## 5 Simulation
### 3D Scenario  
Let us begin by testing the 3D case. As shown in Figure 2, the scenario comprises 3 sensors utilized to locate a moving source, where the noise varies from \(10^{-1}\) m\({}^{2}\) to \(10^{3}\) m\({}^{2}\). The initial positions and velocities of the sensors are listed in Table 3. The source is initially  
Figure 2: Geometric demonstration of 3D scenario (The orange and black arrow represents the moving direction of the source and the sensor, respectively). observed from \(\mathbf{u}=[285,325,275]^{T}\) m with a velocity of \(\dot{\mathbf{u}}=[-20,15,40]^{T}\) m/s. The number of observing times is \(K=16\). The RMSEs of the SDP and GN methods and the corresponding CRLB are shown in Figure 3. The GN method achieves a CRLB accuracy if the noise is not larger than \(10^{2}\) m\({}^{2}\). However, when the noise is at \(10^{3}\) m\({}^{2}\), the GN method encounters divergence problems since the estimation results from SDP are too far from the true values. The RMSE of these estimations using the SDP method fails to attain the CRLB since the effectiveness of the penalty terms is limited when only three sensors exist. This limitation hinders the transformed problem from accurately approximating the original problem and renders the SDP method suboptimal. Considering a practical unmanned aerial vehicle (UAV) object, the aim is to localize the UAV by intercepting its communication and control signal at 2.4 GHz with 10 MHz bandwidth. The observation time is \(10^{-4}\) s, and the SNR is 0 dB. Following the CRLB of TDOA estimation given by Equation (57) in [56], the theoretical noise power in TDOA is approximately 0.1 m\({}^{2}\). Thus, the noise is reasonably fixed at \(\sigma^{2}=10^{-1}\) m\({}^{2}\) in following simulations. In the second numerical experiment, the RMSE performance is assessed as the number of observing times \(K\) increases over the range \([2,16]\). The noise is fixed at \(\sigma^{2}=10^{-1}\) m\({}^{2}\), and other parameters remain consistent with those in Figure 3. As depicted in Figure 4, the performance of the proposed method improves with larger \(K\). This enhancement stems from the relative motion between the source and sensors, which increases the effective baseline length and accumulates more information about the source's position and velocity. Even with only two observations of the source, the GN algorithm remains feasible, achieving an RMSE at the CRLB level. This feasibility arises from the establishment of 8 equations to solve for 6 unknowns. Let us further examine the RMSE of the proposed SDP and GN methods as the source range increases. The initial position of the source is oriented with azimuth and elevation \(\theta=48.75\) deg and \(\phi=32.46\) deg relative to the coordinate origin. The source's initial position is calculated as  
\[\mathbf{u}=d\cdot[\cos\theta\cos\phi,\sin\theta\cos\phi,\sin\phi]^{T}, \tag{54}\]  
Figure 3: RMSE performance as noise power increases: (**a**) Position estimate, (**b**) Velocity estimate. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Sensor No.** & \(\mathbf{x(m)}\) & \(\mathbf{y(m)}\) & \(\mathbf{z(m)}\) & \(\dot{\mathbf{x}}\) (m/s) & \(\dot{\mathbf{y}}\) (m/s) & \(\dot{\mathbf{z}}\) (m/s) \\ \hline
1 & 300 & 100 & 150 & 30 & \(-20\) & 20 \\
2 & 400 & 150 & 100 & \(-30\) & 10 & 20 \\
3 & 300 & 500 & 200 & 10 & \(-20\) & 10 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Positions and velocities of the sensors in the 3D scenario. where \(d\) represents the distance between the source's initial position and the origin. Other settings are the same as above. The results are shown in Figure 5. The performance trends of the proposed algorithms align with the variations in the CRLB. As expected, the GN method can closely match the CRLB regardless of whether the source is near or far. Although SDP fails to reach the CRLB, it closely approximates the GN method in estimating the source's position.