## 2 Materials and Methods
### Signals
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Signal Component** & **Center Frequency/(MHz)** & **Modulation** & **Rate Speed/(sps)** \\ \hline B1C\_data & 1575.42 & BOC (1,1) & 100 \\ B1C\_pilot & 1575.42 & QMBOC (6,1,4/33) & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The characteristics of the B1C signal. Figure 1: Frequency distribution of GNSSs. \(\mathrm{C}^{(i)}_{(\mathrm{B2a})}(t)\) is the ranging code of the data component, and \(\mathrm{C}^{(i)}_{(\mathrm{B1C}_{\mathrm{P}})}(t)\) is the ranging code of the pilot component. The design of this data component and pilot component will greatly improve the ranging ability of the signal. As for the ranging code, it is a layered code structure of B1C that the sub-code chip is strictly aligned with the time of the first code chip of the main code. The code rate of the B1C ranging code is 1.023 Mbps. It is obtained by truncating a Weil code with a length of 10243 chips. The Weil code sequence with length n is defined as follows:  
\[\mathrm{W}(t,w)\mathrm{=L}(t)\ \oplus\ \mathrm{L}(t+w) \tag{3}\]  
\(\mathrm{L}(t)\) is a Legendre sequence of fixed length N, w is the phase difference between two Legendre sequences, and the value is 1 to 5121. The Legendre sequences of length N are defined as follows:  
\[\mathrm{L}(t)= \begin{cases}1;&t\ \
eq\ 0,\mathrm{intx};\ t=\mathrm{x}^{2} \mathrm{modN}\\ 0;&t=0\\ 0;&\mathrm{else}\end{cases}  \tag{4}\]  
The B1C ranging code can be obtained by cyclic interception of the Weil code. B2a is another public navigation service signal broadcast by BDS-3. The BPSK(10) method is selected for its signal modulation in the data component and pilot component. Similarly, according to the spatial interface document of the BDS, we can express the signal in the form of a complex envelope [32]:  
\[\mathrm{S}_{(\mathrm{B2a})}\mathrm{=s}_{(\mathrm{B2a}_{\mathrm{D}})}(t)+ \mathrm{j}\mathrm{s}_{(\mathrm{B2a}_{\mathrm{P}})}(t) \tag{5}\]  
where \(\mathrm{s}_{(\mathrm{B2a}_{\mathrm{D}})}(t)\) represents the data component of the signal, and \(\mathrm{s}_{(\mathrm{B2a}_{\mathrm{P}})}(t)\) represents the pilot component of the data. We can also write the signal in bandpass form [32]:  
\[\mathrm{S}^{(i)}_{(\mathrm{B2a})}(t)=\sqrt{2\mathrm{P}_{(\mathrm{B2a})}}\left[ \mathrm{D}^{(i)}_{(\mathrm{B2a})}(t)\mathrm{C}^{(i)}_{(\mathrm{B2a})}(t)\cos( 2\pi\mathrm{ft})-\mathrm{C}^{(i)}_{(\mathrm{B2a})}(t)\sin(2\pi\mathrm{ft})\right] \tag{6}\]  
In this formula, \(\mathrm{D}^{(i)}_{(\mathrm{B2a})}(t)\) represents the modulated navigation message data, \(\mathrm{C}^{(i)}_{(\mathrm{B2a}_{\mathrm{D}})}(t)\) is the data code of the data component, and \(\mathrm{C}^{(i)}_{(\mathrm{B2a}_{\mathrm{P}})}(t)\) is the data code of the pilot component. \(\mathrm{P}_{(\mathrm{B2a})}\) gives the power of the B2a signal. The code rate of B2a ranging code is 10.23 Mcps and the code length is 10,230. Two thirteen-level linear shift registers \(\mathrm{g}_{1(\mathrm{x})}\) and \(\mathrm{g}_{2(\mathrm{x})}\) are used to generate two extended gold codes of 10,230 chips, and both of them obtain B2a ranging code through modulo-2 addition. The two polynomials are:  
\[\mathrm{g}_{1}(\mathrm{x})\mathrm{=1+x+x^{5}+x^{11}+x^{13}} \tag{7}\]  
\[\mathrm{g}_{2}(\mathrm{x})\mathrm{=1+x^{3}+x^{5}++x^{9}+x^{11}+x^{12}+x^{13}} \tag{8}\]  
for the data component and are:  
\[\mathrm{g}_{1}(\mathrm{x})\mathrm{=1+x^{3}+x^{6}+x^{7}+x^{13}} \tag{9}\]  
\[\mathrm{g}_{2}