## 5 Numerical Results
CFJLAS performs comparably to the RIS and GN when the noise is below 20 dB, but its RMSE deteriorates rapidly as the noise exceeds 20 dB. Regarding the bias, the RIS, GN, and CFJLAS behave at the same level if the noise is not significant. CFJLAS gradually deviates from the others, due to it applying the first-order Taylor expansion that drops the noise terms higher than the first order. The influence of dropping higher-order noise terms is reasonable and does not affect the performance since they are not significant if the noise is relatively small. However, when the noise is large where the higher-order noise terms cannot be neglected, the approximation error may dominate the performance, leading to rapid deterioration. The GN suffers from the divergence problem after 30 dB. The RIS is the best, and the bias of the RIS is better than that of its counterparts in large noise regions. Next, we investigate the performance behavior as the position errors of the ANs change. The noise is fixed at 0 dB, and the variance of the AN position error varies from 10 to \(10^{5}\) m\({}^{2}\). The RMSE and bias of each algorithm are illustrated in Figures 9-12. When the error level is below \(10^{3.5}\), the RMSE of the RIS, GN, and CFJLAS achieve the CRLB. However, if \(\sigma_{\mathrm{s}}^{2}>10^{3.5}\) m\({}^{2}\), the RMSE of the GN diverges from the CRLB, and CFJLAS begins to deviate from the CRLB. The RIS maintains a performance close to the CRLB, demonstrating that it is better than the GN and CFJLAS. The bias behaves similar to the RMSE. The biases of the RIS, GN, and CFJLAS are comparable under the small-error condition, but the RIS shows its superiority if the error is significant. Figure 4: ANs and UN in the simulation scene. Figure 5: The RMSE of the UN location estimation as the noise power increases. Figure 6: The bias of the UN location estimation as the noise power increases. Figure 7: The RMSE of the UN velocity estimation as the noise power increases. Figure 8: The bias of the UN velocity estimation as the noise power increases. \[RMSE(\mathbf{\mathrm{p}}) =\sqrt{\frac{1}{T}\sum_{t=1}^{T}\|\mathbf{\mathrm{p}}^{(t)}-\mathbf{ \mathrm{p}}^{o}\|^{2}} \tag{84}\]0 The bias of the UN location estimation as the position error of the AN increases. \[RMSE(\mathbf{\mathrm{p}}) =\sqrt{\frac{1}{T}\sum_{t=1}^{T}\|\mathbf{\mathrm{p}}^{(t)}-\mathbf{ \mathrm{p}}^{o}\|^{2}} \tag{84}\]1 The RMSE of the UN location estimation as the position error of the AN increases. The simulations validated that the RIS is robust to measurement noise and position errors of ANs compared to the methods. We then investigated the performance of the RIS through convergence, comparing it to the GN, the other iterative solution. Convergence is obtained if the residual between adjacent iterations reaches the threshold \(\|\hat{\mathbf{\mu}}^{(k+1)}(1:2N)-\hat{\mathbf{\mu}}^{(k)}(1:2N)\|<\epsilon\). \(\wp_{\epsilon}\) is the probability of attaining the threshold. The maximum number of iterations was set to 100,000, and the threshold \(\epsilon\) was set to 0.01. In each trial, the initial estimate \(\mathbf{\mu}^{(0)}\) was randomly generated as \(\mathbf{\mu}^{(0)}=\mathbf{\mu}^{o}+e\Delta\mathbf{\hat{s}}\), where \(e\) represents the error coefficient. \(\Delta\mathbf{\hat{s}}\) is a \(2N+2\)-dimensional random vector, where the 1 to \(N\), \(N+1\) to \(2N\), \(2N+1\)-th, and \(2N+2\)-th elements are uniformly distributed within the range \((-0.5,0.5)\) m, \((-0.05,0.05)\) m/s, \((-5,5)\) ns, and \((-0.05,0.05)\) ppm. \(e\) was fixed at 100. The error \(\sigma_{s}\) was set as 0.5 m. Figure 13 reveals the convergence probability as the noise level increases. When the noise is relatively low, such as below 35 dB, both the RIS and GN converge to the threshold \(\epsilon\) with probability 1. However, as the noise increases, the RIS still maintains a significantly high convergence probability that is over 80%.