# A Doppler Frequency-Offset Estimation Method Based on the Beam Pointing of LEO Satellites
## 4 Simulation Analysis
### Simulation of Doppler Frequency Shift When the Minimum Elevation Angle Parameter Is Adjustable  
The Doppler frequency-shift simulation parameters for variable minimum satellite elevation angles are listed in Table 5.  
The simulation process of the Doppler frequency-offset estimation is as follows:  
* According to the simulation parameters, when the minimum elevation angles of the satellite are set at 0\({}^{\circ}\), 10\({}^{\circ}\), 20\({}^{\circ}\), 30\({}^{\circ}\), 40\({}^{\circ}\), and 50\({}^{\circ}\), we use the STK tool to simulate the Doppler frequency-offset changes invisible time. The changes in the Doppler frequency offset and the azimuth and elevation of the satellite are simulated. * Using the satellite beam-pointing method, the Doppler frequency-offset variation is calculated according to the azimuth and elevation angles of LEO satellites. * Compare the error between the Doppler frequency-offset change value of the satellite beam-pointing method and the Doppler frequency-offset change value of the STK simulation. \(\varphi_{max}\) represents the minimum elevation angle of the satellite. \(\varphi_{max}\) is set at 0\({}^{\circ}\), 10\({}^{\circ}\), 20\({}^{\circ}\), 30\({}^{\circ}\), 40\({}^{\circ}\), and 50\({}^{\circ}\). When the azimuth angle is less than 180\({}^{\circ}\), the Doppler frequency-offset \begin{table}
\begin{tabular}{c c c c c c} \hline Orbital Lithitudes & 480 km & 500 km & 600 km & 700 km & 800 km \\ \hline RMSE & 109 Hz & 110 Hz & 83 Hz & 84 Hz & 93 Hz \\ \hline \end{tabular}
\end{table} 
Table 3: The RMSE results of Doppler frequency-offset estimation errors for different orbital altitudes. \begin{table}
\begin{tabular}{c c c c c} \hline \hline Orbital Lithitudes & 480 km & 500 km & 600 km & 700 km & 800 km \\ \hline RMSE & 109 Hz & 110 Hz & 83 Hz & 84 Hz & 93 Hz \\ \hline \end{tabular}
\end{table} 
Table 4: The RMSE results of Doppler frequency-offset estimation errors for different orbital inclinations. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter Name** & **Parameter Value** \\ \hline Orbital altitude, h & 480 km \\ Earth radius, R & 6378 km \\ Eccentricity, e & 0.0 \\ Orbital inclination, i & 50\({}^{\circ}\) \\ Maximum elevation angle, \(\varphi_{max}\) & 0\({}^{\circ}\), 10\({}^{\circ}\), 20\({}^{\circ}\), 30\({}^{\circ}\), 40\({}^{\circ}\), 50\({}^{\circ}\) \\ 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 5: Table of Doppler frequency-shift simulation parameters for variable minimum satellite elevation angles. estimation error of the satellite beam-pointing method is shown in Figure 9. It can be seen from Figure 8 that the frequency-offset estimation error range of the two methods is between [\(-\)200 Hz, \(+\)2000 Hz] in the whole visible period. When the azimuth angle is less than 180\({}^{\circ}\), the positive deviation error of the frequency-offset estimation increases with an increase in the minimum elevation angle of the satellite. We conducted simulations of the RMSE of the frequency-offset estimation for values of 0\({}^{\circ}\), 10\({}^{\circ}\), 20\({}^{\circ}\), 30\({}^{\circ}\), 40\({}^{\circ}\), and 50\({}^{\circ}\). The simulation results are shown in Figure 9. The results indicate that the RMSE of the frequency-offset estimation gradually increases. When \(\varphi_{max}=0^{\circ}\), the RMSE of the frequency-offset estimation does not exceed 100 Hz, and when \(\varphi_{max}=50^{\circ}\), the RMSE is approximately 1900 Hz. The minimum elevation angles of the satellite are represented, and \(\varphi_{max}\) and \(\varphi_{max}\) are set to values such as 0\({}^{\circ}\), 10\({}^{\circ}\), 20\({}^{\circ}\), 30\({}^{\circ}\), 40\({}^{\circ}\), and 50\({}^{\circ}\).