# A Doppler Frequency-Offset Estimation Method Based on the Beam Pointing of LEO Satellites
## 4 Simulation Analysis
* 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 satellite orbital altitudes. * According to the simulation parameters, the STK tool is used to simulate the change value of satellite-ground Doppler frequency offsets under the condition that the minimum elevation angle is 0\({}^{\circ}\) and the satellite orbital inclination \(i\) is set to 40\({}^{\circ}\), 50\({}^{\circ}\), 60\({}^{\circ}\), 70\({}^{\circ}\), 80\({}^{\circ}\), and 90\({}^{\circ}\). The simulation results of the Doppler frequency offset and azimuth and elevation angles of the satellite are output. * 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 satellite orbital inclinations. * Calculate the 95% CI of the Doppler frequency-offset estimation errors based on the RMSEs under different orbital altitudes and orbital inclinations. Due to the small sample size, the t-distribution model is selected for confidence interval analyses. The t-distribution model is \(CI=\overline{x}\pm t_{0,025,n-1}\frac{s}{\sqrt{n}}\), where \(\overline{x}\) is the average value of RMSE, \(t_{0,025,n-1}\) is the distribution values at a 95% confidence level, \(s\) is the sample's standard deviation, and \(n\) is the sample size. When the minimum elevation angle of the satellite is 0\({}^{\circ}\) and the geographical latitude of the terminal is set at 0\({}^{\circ}\), 15\({}^{\circ}\), 30\({}^{\circ}\), and 45\({}^{\circ}\), the Doppler frequency-offset estimation errors of the satellite beam-pointing method are shown in Figure 6. Figure 6a shows that the frequency-offset estimation error range of the two methods is between [\(-\)200 Hz, +200 Hz] in the whole visible period. Figure 6b shows that the RMSE of the frequency-offset estimation for ground terminals at different latitudes is less than 100 Hz. The geographical latitude of the terminal has little effect on the Doppler frequency-offset estimation error. To analyze the adaptability of the DFS-BP algorithm to different satellite orbits, this paper designs simulations of the Doppler frequency-offset estimation and the RMSE of frequency-offset estimation errors for different orbital altitudes and orbital inclinations. Figure 6: Simulation results of frequency-offset estimation error and RMSE for ground terminals at different latitudes. (**a**) The Frequency shift error; (**b**) The Root Mean Squared Error. When the minimum satellite elevation angle is 0\({}^{\circ}\), simulations of the Doppler frequency-offset estimation errors and the RMSE of the frequency-offset estimation errors are conducted for satellites with different orbital altitudes ranging from 480 km to 800 km. The simulation results are shown in Figure 7. The results indicate, based on the DFS-BP algorithm, that the satellite-to-ground frequency-offset estimation errors at different satellite orbital altitudes fall within the range of [\(-\)200 Hz, +200 Hz], and the RMSE of frequency-offset estimation errors is less than 120 Hz. When the minimum satellite elevation angle is 0\({}^{\circ}\), simulations of the Doppler frequency-offset estimation errors and the RMSE of the frequency-offset estimation errors are conducted for satellites with different orbital inclinations ranging from 40\({}^{\circ}\) to 90\({}^{\circ}\).