers or two transmitters and two receivers [36; 37]. These methods leverage measurements from sensors or receivers extracted from one frame of snapshots to establish the equations; thus, they are limited by the actual number of sensors or receivers. Typically, to guarantee that the optimization problem is overdetermined and the solution for source location is unique, it is necessary to incorporate more than \(N+1\) sensors into the localization system in most scenarios [38; 39]. When the number of sensors is limited, one possible method to localize a moving source is known as Target Motion Analysis (TMA). TMA is a technique used in various fields, notably in military and maritime operations, to determine the movement characteristics of a target based on observations from one or more sensors. The primary goal of TMA is to estimate the position, velocity, and possibly other relevant parameters of a target, such as its course and acceleration. TMA incorporates a kinetic model, allowing it to handle moving source localization with fewer sensors. The early development of TMA evolved from bearing-only (BO) measurements [40]. The team led by [PERSON] made significant contributions to BO-TMA closed-form solutions, addressing issues such as two-dimensional single-station TMA [41], bias reduction in target motion parameter estimation [42], and the effect of receiver station location errors [43]. These research findings were further extended to three-dimensional targets [44]. TMA is not only designed for bearing-only measurements but is also applicable to TDOA [45] and FDOA [46]. In a scenario where the number of sensors is smaller than the dimension of space, leveraging TMA can help handle constant velocity moving source localization using TDOA and FDOA measurements. This paper focuses on the TDOA-FDOA localization problem when the number of sensors is less than \(N+1\) for an \(N\)-dimensional scenario. By incorporating the kinetic model of the source, which is an effective method involving TMA, it becomes feasible to solve for \(2N\) unknowns, such as position and velocity. The sensors observe multiple times while the source is moving, obtaining multiple sets of TDOAs and FDOAs. This paper first formulates the moving source localization as a MLE problem, which is nonlinear and nonconvex. The MLE is then implemented by Gauss-Newton (GN) iteration. To find an initialization to start the GN iteration, the original MLE is then relaxed to a convex optimization problem through SDR technology, resulting in an SDP problem with penalties.