## 3 Methodology of the Resampling Strategy and Variable Circular Correlation Time
### Realization of the Resampling Strategy for Signal Acquisition
\tag{6}\]  
Here, \(n=1,2,\ldots,N\). And \(N=\mathcal{Z}^{-}(f_{du}/B_{d})\). The center of the acceptable sampling frequency range \([f_{sl},\ f_{su}]\) is regard as the resampling frequency \(f_{ds}\):  
\[f_{ds}\ =\ \frac{f_{sl}+f_{su}}{2} \tag{7}\]  
The parameters of the resampling strategy, \(f_{sl},\ f_{su}\) and \(f_{ds}\), are illustrated as the blue, red and green curves in Figure 2, respectively. The number \(n\) is chosen as the maximum integer \(N\) to decrease the resampling frequency as much as possible. For that \(S_{L2}(t)\) is the high-frequency narrowband signal, the conventional sampling frequency \(f_{s}\) is much higher than the bandwidth (\(f_{s}\geq 2f_{u}\gg\) B). As the red curve in Figure 2, the resampling frequency \(f_{ds}\) is maintained in the lower frequency range \([2B_{d},\ 6B_{d}],\ (\ B_{d}\ll f_{u})\). (2) Resample the Main Lobe Signal with the Resampling Strategy  
When applying the resampling filter to the main lobe signal \(S_{L2}(t)\), the sampling frequency and intermediate frequency are updated to \(f_{ds}\) and \(f_{dIF}\). Especially, \(f_{ds}\) is significantly decreased compared to the conventional one \(f_{s}\). The original intermediate frequency \(f_{IF}\) also is equivalent to the much lower one \(f_{dIF}\), which is the remainder of \(f_{IF}\) and \(f_{ds}\), as Equation (8). The resampled sequence \(S_{d}(k)\) is expressed as Equation (9). \[f_{dIF}\ =\ \mathcal{M}(f_{IF},\ f_{ds}). \tag{8}\]  
\[\left\{ \begin{array}{ll}S_{d}(k)\ =\ S_{s}(k)\\ k\ =\ \mathcal{Z}^{+}\left(\frac{i\times f_{s}}{f_{ds}}\right)\end{array} \right. \tag{9}\]  
\[L_{d}\ =\ \mathcal{Z}^{-}\left(\frac{L\times f_{ds}}{f_{s}}\right) \tag{10}\]  
where \(\mathcal{Z}^{+}(x)\) denotes the minimum integer bigger than \(x\). \(S_{s}(\cdot)\) and \(S_{d}(\cdot)\) are the sequences obtained by sampling \(S_{L2}(t)\) with \(f_{s}\) and \(f_{ds}\), respectively. \(i\) is the index of samples in the sequence \(S_{s}(i)\), \(i\ =\ 1,2,\ldots,L\). \(L\) denotes the length of \(S_{s}(i)\), while \(k\) is that of the resampled sequence \(S_{d}(k)\), \(k\ =\ 1,2,\ldots,L_{d}\). Due to \(f_{ds}\ll f_{s}\), \(L_{d}\) is much shorter than \(L\ (L_{d}\ll L)\). Therefore, \(S_{d}(k)\) is extracted from \(S_{s}(i)\) to equivalently express the main lobe signal \(S_{L2}(t)\). Though the length is greatly reduced, \(S_{d}(k)\) is sufficient to reconstruct the main lobe signal. It is beneficial for reducing the computation and time consumption of signal acquisition. (3) Restore the Resampling Acquisition Results to that of Conventional Acquisition Algorithms  
The parameters of the resampling strategy, \(f_{ds}\) and \(f_{dIF}\), and the resampled sequence \(S_{d}(k)\) are utilized to obtain the acquisition results, including actual carrier frequency \(f_{dc}\), Doppler frequency shift \(f_{dD}\), and code phase offset \(\theta_{d}\). In order to be applied to signal tracking in the conventional way, the acquisition results of the resampling strategy are restored equivalently to the carrier frequency \(f_{c}\) and code phase offset \(\theta\) of the conventional acquisition algorithm. In this way, the capability of the proposed resampling strategy to combine with conventional acquisition algorithms has been proven. \[f_{dD}=\ f_{dc}-f_{dIF} \tag{11}\]  
\[f_{c}\ =\ f_{dD}\ +\ f_{IF} \tag{12}\]  
\[\theta=\ \mathcal{Z}^{-}\left(\frac{\theta_{d}\times f_{s}}{f_{ds}}\right)\ +1 \tag{13}\]  
With the resampling strategy acting on the original broadband signal, the signal flow chart is illustrated in Figure 3. Remarkably, the data size of the resampled signal is decreased from \(L\) to \(L_{d}\), \((L_{d}\ll L)\). Therefore, the resampling strategy is capable to greatly reduce computation and time consumption of signal acquisition.