## 3 Methodology of the Resampling Strategy and Variable Circular Correlation Time
### Realization of the Resampling Strategy for Signal Acquisition  
Sampling the time-continuous signal \(s(t)\) with the frequency \(f_{s}\), the frequency spectrum of the sampled sequence \(S(kT_{s})\) is the same as that of the original signal \(s(t)\) in the frequency domain, periodically extended with the period \(f_{s}\). Hence, with the appropriate sampling frequency, expanding components of the sampled sequence will not fold with the frequency spectrum of the original signal  
Figure 2: Acceptable sampling frequency (cyan areas) based on the bandpass sampling theory. Blue and green lines are lower and upper boundaries of the acceptable sampling frequency. The red line indicates the resampling frequency of the proposed strategy. in \([-f_{u},-f_{l}]\ \cup\ [f_{l},\ f_{u}]\). In this way, the original signal can be recovered from the sampled sequence without loss of signal quality. Based on the bandpass sampling theory, the proposed resampling strategy is realized as Algorithm 1 for signal acquisition of GNSS receivers. ```
Input: The received broadband satellite signal; Output: Doppler frequency and code phase offset of the received satellite signal. 1. Filter out side lobes of the original broadband satellite signal;
2. Update the parameters of the resampling strategy;
3. Resample the main lobe signal with the resampling frequency;
4. Restore the resampling acquisition results to that of conventional acquisition algorithms;
```  
**Return**: the acquired Doppler frequency and code phase offset of the received satellite signal. ```  
**Algorithm 1****Realization of the Resampling Strategy for GNSS Signal Acquisition**  
Note the broadband signal received by multimode multiband GNSS receivers as \(\overline{S}(t)=\{S_{L1}(t)\), \(S_{L2}(t)\), \(S_{L5}(t)\), \(S_{E5}(t)\), \(\ldots\}\). It includes GPS L1, L2, L5 signals, as well as signals transmitted by other satellite systems, such as Galileo, GLONASS, and BeiDou. We intend to assume that the main lobe refers to the GPS L2C signal, and other signals are regarded as the side lobe signals. A bandpass filter is designed to extract the main lobe signal \(S_{L2}(t)\) from the broadband signal \(\overline{S}(t)\). \[\left\{ \begin{array}{c}f_{l}=\ f_{lF}-\frac{B}{2}\\ f_{u}=\ f_{lF}+\frac{B}{2}\\ \omega_{p}=\left[\frac{2f_{l}}{f_{s}}-\Delta\omega_{f}\ \frac{2f_{u}}{f_{s}}+\Delta\omega_{f}\right]\end{array} \right. \tag{3}\]  
where \(f_{l}\) and \(f_{u}\) are the lower and upper cutoff frequencies of the main lobe signal \(S_{L2}(t)\). \(f_{s}\) is the conventional sampling frequency for the broadband signal \(\overline{S}(t)\). \(f_{lF}\) and \(B\) represent the intermediate frequency and bandwidth of \(S_{L2}(t)\). It is guaranteed that \(f_{s}\gg 2f_{u}\) in order to effectively sample \(S_{L2}(t)\). The bandpass filter is designed as an \(n\)-order Hamming-window linear-phase finite impulse response (FIR) filter. The pass band \(\omega_{p}\) is the vector of the normalized cutoff frequencies. \(\Delta\omega_{f}\) is the margin to ensure the main lobe signal reserved completely. By employing the designed bandpass filter on the broadband signal, the side lobe signals are filtered out while the main lobe remains. This plays as a foundational block to avoid frequency aliasing and reduce the data size of the sampled signal. (1) Update the Parameters of the Resampling Strategy  
A resampling filter is designed to resample the main lobe signal \(S_{L2}(t)\), obtained by the bandpass filter. The lower and upper cutoff frequencies and bandwidth of the resampling filter, \(f_{dl}\), \(f_{du}\) and \(B_{d}\), are expressed as  
\[\left\{ \begin{array}{c}f_{dl}=\ f_{l}\\ f_{du}=\ f_{u}\\ B_{d}=f_{du}-f_{dl}\end{array} \right. \tag{4}\]  
Based on the bandpass sampling theory in Theorem 1, the lower and upper boundaries of the acceptable sampling frequency, \(f_{sl}\) and \(f_{su}\), are determined as  
\[f_{sl}=\ \left\{\begin{array}{c}2f_{du},\ \  
\[f_{su}=\left\{ \begin{array}{ll}f_{sl},&\quad if\ n\leq 1\\ \frac{2f_{sl}}{n-1},&\quad if\ n>1\end{array} \right.