## 3 Algorithm Simulation Comparison
### Simulation Comparison of ASD Estimation Algorithms  
Simulations of mid-to-low frequency noise vibrations were conducted to evaluate the algorithms' performance in spectral resolution and computational complexity. On one hand, sinusoidal stimuli were simulated, representing the incident laser undergoing simultaneous yaw jitter at frequencies of 100 mHz (amplitude 50 \(\upmu\)rad), 10 mHz (amplitude 50 \(\upmu\)rad), and 1 mHz (amplitude 50 \(\upmu\)rad), with Gaussian noise added to emulate real-world environmental interference. On the other hand, simulations were performed to estimate the spectral density of quantification metrics for system noise. Ten experimental sets were designed to evaluate the algorithms and obtain the true spectral density and covariance function. The mean ASD estimates derived from the standard algorithms were used as the benchmark reference and compared with the PSD estimation outcomes produced by various algorithms. By considering the mean squared error (MSE) and computational speed of each algorithm, this study provides an objective evaluation of their performance. As illustrated in Figure 2a, the simulation results reveal that the improved adaptive sine multi-taper algorithm demonstrates exceptional performance at several critical frequency points. From Figure 2b, it can be observed that the taper numbers across the entire frequency domain are significantly reduced after the algorithm's enhancement, with the reduction being particularly pronounced in the low-frequency range. Notably, near 1 mHz and 10 mHz, the resolution of the improved adaptive sine multi-taper algorithm is markedly superior to that of the original version. Furthermore, at the peak frequency of 1 mHz, the improved algorithm significantly mitigates spectral leakage compared to the LPSD and LISA-LPSD algorithms. These findings provide preliminary qualitative validation of the potential and superiority of the improved adaptive sine multi-taper algorithm in applications within this field. To thoroughly assess the performance of the algorithms, further quantification was performed through metrics such as bias, variance, and overall MSE, while also comparing the computational speed of several algorithms. Figure 3a shows that the mean PSD from ten sine stimuli, with added Gaussian noise, represents the true value. Near 1 mHz, the improved adaptive sine multi-taper algorithm demonstrates a reduction in MSE compared  
Figure 2: Comparative analysis of theoretical ASD and simulation results for LPSD, Adaptive Sine Multi-Taper, Original Adaptive Sine Multi-Taper, and LISA-LPSD Algorithms in sine excitation simulation. (**a**) Performance comparison; (**b**) Taper number comparison before and after improvements for Adaptive Sine Multi-Taper Algorithm. to the original algorithm, and it exhibits a smaller MSE relative to the LPSD algorithm. At the 1 mHz point, the adaptive sine multi-taper algorithm's MSE is approximately 55% of that of the original algorithm, reflecting a reduction of 45%. Furthermore, when compared to the LPSD algorithm, its MSE shows a reduction of approximately 34%. The LISA-LPSD algorithm's MSE is approximately 6.5 times that of the LPSD algorithm, while the improved adaptive sine multi-taper algorithm achieves an MSE approximately 10% of that of the LISA-LPSD algorithm, reflecting a substantial reduction of 90%, as shown in Table 1 in 'Sinusoidal Simulation: Adaptive Sine Multi-Taper vs. Others'. Additionally, as shown in Figure 3b, when the data sampling time is less than one hour, the original adaptive sine multi-taper algorithm runs faster than the LISA-LPSD algorithm; however, as the data sampling time increases, the advantages of the LISA-LPSD algorithm become evident. The execution time of the modified adaptive algorithm is markedly shorter than that of the LPSD algorithm, and it also surpasses the original adaptive algorithm and the LISA-LPSD algorithm in terms of efficiency.