**5. Simulation Results**

In this section, the efficiency and superiority of the proposed solution will be corroborated through Monte Carlo simulations. Amiri's method presented in [14], which does not consider the transmitter and receiver position error and Zhao's method proposed in [22], which considers the statistical distributions of transmitter/receiver position error but does not use any calibration targets, are chosen as references for comparison. The exact positions of transmitters/receivers/calibration targets are the same as those in Table 1. Localization accuracy is quantitatively evaluated using root mean squares error (RMSE), which comes from 1000 independent Monte Carlo runs. In each run, the zero-mean Gaussian random errors with covariance matrices **Qr** = σ<sup>2</sup> <sup>r</sup>**Vr**, **Qr**<sup>c</sup> = σ<sup>2</sup> rc**Vr**c, **Qs** = σ<sup>2</sup> <sup>s</sup>**Vs** and **Qc** = σ<sup>2</sup> <sup>c</sup>**Vc** are added to the BRs from unknown target, BRs from calibration targets, actual transmitter and receiver positions, and actual calibration target positions, respectively, in order to simulate a real localization scenario. The setting of **Vr**, **Vr**c, **Vs** and **Vc** are also the same as that in Example 1.

First of all, in order to intuitively show the difference between target localization with and without the use of calibration targets, we plot in Figure 4 the estimated target positions from each Monte Carlo run, which forms a scatter plot for target position estimation. For comparison, the scatterplots of Amiri's method and Zhao's method are also plotted. The transmitter/receiver position error level is set to σ<sup>s</sup> = 20 m, the noise level of BR measurements from unknown target and calibration targets is set as σ<sup>r</sup> = σrc = 10 m, and the calibration target position error level is set to be σ<sup>c</sup> = 10 m. The true position of the unknown target is **u**<sup>o</sup> = [50000, 15000, 5000] Tm, which is marked with red pentagram in Figure 4 for comparison. By comparing the scatterplots of the methods, we find that with the use of calibration targets, the scattered dots of target position estimation are more closely around the target's true position, which intuitively illustrates the performance gain from the use of calibration targets. Without the use of calibration targets, considering the statistical distributions of transmitter/receiver position error can also reduce dispersion of estimated target position dots to some extent, but compared to using calibration targets, this degree of reduction in dispersion is not sufficiently impressive.

**Figure 4.** Scatter plots of estimated positions from different methods.

Now, in order to quantitatively evaluate the localization accuracy of the methods, we calculate the RMSE of the proposed solution under different error or noise conditions, and compare it with Amiri's method, Zhao's method, as well as the CRLB. As mentioned in Section 4.2, the localization accuracy of the proposed solution is related to the distance between the target and MPR system. Hence, in order to achieve a more comprehensive insight on the performance of the proposed solution, we consider two cases, i.e., the near-field case where the target is close to the MPR system, and the far-field case where the target is far away from the MPR system. The exact positions of transmitters/receivers/calibration targets remain the same as before. We first address the far-field target, whose position is set to **u**<sup>o</sup> = [120000, 120000, 12000] Tm. The results are presented in Figure 5.

**Figure 5.** Comparison of the RMSEs among different localization methods in the far-field case: (**a**) with different BR measurement noise level σ<sup>r</sup> and σ<sup>s</sup> = 20 m, σ<sup>c</sup> = 10 m; (**b**) with different transmitter/receiver position error level σ<sup>s</sup> and σ<sup>r</sup> = 10 m, σ<sup>c</sup> = 10 m; (**c**) with different calibration target position error level σ<sup>c</sup> and σ<sup>r</sup> = 10 m, σ<sup>s</sup> = 20 m.

Figure 5a plots the RMSE curves of the methods versus the BR measurement noise level. It shows that the localization RMSE of the proposed solution matches the CRLB very well and is about an order of magnitude lower than that of Amiri's method and Zhao's method at a low-to-moderate BR measurement noise level. Although it deviates from the CRLB when the BR measurement noise level is large, it is still much smaller than that of other two methods. The deviation from the CRLB, known as the thresholding phenomenon, is due to the ignored second order error terms in the design of the solution, which is invalid for large error levels. Owing to considering the statistical distributions of the transmitter/receiver position error, the RMSEs produced by Zhao's method is generally lower than that by Amiri's method. But compared with the use of the calibration targets in the proposed solution, the localization accuracy improvement brought by the consideration of transmitter/receiver position error in Zhao's method is not so significant. Figure 5b gives the RMSE curves of the methods versus the transmitter/receiver position error level. It can be seen that, the superiority of the proposed solution in localization accuracy is mainly reflected at moderate to high transmitter/receiver position error level. When the transmitter/receiver position error is small, the localization accuracy of the proposed solution and the other two methods is comparable. This again agrees very well with the theoretical performance in Section 3. Figure 5c compares the RMSEs from the methods with respect to different calibration target position error levels. As is illustrated in Figure 5c, the proposed solution always offers a remarkable advantage over the other two methods at different calibration target position error level, even when the calibration target position error is extremely large. This is in agreement with the previous simulation results for the CRLB in Section 3.

Next, the same set of simulations was repeated for a near-field target, whose position is set to be **u**<sup>o</sup> = [12000, 1200, 1200] Tm. The results are provided in Figure 6, from which we observe that the proposed solution still performs much better than the other methods. However, comparing with the corresponding results in Figure 5, we find the localization accuracy for near-field target is generally better than a far-field target, given the same noise and error levels. One reason may be that, when the target is close to the MPR system, the transmitters/receivers are far apart relative to the distance between the target and the MPR system. Thus, the localization geometry would become more regular and the corresponding geometric dilution of precision (GDOP) value would be smaller compared to the far-field case. However, on the other hand, comparing the thresholding values in Figures 5 and 6 indicates that the RMSE curves for the near-field target deviate from the CRLB at smaller values than those for the far-field target. This phenomenon is consistent with the analysis under (83) that the

equivalency between the estimate variance and the CRLB is more affected by the BR measurement noises when the target is close to the MPR system.

**Figure 6.** Comparison of the RMSEs among different localization methods in the near-field case: (**a**) with different BR measurement noise level σ<sup>r</sup> and σ<sup>s</sup> = 20 m, σ<sup>c</sup> = 10 m; (**b**) with different transmitter/receiver position error level σ<sup>s</sup> and σ<sup>r</sup> = 10 m, σ<sup>c</sup> = 10 m; (**c**) with different calibration target position error level σ<sup>c</sup> and σ<sup>r</sup> = 10 m, σ<sup>s</sup> = 20 m.

At an intuitive level, the more calibration targets are used, the better the localization accuracy is. In what follows, we will quantitatively analyze the effect of number of calibration targets on the localization accuracy by varying the number of calibration targets from 1 to 10. The positions of the transmitters and receivers remain the same as before. The positions of calibration targets and unknown target are chosen randomly from the 50 km × 40 km × 5 km volume as presented in Figure 2. The simulation results are depicted in Figure 7.

**Figure 7.** Localization accuracy versus the number of calibration targets.

Figure 7 shows the RMSE, as well as the CRLB, versus the number of calibration targets. As expected, when the number of calibration targets is small, the localization accuracy improves significantly as the number of calibration targets increases. However, it is seen that there is no obvious dependence on the number of calibration targets as soon as the number of calibration targets is larger than 3. This indicates that when the number of calibration targets reaches 3, the use of more calibration targets would only increase the computational expense and not remarkably enhance the localization

accuracy. Therefore, in the absence of any other consideration, it is reasonable to set the number of calibration targets as 3.
