# A Cumulant-Based Method for Acquiring GNSS Signals
## 3 Cumulant-Based Acquisition Method
### FFT-Cumulant Method for GNSS Signal Acquisition
Therefore, Equation (14) simplifies to  
\[FFT\{\widetilde{y}(t)\}=FFT\Big{\{}c(t-\tau)^{2}\Big{\}}+FFT\{\hat{w}(t)\} \tag{15}\]  
Then, from the cumulant property 1 (Cumulant additivity):  
\[C_{3}\{FFT\{\widetilde{y}(t)\}\}=C_{3}\Big{\{}FFT\Big{\{}c^{2}(t-\tau)\Big{\}} \Big{\}}+C_{3}\{FFT\{\hat{w}(t)\}\} \tag{16}\]  
In this context, \(FFT\{\bullet\}\) denotes the fast Fourier transform operation. Since \(\hat{w}(t)\) is a real signal, the second term \(FFT\{\hat{w}(t)\}\) in the frequency domain is a symmetric function, which can be disregarded due to the properties of cumulants. While this method offers strong noise suppression capabilities, it involves conducting two separate searches (for code phase and carrier), which leads to a significant computational burden. To enhance computational efficiency, we propose a second algorithm, described below. Approach #2:  
1. The signal from Equation (11) is multiplied by a locally generated PRN code modulated with the BOC subcarrier reciprocal. 2. A Fourier Transform is performed on the resulting signal. 3. The cumulant is estimated. The implementation of this approach is straightforward and can be directly executed using the steps outlined above, following the block diagram shown in Figure 3b. Multiplying the signal in Equation (11) by the replica of the PRN sequence results in  
\[\widetilde{y}(t)c(t-\hat{\tau})c_{sc}^{-1}(t)=c(t-\tau)c_{sc}(t)c(t-\hat{\tau} )c_{sc}^{-1}(t)\text{cos}(2\pi(f_{IF}+f_{D})\,t+\theta)+w(t)c(t-\hat{\tau}) \tag{17}\]  
If \(\hat{\tau}\) matches with \(\tau\), the term \(c(t-\tau)c_{sc}(t)c(t-\hat{\tau})c_{sc}^{-1}(t)\) becomes one, and (15) can be rewritten as  
\[\widetilde{y}(t)=\text{cos}(2\pi(f_{IF}+f_{D})t+\theta)+\hat{w}(t) \tag{18}\]  
Equation (18) is simply a sinusoidal wave added with an unknown noise. Taking Fourier transform on (18), we obtain  
\[y(t)=c(t-\tau)c_{sc}(t)\cos(2\pi(f_{1}+f_{D})t+\theta)+w(t) \tag{10}\]0  
\[y(t)=c(t-\tau)c_{sc}(t)\cos(2\pi(f_{1}+f_{D})t+\theta)+w(t) \tag{10}\]1 (**a**) Block diagram of the FFT-Cumulant method; (**b**) Block diagram of the Modified FFT-Cumulant method. The first term on the right-hand side exhibits two peaks in the frequency domain. Consequently, after performing the cumulant estimation on Equation (19), we obtain a large value, which can be expressed as  
\[y(t)=c(t-\tau)c_{sc}(t)\cos(2\pi(f_{1}+f_{D})t+\theta)+w(t) \tag{10}\]2  
As noted earlier, the second term on the right-hand side equals zero due to the properties of the cumulant. Compared to the previous approach, this modified FFT-cumulant method reduces the search space by focusing only on finding code phases, eliminating the need for a local oscillator. As shown in Figure 3b, Approach 2 simplifies the process by reducing several blocks compared to the previous method. This approach can also be easily extended to the GPS signal case by rewriting Equation (11) as follows:  
\[y(t)=c(t-\tau)c_{sc}(t)\cos(2\pi(f_{1}+f_{D})t+\theta)+w(t) \tag{10}\]3  
In our study, we processed the data in batch mode, which presents challenges in providing precise quantitative measurements of the speed difference between the two approaches. Our typical dataset consisted of a few to several tens of seconds of sample data, approximately equivalent to a few hundred megabytes. To estimate the computational time, we utilized Matlab's tic-toc function. While this method doesn't provide extremely precise measurements, it does offer a reasonable approximation of the processing duration. Our observations showed that Approach 1 typically required between 10 and 20 min of processing time for our dataset, while Approach 2 consistently completed in less than 10 min for the same dataset. These results suggest that Approach 2 is approximately 1.5 to 2 times faster than Approach 1 in our implementation. However, it is important to note that the exact speed improvement can vary depending on factors such as the specific hardware used, the size of the dataset, and the particular signal characteristics being analyzed.