## 4 Simulation and Analysis of EKF-GAPF
### Parameter Estimation for Two Multipath Signals  
To further verify the feasibility of the experiment, the scenario is extended to include two multipath signals. The direct signal is assumed to have an amplitude of 1 and a delay of 0.1, while the two multipath signals have amplitudes of 0.7 and 0.5, with delays of 0.3 and 0.5, respectively. Under these conditions, the real values of the estimated parameters can be expressed as \(x=[1;0.7;0.5;0.1;0.3;0.5]\). Accordingly, modifications to certain experimental parameters are required compared to the single multipath scenario. In Equation (10), the number of multipath signals is set to \(M=2\), leading to an updated measurement matrix. For all algorithms, the state transition matrix is defined as \(F=\text{eye}(6)\), the system noise covariance \(Q\) is set to 0.0001 \(\times\) eye(6). All other experimental conditions remain unchanged. Similarly to the experiment with only a single multipath, this experiment randomly sets an initial value different from the true value and performs 1000 Monte Carlo simulations to comprehensively evaluate the algorithm's convergence speed and stability. The selected initial value is \(x_{3}=[0.96;0.52;0.41;0.09;0.21;0.45]\). Figure 9 displays the run closest to the average performance, representing the typical behavior of the algorithms. Table 3 presents the RMSE mean and standard deviation statistics for the direct signal and two multipath signals, based on 1000 Monte Carlo experiments. The results include both the mean and the standard deviation, offering a comprehensive evaluation of the algorithms' performance under the given experimental conditions. Additionally, similar performance trend is observed under different initial value conditions. As shown in Figure 9 and Table 3, EKF-GAPF maintains good parameter estimation performance even as the number of multipath components increases. Compared to the other algorithms, EKF-GAPF exhibits the smallest average RMSE, indicating better parameter estimation accuracy. Moreover, it has the smallest standard deviation, highlighting its \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Algorithm**} & \multirow{2}{*}{\begin{tabular}{c} **Amplitude of** \\ **Direct Signal** \\ \end{tabular} } & \begin{tabular}{c} **Amplitude of** \\ **the First** \\ **Multipath** \\ **Signal** \\ \end{tabular} & \begin{tabular}{c} **Amplitude of** \\ **the Second** \\ **Signal** \\ \end{tabular} & \begin{tabular}{c} **Delay of Direct** \\ **Signal** \\ \end{tabular} & \
\begin{tabular}{c} **Delay of the** \\ **Second** \\ **Multipath** \\ **Signal** \\ \end{tabular} &
\begin{tabular}{c} **Delay of the** \\ **Multipath** \\ **Signal** \\ \end{tabular} \\ \hline PF [14] & 0.0451 \(\pm\) 0.0270 & 0.1416 \(\pm\) 0.0476 & 0.0972 \(\pm\) 0.0415 & 0.0546 \(\pm\) 0.0336 & 0.0776 \(\pm\) 0.0271 & 0.0703 \(\pm\) 0.0359 \\ EKF [18] & 0.0199 \(\pm\) 0.0028 & 0.0539 \(\pm\) 0.0400 & 0.0505 \(\pm\) 0.0433 & 0.0062 \(\pm\) 0.0005 & 0.0388 \(\pm\) 0.0408 & 0.0499 \(\pm\) 0.0420 \\ EPF [20] & 0.0137 \(\pm\) 0.0049 & 0.0427 \(\pm\) 0.0278 & 0.0428 \(\pm\) 0.0305 & 0.0057 \(\pm\) 0.0004 & 0.0353 \(\pm\) 0.0228 & 0.0428 \(\pm\) 0.0280 \\ GAPF & 0.0422 \(\pm\) 0.0391 & 0.1043 \(\pm\) 0.0512 & 0.1005 \(\pm\) 0.0574 & 0.0165 \(\pm\) 0.0087 & 0.0650 \(\pm\) 0.0342 & 0.0873 \(\pm\) 0.0423 \\ EKF-GAPF & 0.0092 \(\pm\) 0.0008 & 0.0224 \(\pm\) 0.0093 & 0.0232 \(\pm\) 0.0108 & 0.0054 \(\pm\) 0.0001 & 0.0073 \(\pm\) 0.0008 & 0.0127 \(\pm\) 0.0031 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: RMSE mean and standard deviation statistics for the direct signal and two multipath signals based on 1000 Monte Carlo experiments.