# A Cumulant-Based Method for Acquiring GNSS Signals
## 4 Simulation and Experimental Results
### Simulated Signal Acquisition  
In the first part of this section, we present some simulation results. The simulations were conducted on a standard desktop computer using MATLAB programs. For GPS signals, the results of the traditional method and the proposed method are shown in Figures 4 and 5, respectively. At first glance, it might appear that the proposed method offers no clear advantages. However, it is important to note that our method exhibits a relatively higher processing gain. This suggests that our approach may have greater potential for applications involving weak signals. More details will be given at the end of this section. The simulation procedure for the Galileo signal is similar to that used for the GPS signals. In this case, we selected the E1-C channel as our simulation signal (i.e., CBOC(6,1,1/11)). For Galileo PRN 11, the results obtained using the traditional method and the proposed method are displayed in Figures 6 and 7, respectively. Figure 8 provides a zoomed-in view of the acquisition results. As shown in these figures, the proposed method demonstrates effective side-peak suppression when acquiring the Galileo signal, indicating its superior efficiency. In the following section, we present several simulations to evaluate the performance of the different methods. The first case considers a multipath environment. In this scenario, the signal is assumed to follow a two-path channel model, as described in [23]:  
\[h(t)=\delta(t)+\alpha\delta(t-\beta) \tag{21}\]  
where \(\alpha\) is the attenuation parameter of the second path, \(\beta\) is the time delay of the second path relative to the first path, and \(\delta(t)\) is the Dirac-delta function. A Monte Carlo simulation is conducted, with \(\alpha\) set to 0.6 and \(\beta\) randomly assigned at each iteration. Figure 9 shows the results of the probability of detection against SNR. In this simulation, acquisition is considered successful if the following conditions are met:  
1. The ratio between the main peak and the second peak is greater than 2.5. 2. The position of the main peak corresponds to the correct code phase delay. Figure 4: Acquisition results of the traditional method for GPS PRN 5. Figure 5: Acquisition results of the proposed method for GPS PRN 5. Figure 10 presents the results of the probability of detection in a multipath environment. In both scenarios, it is evident that the proposed method outperforms the traditional approach. Notably, the proposed method maintains a detection rate of over 50% when the signal level is around 23 dB-Hz. This signal level is significantly below what is typically required for off-the-shelf receivers, indicating that our method may have considerable potential for indoor applications where signal strength is often weaker. Figure 6: Acquisition results of the traditional method for Galileo PRN 11. Figure 7: Acquisition results of the proposed method for Galileo PRN 11. In summary, as shown in Figure 8a,b, our proposed method effectively eliminates the side-peaks that are present in the traditional method's acquisition results. This sup  
Figure 8: (a) Partial enlarged view of Figure 6. (b) Partial enlarged view of Figure 7.  
Figure 10: Probability of detection for the two methods (with multipath). Figure 9: Probability of detection for the two methods (without multipath). pression of side-peaks significantly reduces the risk of false acquisition. Referring to Figures 9 and 10, we can quantify the improvement in detection probability as follows:  
* In a non-multipath environment (Figure 9), at a signal level of 25 dB-Hz, the proposed method achieves a detection probability of approximately 90%, compared to about 70% for the traditional method. This represents a 28.6% improvement in detection probability. * In a multipath environment (Figure 10), the improvement is even more significant. At 25 dB-Hz, the proposed method maintains a detection probability of about 80%, while the traditional method drops to approximately 50%.