## 3 Methodology of the Resampling Strategy and Variable Circular Correlation Time
### Coarse Acquisition with Variable Circular Correlation Time  
Conventional acquisition methods usually set the circular correlation time as one period of PRN code. Those are low-efficiency, time-consuming, and of large computation. Especially, code periods of GPS L2C signal are as long as 20 ms for CM code and 1.5 s for CL code. Actually, the shorter circular correlation time could quicken signal acquisition if the received GPS signal is strong enough. For weak signals in noisy environments, the longer circular correlation time ensures the acquisition performance. For the purpose of exploring the effects of different circular correlation times on the performance of signal acquisition, we introduce variable circular correlation time in the coarse acquisition. The reference signal \(S_{r}(i)\) of length \(L_{r}\) is extracted from the resampled signal \(S_{d}(i)\). The zero-padding PRN code \(C_{r}(i)\) is introduced to calculate circular correlation with the reference signal to obtain accurate carrier frequency and code phase. \[\left\{ \begin{array}{l}P_{r}=\ P_{c}+P_{x}\\ L_{r}=\ L_{c}+L_{x}\end{array} \right.,\ P_{x}\in(0,\ P_{c}] \tag{14}\]  
\[S_{r}(i)\ =\ S_{d}(i),\ i=1,2,\ \ldots,\ L_{r}. \tag{15}\]  
\[C_{r}(i)\ =\ \left\{ \begin{array}{l}C_{M}(i),\ i=1,\ 2,\ \ldots,\ L_{x}\\ 0,\ i=L_{x}+1,\ L_{x}+2,\ \ldots,\ L_{x}+L_{c}.\end{array} \right. \tag{16}\]  
Figure 3: Signal flow chart by applying the resampling strategy and the convolutional method to the received broadband satellite signal. The green, dark brown, and orange represent frequency spectra of GPS L2C, P(Y), and M code signals, respectively. Specifically, \(P_{c}\) is the period of PRN code and \(P_{x}\) represents variable circular correlation time, \(P_{x}\in(0,\ P_{c}]\). If \(P_{x}=P_{c}\), the variable circular correlation time method is the same as conventional acquisition algorithms. \(L_{r},\ L_{c}\) and \(L_{x}\) are the lengths of signals with periods \(P_{r},\ P_{c}\), and \(P_{x}\). At frequency search step \(k\), the reference signal \(S_{r}(i)\) is multiplied with local carrier replica \(W_{l}(k)\) of frequency \(f(k)\) to remove carrier wave from \(S_{r}(i)\) and obtain the baseband signal \(S_{w}(k)\)  
\[W_{l}(k)=e^{i2\pi f(k)} \tag{17}\]  
\[S_{w}(k)=\ W_{l}(k)\cdot S_{r}(i) \tag{18}\]  
With different code phase offsets, the baseband signal \(S_{w}(k)\) is circular correlated with the local zero-padding code \(C_{r}(p)\), as illustrated in Figure 4. \(r_{x}(k,\ p)\) is the circular correlation result of \(S_{w}(k)\) and \(C_{r}(p)\). FFT is adopted to quicken the correlation calculation in the time domain, as Equation (20). \(\mathcal{F}[x]^{*}\) refers to the complex conjugate value of \(\mathcal{F}[x]\). \[r_{x}(k,p)=\sum_{i=1}^{L_{r}}S_{w}(k,i)C_{r}(p+i),\ p=1,\ 2,\ \ldots,\ L_{r}. \tag{19}\]  
Figure 4: Circular correlation results of the baseband signal \(S_{w}(k)\) and the local zero-padding code \(C_{r}(p)\): (**a**) code offset between \(S_{w}(k)\) and \(C_{r}(p)\) is 0 (aligned); (**b**) code offset between \(S_{w}(k)\) and \(C_{r}(p)\) is less than \((L_{c}-L_{x})\) samples; (**c**) code offset between \(S_{w}(k)\) and \(C_{r}(p)\) is more than \((L_{c}-L_{x})\) samples. \[\mathcal{F}[r_{x}(k,\ p)](n)=\sum_{p=1}^{L_{r}}\sum_{i=1}^{L_{r}}S_{w}(k,i)C_{r}(p+ i)e^{\frac{-2\pi np}{L_{r}}}\ =\ \mathcal{F}[S_{w}(k)]\cdot\mathcal{F}[C_{r}(p)]^{*} \tag{20}\]  
Correlation magnitude reflects the self-/cross-correlation relationship of the received satellite signal and the local code replica. When they are completely aligned, a much higher correlation peak is gained. In the search space of carrier frequency and code phase, the correlation magnitude \(M_{x}(k,p)\) is formulated as  
\[M_{x}(k,\ p)=|r_{x}(k,p)|=\sqrt{R^{2}[r_{x}(k,p)]+I^{2}[r_{x}(k,\ p)]} \tag{21}\]  
where \(R(\cdot)\) and \(I(\cdot)\) denote the real and imaginary parts of a complex data. By Equations (19)-(21), it is very easy to find out the highest correlation peak in the search space.