## 3 Frequency Shift and Code Phase Estimation Theory  
### Frequency Shift Estimation  
In the spread of GNSS signal from satellite to receiver, the main factors of Doppler frequency shift are the motion of the receiver relative to the satellite, the receiver clock frequency drift and the satellite clock frequency drift. Therefore, the carrier Doppler frequency shift can be expressed as:  
\[f_{d}\ =\ f_{r}-f_{s}+f_{r,c}-f_{s,c} \tag{6}\]  
where \(f_{r}\) is the Doppler shift on the carrier due to the motion of the receiver with respect to the satellite and \(f_{s}\) is the Doppler shift on the carrier due to the motion of the satellite with respect to the receiver. \(f_{r,c}\), \(f_{s,c}\) are the Doppler frequency shift of the receiver clock frequency drift and satellite clock frequency drift, respectively.  
Figure 3: INS-aided acquisition scheme.  
Due to the fact that the satellite uses the atomic clock which has high stability and high precision as time reference and the receiver clock frequency offset can be calibrated locally, the clock error of the satellite \(f_{s,c}\) and receiver \(f_{r,c}\) can be ignored, so the estimation of Doppler frequency shift in LOS direction can be expressed as:  
\[f_{d}=\boldsymbol{E}\cdot(\boldsymbol{V}_{r}-\ \boldsymbol{V}_{s})/\lambda_{carrier}\ =\ f_{r}-f_{s} \tag{7}\]  
where \(\boldsymbol{E}\) is the unit vector of the (LOS) and it can be calculated by the relative position of satellite and carrier, \(\boldsymbol{V}_{r}\) is the velocity of the receiver, \(\boldsymbol{V}_{s}\) is the velocity of the satellite, \(\lambda_{carrier}\) is the carrier wavelength and it can be calculated by \(\lambda_{carrier}\ =\ c/f_{carrier}\), \(c\) is the velocity of light and \(f_{carrier}\) is the carrier frequency.  
Taking GPS L1 frequency as an example, the maximum radial velocity component of relative motion between the stationary receiver and satellite is about 929 m/s, leading to the maximum Doppler frequency shift is:  
\[f_{\text{dM}}\ =\ V_{\text{dM}}\frac{f_{\text{L1}}}{c}\ \approx\ \frac{929\times 1575.2\times 10^{6}}{3 \times 10^{8}}\ \approx\ 4878.8\ \text{Hz} \tag{8}\]  
As for the Doppler frequency shift of BDS, it is more complex than GPS. On the one hand, the carrier frequency of BDS is different from GPS. On the other hand, the BDS space constellation includes three kinds of satellites: GEO, IGSO, MEO, and different types of satellites lead to the different maximum Doppler frequency shift. According to the orbit parameters of BDS broadcast, the maximum satellite operation speed can be calculated. The maximum radial velocities of GEO, IGSO and MEO satellite are 13.6 m/s, 424.2 m/s and 864.1 m/s respectively. The maximum Doppler frequency can be calculated by [1]:  
\[f_{\text{dM,GEO}}=V_{\text{dM,GEO}}\frac{f_{\text{B1}}}{c}\approx\frac{13.6 \times 1561.098\times 10^{6}}{3\times 10^{8}}\approx 70.8\ \text{Hz} \tag{9}\]  
\[f_{\text{dM,IGSO}}=V_{\text{dM,IGSO}}\frac{f_{\text{B1}}}{c}\approx\frac{424. 2\times 1561.098\times 10^{6}}{3\times 10^{8}}\approx 2207.4\ \text{Hz} \tag{10}\]  
\[f_{\text{dM,MEO}}=V_{\text{dM,MEO}}\frac{f_{\text{B1}}}{c}\approx\frac{864.1 \times 1561.098\times 10^{6}}{3\times 10^{8}}\approx 4496.5\ \text{Hz} \tag{11}\]  
From the above analysis, the maximum Doppler shift caused by the satellite motion is about 5 kHz for a stationary receiver in the earth's surface. As shown in Equation (7), the Doppler shift caused by the receiver motion can be calculated by the velocity and position of INS.