## 4 Simulation Results
### Acquisition Performance  
Since it is hard to give an analysis in a meantime-SNR graph, only theoretical results were given in this paper. Moreover, we assumed that the unit of mean acquisition time is the time taken by one correlation calculation.  
Figure 6 plots the theoretical mean acquisition time of DC-XFAST with different verification times (Nth). As discussed in the third part, analysis models for mean acquisition time of these methods are all based on Pd and Pfa, and the greater the threshold, the lower the false alarm probability and detection probability. The figure shows that even when the false alarm probability is lower, the low detection probability of models with higher verification times could not be counteracted, thus the mean acquisition time is longer. Thus, the application of DC-XFAST should balance false alarm and time cost. Figure 7 illustrates the acquisition performance of three folding methods as mean acquisition time. The four figures all show that when the incoming signal is weak, that is, the incoming SNR is between \(-\)25 dB and \(-\)10 dB, folding degrades the PN-code correlation properties and the degradation is reflected by the mean acquisition time as well. By the position verification of multi-peaks, DC-XFAST decreases the false alarm probability and thus diminishes the degradation of folded PN-code correlation properties. With the same local folding number, and when the SNR is not  
Figure 5: Theoretical (“T”, continuous line) and simulation (“S”, dotted line) results for detection probability under 1–4 verification times (Nth) of DC-XFAST (“DX”), DF-XFAST (“DF”) and XFAST (“XF”) with folding number equals to 5, 10 and 20 and Nth equals to 1(a), 2(b), 3(c), 4(d).  
larger than \(-15\) dB, the mean acquisition time of DC-XFAST is about 1/10 that of XFAST, and about 1/5 that of DF-XFAST when Nth is 3 and 4. When the incoming SNR is larger than \(-10\) dB, with a larger folding number, the code phases searched in parallel are increased and the benefit of folding is represented in the decrease of mean acquisition time.  
Figure 6: Theoretical results for mean acquisition time under 1–4 verification times (Nth) of DC-XFAST (“DX”).  
Figure 7: Theoretical results for mean acquisition time of DC-XFAST (“DX”), DF-XFAST (“DF”) and XFAST (“XF”) with N equals to 4096 and Nth equals to 1(a), 2(b), 3(c), 4(d).  
Figure 8 focuses on the acquisition performance analysis between the folding method and unfolding method. By folding the local signal, folding methods like XFAST, DF-XFAST and DC-XFAST decrease the cardinality of the code phase, hence, when the SNR is higher than \(-15\) dB, folding methods consume less acquisition time than ZP. However, when the SNR is lower than \(-18\) dB, the benefit of folding decreases and the cost appears. As shown in Figure 8, XFAST and DF-XFAST with weak signal have longer mean acquisition time than ZP. With respect to the proposed DC-XFAST method, DC-XFAST shows a noticeable advantage over XFAST and DF-XFAST in a low SNR environment. Especially when \(\mathrm{N_{th}}\) equals to 1 and 2, the mean acquisition time of DC-XFAST is about 1/50 that of ZP. When \(\mathrm{N_{th}}=3\) and 4, the advantage of multi-peak verification cannot compensate for the degradation of folding, DC-XFAST shows a slight disadvantage compared with ZP when the SNR is less than \(-20\) dB, but as shown in Figure 8c,d, when the SNR is between \(-20\) dB and \(0\) dB, DC-XFAST consumes much less acquisition time than ZP.