## 3 Experimental Results and Analysis
### Simulated Signal Experiments
#### 3.1.1 Fine Satellite Geometry  
We first consider an open-sky reception scenario with eight satellites, using a 20\({}^{\circ}\) elevation mask at a \(C/N_{0}\) of 30 dB/Hz. The corresponding PRN code numbers, azimuths, and elevations of the satellites are summarized in Table 6.  
The position error (PE), clock bias error (CBE), number of evaluated points (PN), and execution time (ET) of the ten algorithms were compared, as shown in Table 7. Due to fundamental differences in processing architecture between traditional 2 SP and DPE, the execution time of 2 SP is not included in the analysis. In 2 SP, pseudoranges are first obtained through acquisition and tracking, followed by near real-time least-squares solving. In contrast, DPE directly estimates the position by correlating raw signals without intermediate measurements. Therefore, the analysis focuses on evaluating the efficiency of different search algorithms within the DPE framework. As demonstrated in Table 7, the MRG, ARS, GWO, DBO, and ADBO algorithms achieve comparable positioning performance to 2 SP, highlighting the effectiveness of the proposed DPE-SI framework. However, even in this ideal case, 2 SP still provides the best results. Achieving similar results with DPE-SI methods would likely require either a finer grid or an increase in the number of populations and iterations, though this comes with higher computational costs. Among the DPE-SI methods, GWO, DBO, and ADBO achieve comparable positioning accuracy to the MRG and ARS but require significantly fewer candidate points to evaluate. Gr \begin{table}
\begin{tabular}{c c c c c c c c} \hline
**PRN** & **2** & **4** & **6** & **23** & **25** & **26** & **29** & **31** \\ \hline Azimuth & 27.5 & 195.4 & 341.3 & 291.6 & 79.1 & 205.4 & 111.2 & 152.4 \\ Elevation & 29.9 & 46.5 & 20.3 & 39.7 & 32.2 & 38.5 & 41.5 & 53.2 \\ \hline \end{tabular}
\end{table} 
Table 6: Azimuth and elevations of satellites with fine geometry. \begin{table}
\begin{tabular}{c c c c c} \hline
**Item** & **Step 1** & **Step 2** & **Step 3** & **Step 4** \\ \hline North/East range (m) & \(\pm\)3000 & \(\pm\)1000 & \(\pm\)300 & \(\pm\)100 \\ N/E step Size (m) & 100 & 50 & 25 & 10 \\ Up range (m) & \(\pm\)600 & \(\pm\)400 & \(\pm\)200 & \(\pm\)100 \\ Up Step Size (m) & 100 & 50 & 25 & 15 \\ Clock-bias range (m) & \(\pm\)15000 & \(\pm\)5000 & \(\pm\)2500 & \(\pm\)1000 \\ Bias Step Size (m) & 150 & 50 & 25 & 12.5 \\ Points number (\(10^{3}\)) & 144 & 128 & 76.8 & 51.2 \\ \hline \end{tabular}
\end{table} 
Table 5: Description of the search space stage used in MRG. exhaustive evaluation of fixed-resolution candidate points, while random search algorithms rely on sufficient iterations, making them computationally expensive. In contrast, swarm intelligence algorithms employ populations of agents that intelligently explore the search space, focusing on promising regions and avoiding unnecessary evaluations. This efficiency leads to a substantial reduction in computation time for DPE compared with MRG and ARS. Nonetheless, certain algorithms like PSO, HHO, SSA, and PIO are sensitive to zeros and often converge to an incorrect search space, typically the zero point in the optimization problem. Consequently, these algorithms yield similar positioning errors. On the other hand, the PO algorithm frequently produces large errors due to its tendency to fall into local optima, resulting in a relatively high average positioning error. To visually demonstrate the performance of the swarm intelligence algorithms, the convergence curves and positioning error curves are presented in Figure 6. According to the results in Figure 6a, PSO and SSA converge relatively slowly, whereas the other algorithms achieve low fitness values within as few as 50 iterations. This is consistent with previous findings that PSO and SSA are better suited for unimodal problems, whereas the PIO algorithm exhibits poorer overall performance. Unfortunately, the majority of algorithms, including PO and HHO, suffer from falling into local optima, leading to stagnation.