## 4 Simulation and Analysis of EKF-GAPF
### Comparison of Signal Estimation Accuracy
Specifically, the estimation errors for the delays of the direct and multipath signals converge to 0.004, corresponding to approximately 1.2 m, and 0.006, corresponding to approximately 1.8 m. Table 2 shows the mean and standard deviation of the RMSE of five algorithms under this randomly \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Algorithm**} & **Amplitude of** & **Amplitude of** & **Delay of** & **Delay of** \\  & **Direct Signal** & **Multipath** & **Direct Signal** & **Multipath** \\ \hline PF [14] & 0.2038 & 0.2037 & 0.0342 & 0.0534 \\ EKF [18] & 0.0478 & 0.0517 & 0.0119 & 0.0238 \\ EPF [20] & 0.0493 & 0.0479 & 0.0095 & 0.0202 \\ GAPF & 0.0439 & 0.0453 & 0.0130 & 0.0141 \\ EKF-GAPF & 0.0290 & 0.0285 & 0.0065 & 0.0064 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: RMSEs of estimation for direct and multipath signals. Figure 7: Absolute error of estimation for direct and multipath signals (under true initial values): **(a)** Absolute error of amplitude estimation for direct signals; **(b)** absolute error of amplitude estimation for multipath signals; **(c)** absolute error of delay estimation for direct signals; **(d)** absolute error of delay estimation for multipath signals. set initial value, where EKF-GAPF outperforms the other algorithms across all evaluation metrics. In the estimation of the direct signal amplitude, EKF-GAPF improves upon PF, EKF, GAPF, and EPF by 69.7%, 51.9%, 52.2%, and 51.8%, respectively. For the amplitude of multipath signal, EKF-GAPF shows improvements of 70.3%, 37.8%, 62.0%, and 53.7%, respectively. In terms of direct signal delay estimation, EKF-GAPF achieves improvements of 65.3%, 21.5%, 58.8%, and 33.7% over the other four algorithms. For the multipath signal delay estimation, EKF-GAPF shows improvements of 77.1%, 42.9%, 71.0%, and 69.1%. Moreover, the RMSE standard deviation of EKF-GAPF is significantly lower than that of other algorithms, indicating greater stability under random initial conditions. Additionally, a similar performance trend is observed under different initial value conditions. Therefore, EKF-GAPF proposed in this paper outperforms the other four algorithms in both estimation accuracy and stability. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Algorithm**} & **Amplitude of** & **Amplitude of** & **Delay of** & **Delay of** \\  & **Direct Signal** & **Multipath** & **Direct Signal** & **Multipath** \\ \hline PF [14] & 0.1264 \(\pm\) 0.0441 & 0.1747 \(\pm\) 0.0357 & 0.0739 \(\pm\) 0.0063 & 0.0789 \(\pm\) 0.0159 \\ EKF [18] & 0.0797 \(\pm\) 0.0250 & 0.0834 \(\pm\) 0.0237 & 0.0326 \(\pm\) 0.0023 & 0.0317 \(\pm\) 0.0021 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: RMSE mean and standard deviation statistics for direct signal and one multipath signal based on 1000 Monte Carlo experiments. Figure 8: Absolute error of estimation for direct and one multipath signal (under random initial values): (**a**) Absolute error of amplitude estimation for direct signals; (**b**) absolute error of amplitude estimation for multipath signals; (**c**) absolute error of delay estimation for direct signals; (**d**) absolute error of delay estimation for multipath signals.