# Radar Target Localization with Multipath Exploitation in Dense Clutter Environments
## 4 Simulation Results
However, in the case of high noise levels displayed in the blue lines, the NLS-based approach shows little improvement due to the incorrect multipath propagation model estimated in the first step. Furthermore, the impact of measurement noise is analyzed by Monte Carlo simulation of a target located at (110.5, 125.5). The root mean square error (RMSE) of localization is obtained by 100 simulations as  
\[RMSE=\sqrt{\frac{1}{100}\sum_{i=1}^{100}\|\mathbf{\hat{x}}_{i}-\mathbf{x}\|^{ 2}}. \tag{57}\]  
Figure 5: Number of round-trip multipaths in the study area: (**a**) Reflection is limited to single-bounce; (**b**) reflection is limited to double-bounce. Figure 6: Ground truth and measurements with clutter: (**a**) target trajectory in the measurement space; (**b**) measurements versus target index. Figure 8 shows the RMSE with respect to different distance and angle noise STD, where Figure 8a is the grid-based result, and Figure 8b is the NLS-based result. The simulation result in Figure 8a shows that the first step of the proposed algorithm is more sensitive to distance measurement than angle measurement. Accurate localization results are obtained regardless of angle error in the condition that the STD of distance noise is less than one. However, the localization error increases along with the STD of distance when the STD of angle is less than one. The reason is that nearby samples share similar received angles, while the distance varies greatly. Therefore, if the distance measurement is precise, measurements with large angle errors are not associated with nearby samples. However, if the angle measurement is accurate and the distance measurement is inaccurate, the measurement is more likely to be associated with a nearby sample with similar angles and result in a wrong estimated location. Results illustrated in Figure 8b validate the effectiveness of the NLS-based re-localization process when the first step estimates the correct sample location. The detailed performance of the proposed algorithm is illustrated in Figure 9. The NLS-based method improves the localization accuracy when the STD of distance is less than 1.5. When the STD of the angle equals 0.5, the proposed algorithm can locate the target in the high clutter environment with precision less than the off-grid error. When the STD of the angle becomes larger, the proposed algorithm still has fairly good performance, although the NLS-based method has no more improvement. If the radar distance measurement is accurate while the angle measurement is inaccurate, which is often the case, the proposed algorithm provides precise location results as shown in Figure 9b. Figure 8: RMSE of localization versus STD of the measurement: (**a**) grid-based results; (**b**) NLS-based results. Figure 7: Localization error of the proposed algorithm at different locations. Existing multipath localization algorithms do not consider detections with clutter, and most of them have strict restrictions on the target scene, such as in the corner of a corridor [28] or in an urban canyon [25]. We choose an NLOS target localization algorithm based on grid matching of TOA returns proposed in [45] to verify the performance of the proposed algorithm. Since the original referenced algorithm does not consider measurements with clutter, the performance of the referenced algorithm is evaluated by Monte Carlo simulations with clean measurements as well as cluttered measurements. Since the referenced TOA-only algorithm does not use angle measurements, we restrict the proposed algorithm by fixed STD of angle measurements and analyze the performance of the two algorithms with varying STD of distance measurement. The Monte Carlo results illustrated in Figure 10 show that the proposed algorithm of our paper outperforms the referenced algorithm under both conditions even if the STD of angle measurement is set up to four. Table 1 shows the RMSE of the two localization algorithms.