## 6 Simulation Results and Discussion
### Error Estimation for Satellite Vibration Effects  
Figure 12(a) shows the PSD of the satellite platform vibration model used by ESA during the SILEX program, as described by Equation (65) for the specified frequency range. This model can be used for satellite vibration analysis and to generate vibration signals in both the vertical and horizontal directions. Figure 12(b),c show the estimated PSDs for the vibration signals in the vertical and horizontal directions, respectively. The estimated PSD is consistent with the PSD model shown in Figure 13, validating its use for generating vibration signals in both directions. These signals were used to compute the radial vibration signal and to ultimately estimate the standard deviation using the sliding window method, as depicted in Figure 14.  
The algorithm to generate time-series data from the PSD involves seven steps. The first step is to define the time length of the time-series data, which is then used to retrieve the frequency resolution. The second step is to define the desired frequency range. To generate the time array, a sampling frequency is required, which can be calculated using the final frequency of the PSD curve according to the Nyquist theorem. The third step involves sampling the PSD curve within the selected range. The fourth step is to compute the single-sided and normalized amplitude of the discrete Fourier transform (DFT). To build the time-series data, phase information is also needed, which is absent in the power spectral density; therefore, in the fifth step, random phases are generated for each value of the DFT. This step is crucial as it enables the creation of infinite time-domain signals. The sixth step is to retrieve the DFT coefficients in both real and imaginary formats. The final step is to compute the inverse DFT to generate the time-domain vibration signals in both vertical and horizontal directions.  
Figure 12: Mean acquisition time analysis for hexagon scan path. (**a**) Mean acquisition time vs. uncertainty scan area size for hexagon scan path. (**b**) Mean acquisition time vs. receiver satellite position error for hexagon scan path.  
Figure 14 illustrates the performance of the maximum likelihood estimator using a sliding window of size 200 to estimate different standard deviations, i.e., \(\sigma_{\text{vib}}=[1,5,10]\)\(\upmu\)rad, where \(\sigma_{\text{vib}}=10\)\(\upmu\)rad is associated with the PSD model described in Equation (65). The estimator's performance is quite stable with the current window size, as the errors in both the vertical and horizontal directions are independent and identically distributed (i.i.d.) Gaussian random variables with zero mean and the same variance. The estimation  
Figure 14: Performance of the maximum likelihood estimator for standard deviation estimation.  
Figure 13: Power spectral model and its estimation performance. (**a**) Power spectral model of SILEX program. (**b**) Estimation of PSD for vibration signal in vertical direction. (**c**) Estimation of PSD for vibration signal in horizontal direction.  
accuracy, evaluated using the root mean square error (RMSE), was [0.03, 0.3, 0.5] \(\upmu\)rad for \(\sigma_{\text{vib}}=[1,5,10]\)\(\upmu\)rad, respectively. These RMSE values confirm that the estimator maintains reliable performance across varying standard deviations, with minimal deviation around each target value.