## 5 Simulation Results  
In this section, the efficiency and superiority of the proposed solution will be corroborated through Monte Carlo simulations. [PERSON]'s method presented in [14], which does not consider the transmitter and receiver position error and [PERSON]'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 \(\mathbf{Q}_{\mathrm{r}}=\sigma_{\mathrm{r}}^{2}\mathbf{V}_{\mathrm{r}} \mathbf{\prime}_{\mathrm{r}}\), \(\mathbf{Q}_{\mathrm{c}}=\sigma_{\mathrm{r}}^{2}\mathbf{V}_{\mathrm{rc}}\), \(\mathbf{Q}_{\mathrm{s}}=\sigma_{\mathrm{s}}^{2}\mathbf{V}_{\mathrm{s}}\) and \(\mathbf{Q}_{\mathrm{c}}=\sigma_{\mathrm{c}}^{2}\mathbf{V}_{\mathrm{c}}\) 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 \(\mathbf{V}_{\mathrm{r}}\), \(\mathbf{V}_{\mathrm{rc}}\), \(\mathbf{V}_{\mathrm{s}}\) and \(\mathbf{V}_{\mathrm{c}}\) 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 [PERSON]'s method and [PERSON]'s method are also plotted. The transmitter/receiver position error level is set to \(\sigma_{\rm s}=20\) m, the noise level of BR measurements from unknown target and calibration targets is set as \(\sigma_{\rm r}=\sigma_{\rm rc}=10\) m, and the calibration target position error level is set to be \(\sigma_{\rm c}=10\) m. The true position of the unknown target is \(\mathbf{u}^{\rm o}=[50000,15000,5000]^{\rm T}\)m, 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. 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 [PERSON]'s method, [PERSON]'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 \(\mathbf{u}^{\rm o}=[120000,120000,12000]^{\rm T}\)m. The results are presented in Figure 5.  
Figure 4: Scatter plots of estimated positions from different methods. Figure 5a plots the RMSE curves of the methods versus the BR measurement noise level.