## 2 The Principle of INS-Aided Acquisition
### INS-Aided Acquisition Methodology  
As shown in Figure 1, the GPS and BDS signal acquisition process is a three-dimensional search, including pseudo-random noise code (PRN), code phase and Doppler shift. The GPS satellites are differentiated by 32 different PRN sequences, while the BDS satellites are differentiated by 35 (the number before 2020 is 14) different PRN sequences [1]. From the code phase search direction, the C/A code in GPS L1 frequency contains 1023 chips and the BDS B1I has 2046 chips, however the Doppler shift range is the possible maximum frequency shift. Take the BDS B1I signal for example, there exist 4092 code bins and 21 frequency bins, assuming that the bin width of the code phase search is 1/2 chip, the step size of the Doppler search is 500 Hz and the Doppler span is \(\pm\)5 kHz. Thus this results in a total of 85,932 search cells, so it is necessary to reduce the range in order to achieve faster acquisition. The GNSS signal acquisition can be successfully achieved by exceeding the correlation threshold between the local signal and the received signal. Taking the BDS B1I signal for example, the integration output \(\overline{I}_{\mathrm{P}}(n)\) and \(\overline{Q}_{\mathrm{P}}(n)\) are described as follows:  
\[\overline{I}_{\mathrm{P}}(n)=AD(n)R(\Delta\tau)\mathrm{sinc}(\pi\Delta fT_{coh })\cos(\pi\Delta fT_{coh}+\delta\varphi_{0})+\overline{N}_{I} \tag{3}\]  
\[\overline{Q}_{\mathrm{P}}(n)=AD(n)R(\Delta\tau)\mathrm{sinc}(\pi\Delta fT_{ coh})\sin(\pi\Delta fT_{coh}+\delta\varphi_{0})+\overline{N}_{Q} \tag{4}\]  
where \(A\) is the signal amplitude; \(D(n)\) is the navigation data; \(R(\Delta\tau)\) is the I autocorrelation function; \(\Delta\tau\) is the code phase error between received signal and local signal; \(\Delta f\) is the frequency difference between the received signal and local signal; \(\delta\varphi_{0}\) is the carrier phase error; \(T_{coh}\) is the coherent integration time; \(\overline{N}_{I}\) and \(\overline{N}_{Q}\) are the white noise. From Equations (3) and (4), the correlation output is \(z_{I}(n)=\overline{I}_{\mathrm{P}}(n)+j\cdot\overline{Q}_{\mathrm{P}}(n)\). The magnitude of \(z_{I}(n)\) can be written as:  
\[V_{i}=\sqrt{I_{i}^{2}(n)+Q_{i}^{2}(n)}=AR(\Delta\tau)|\mathrm{sinc}(\Delta fT_ {coh})|+N_{R} \tag{5}\]  
Equation (5) shows that the smaller are the code phase error \(\Delta\tau\) and the frequency shift \(\Delta f\), the higher is the detection probability. Under the conditions of \(T_{coh}\) = 1 ms and false alarm rate \(P_{fa}\) = 0.01, Figure 2a,b show the detection probability when the \(\Delta\tau\) is 0, 1/8, 1/4, 1/2 chip and the \(\Delta f\) is 0, 200, 500, 800, 1200 and 1500 Hz, respectively. From Figure 2, in the same C/N\({}_{0}\), the detection probability is gradually reduced with the increase of the local carrier frequency estimation error and the code phase error. Considering that it is difficult to improve C/N\({}_{0}\), an appropriate means of ensuring the detection probability is by reducing the error of local carrier frequency estimation and code phase estimation. In the INS-aided acquisition, according to the velocity, position information provided by INS and ephemeris, the Doppler frequency shift and code phase can be calculated. The estimated carrier frequency and code phase offset are set as the center of the search range, and the search boundary is set according to the uncertainty of INS information, so this can greatly reduce the carrier frequency and code phase search range. At the same time, the INS-aided acquisition can reduce the error of local carrier frequency estimation and code phase estimation to enhance the detection probability. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **BDS B1 Signal** & **GPS L1 Signal** \\ \hline Center Freq. (MHz) & 1561.098 MHz & 1575.42 MHz \\ Modulation Type & QPSK-CDMA & BPSK-CDMA \\ Code Rate (MHz) & 2.046 & 1.023 \\ Code Length (chip) & 2046 & 1023 \\ Symbol Rate & GEO 500 bps, MEO/IGSO 50 bps & 50 bps \\ \hline \hline \end{tabular}
\end{table}