# A Doppler Frequency-Offset Estimation Method Based on the Beam Pointing of LEO Satellites
## 4 Simulation Analysis
When the minimum elevation angle of the satellite is 0\({}^{\circ}\) and the geographical latitude of the terminal is 0\({}^{\circ}\), the Doppler frequency-offset estimation of the satellite beam-pointing method is shown in Figure 4. Figure 4 compares the simulation results of the satellite beam-pointing method with STK simulation results. Within the visible time of the satellite, the range of frequency-offset estimation error is [\(-\)150 Hz, \(+\)150 Hz]. When the minimum satellite elevation angle is 0\({}^{\circ}\) and the terminal geographic latitude is 0\({}^{\circ}\), the satellite's geocentric latitude \(\beta\) is set to 0\({}^{\circ}\), 10\({}^{\circ}\), 20\({}^{\circ}\), 30\({}^{\circ}\), 40\({}^{\circ}\), and 50\({}^{\circ}\). The error values of the Doppler frequency-offset estimation by the DFS-BP method are shown in Figure 5a. The RMSEs of the Doppler frequency-offset estimation errors at different satellite geocentric latitudes \(\beta\) are shown in Figure 5b. When the satellite's geocentric latitude \(\beta\) is set to different values, the RMSE of the Doppler frequency-offset estimation errors of the three sub-equations of the DFS-BP method are all less than 100 Hz, and the maximum difference in their RMSE values is 30 Hz. Figure 4: Simulation of Doppler frequency shifts when the minimum elevation angle is 0\({}^{\circ}\) and the geographical location of the terminal is 0\({}^{\circ}\). (**a**) The Frequency shift; (**b**) The Frequency shift error. Simulation of Doppler Frequency Shift When the Minimum Elevation Angle Is 0\({}^{\circ}\) and the Geographical Location of the Terminal Is Adjustable  
The Doppler frequency-shift simulation parameters of different terminal geographic positions and satellite orbits are listed in Table 2.  
The simulation process of Doppler frequency-shift estimations is as follows:  
* According to the simulation parameters, the STK tool is used to simulate the change value of the satellite-ground Doppler frequency offset under the condition that the minimum elevation angle is 0\({}^{\circ}\) and the geographic latitude of the terminal is set to 0\({}^{\circ}\), 15\({}^{\circ}\), 30\({}^{\circ}\), and 45\({}^{\circ}\). The simulation results of the Doppler frequency offset and azimuth and elevation angles of the satellite are outputted. * Using the satellite beam-pointing method, the Doppler frequency offset is calculated according to the azimuth and elevation angles of the satellite. Compare the errors with the Doppler frequency-shift variations simulated by STK, analyze the RMSE of the Doppler frequency shift, and verify the adaptability of the DFS-BP method to different terminal geographic latitudes. * According to the simulation parameters, the STK tool is used to simulate the change value of the satellite-ground Doppler frequency offsets under the condition that the minimum elevation angle is 0\({}^{\circ}\) and the satellite orbital altitude h is set to 480 km, 500 km, 600 km, 700 km, and 800 km. The simulation results of the Doppler frequency offset and azimuth and elevation angles of the satellite are outputted. \begin{table}
\begin{tabular}{c c} \hline
**Parameter Name** & **Parameter Value** \\ \hline Orbital altitude, h & 480 km, 500 km, 600 km, 700 km, 800 km \\ Earth radius, R & 6378 km \\ Eccentricity, e & 0.0 \\ Orbital inclination, i & 40\({}^{\circ}\), 50\({}^{\circ}\), 60\({}^{\circ}\), 70\({}^{\circ}\), 80\({}^{\circ}\), 90\({}^{\circ}\) \\ Geographical latitude of the terminal, Lat & 0\({}^{\circ}\), 15\({}^{\circ}\), 30\({}^{\circ}\), 45\({}^{\circ}\) \\ Earth’s rotation period, T & 86,164 s \\ Signal propagation speed, c & 2.998 \(\times\) 10\({}^{8}\) m/s \\ Signal carrier frequency, Fc & 2.0 GHz \\ \hline \end{tabular}
\end{table} 
Table 2: Doppler frequency-shift simulation parameter table for different terminal geographic positions and satellite orbits. Figure 5: Doppler frequency-shift simulation and error statistics at different satellite geocentric latitudes. **(a)** The Frequency shift error; **(b)** The Root Mean Squared Error.