## 4 Simulation and Analysis of EKF-GAPF
### Comparison of Signal Estimation Accuracy  
To verify the performance of EKF-GAPF in signal estimation, under the condition that the power of the received signal and the noise signal are equal and the initial value is true, that is, \(x_{2}\) = [1; 0.7; 0.1; 0.3], the estimation results of EKF-GAFF are compared with those of PF, EKF, and EPF proposed in the previous literature and GAPF derived by AGA introduced in this paper. In the experiment, the Root Mean Square Error (RMSE) [31] is used to evaluate the accuracy of algorithms in signal estimation, which is calculated as follows:  
\[\text{RMSE}=\sqrt{\sum_{k=1}^{n}\left(x_{k}-x_{k}^{*}\right)^{2}/n} \tag{22}\]  
where \(n\) denotes the number of iterations, \(x_{k}\) is the estimated value, and \(x_{k}^{*}\) is the true value. Figure 6 shows the comparison of estimation results for direct and multipath signals, Figure 7 shows the absolute error of estimations, and Table 1 shows the RMSEs of estimations. These results correspond to a single run of the experiment. Compared with GAPF proposed in [16], which can only estimate signal delay, the signal model proposed in this paper can not only accurately estimate signal delay but also estimate amplitude. As shown in Figures 6 and 7, estimations of EKF-GAPF have smaller fluctuations and is closest to the true value, indicating that EKF-GAPF performs well. From Table 1, it can be seen that EKF-GAPF performs best on all four evaluations metrics, followed by EPF and GAPF, while EKF and PF perform relatively poorly. In the estimation of the direct signal amplitude, the EKF-GAPF outperforms PF, EKF, GAPF, and EPF by 85.8%, 39.3%, 33.9%, and 41.2%, respectively. For the amplitude of multipath signal, EKF-GAPF shows improvements of 86.0%, 44.9%, 37.1%, and 40.5%, respectively. In terms of direct signal delay estimation, EKF-GAPF improves by 81.0%, 45.4%, 50%, and 31.6%, with the RMSE of the direct signal delay reduced to 0.0065, corresponding to about 1.95 m. For the multipath signal delay estimation, EKF-GAPF achieves improvements of 88.0%, 73.1%, 54.6%, and 68.3%, with the  
Figure 5: Distribution of particle weights after 500 iterations: (**a**) Particle weight distribution using the particle filter (PF); (**b**) particle weight distribution using the EKF-GAPF. To verify the convergence and parameter estimation capabilities of different algorithms in the absence of known true signal parameters, this study randomly selects an initial value and conducts 1000 Monte Carlo simulations under the same initial conditions for each algorithm. The mean and standard deviation of the RMSE are computed to evaluate the estimation accuracy and stability of the algorithms. In the experiment, the selected initial value is \(x_{2}=\) [ 0.7415; 0.0529; 0.4197; 0.5240], which deviates from the real state value, thereby better reflecting the uncertainty conditions commonly encountered in practical applications. Figure 8 shows the estimated absolute error of the direct signal and multipath signal with a randomly set initial value. These figures present the run that is closest to the average performance, representing the typical behavior of the algorithm. It can be seen that in terms of amplitude estimation, EKF-GAPF performs well throughout the entire iterative process, with estimation errors continuously decreasing and eventually reaching the minimum value (the estimation error of direct signal amplitude can converge to 0.011, and the estimation error of multipath signal amplitude can converge to 0.002). The initial estimation error of EKF decreases rapidly, but the convergence effect is limited in the later period, and it fails to reach the minimum error. The overall performance of PF is poor. Although the initial error has decreased, its convergence speed is significantly lower than other algorithms. In terms of delay estimation, PF and EKF perform poorly and fluctuate significantly. In comparison, EKF-GAPF has the smallest estimation error in the entire iterative process and obtains the best convergence result.