## 4 Validation and Discussion
Additionally, its MSE is significantly smaller than that of the LPSD algorithm. At 1 mHz, the MSE of the adaptive sine multi-taper algorithm is approximately 19% of that of the original algorithm, reflecting a reduction of 81%, and 9.8% of that of the LPSD algorithm, representing a substantial reduction of 90%. The LISA-LPSD algorithm's MSE is about 46% of that of the LPSD algorithm, while the adaptive sine multi-taper algorithm's MSE is roughly 21% of that of the LISA-LPSD algorithm, indicating a 79% reduction, as shown in Table 2 in 'Sinusoidal Excitation Experiment: Adaptive Sine Multi-Taper vs. Others'. As shown in Figure 9b, when the data sampling duration is less than one hour, the original adaptive sine algorithm executes faster than its improved version. However, as the sampling duration increases, the LISA-LPSD algorithm's advantages become more evident. For the longest processing time of 5 h, the improved adaptive sine algorithm's execution time is reduced to approximately 47% of that of the original algorithm, reflecting a reduction of 53%, 10% of that of the LPSD algorithm, indicating a reduction of 90%, and 65% of that of the LISA-LPSD algorithm, corresponding to a reduction of 35%, as shown in Table 2 in 'Sinusoidal Excitation Experiment: Adaptive Sine Multi-Taper vs. Others'. To further evaluate the algorithm's performance, the mechanism was deflected to 15 mrad and held in a static state, as illustrated in Figure 10a, during the noise spectral density estimation process, the input signal exhibits significant sinusoidal noise near 1 Hz, which poses challenges for accurate spectral resolution. To address this issue, the improved \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Algorithm**} & \multicolumn{2}{c}{**In sinusoidal Excitation Experiment Adaptive**} & \multicolumn{2}{c}{**Under Noise Metric Experiment Adaptive**} \\  & \multicolumn{2}{c}{**Sine Multi-Taper vs. Others**} & \multicolumn{2}{c}{**Sine Multi-Taper vs. Others**} \\  & **MSE Reduction (\%)** & **Runtime Reduction (\%)** & **MSE Reduction (\%)** & **Runtime Reduction (\%)** \\ \hline Original & & & & \\ Adaptive Sine & 81 & 53 & 46 & 67 \\ Multi-taper & & & & \\ LPSD & 90 & 90 & 55 & 86 \\ LISA-LPSD & 79 & 35 & 41 & 30 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Performance: Adaptive Sine Multi-Taper Algorithm vs. others in sinusoidal excitation experiment and under noise metric experiment. Figure 9: Performance analysis of PAAM in sinusoidal excitation experiment. (**a**) Comparison of Squared Bias, Variance, and MSE before and after improvements using LPSD, LISA-LPSD, and Adaptive Sine Multi-Taper Algorithm; (**b**) Comparison of Computation time for LPSD, Adaptive Sine Multi-Taper, Original Adaptive Sine Multi-Taper, and LISA-LPSD Algorithms in PAAM. algorithm reduces the taper number in the vicinity of 1 Hz, as shown in Figure 10b, effectively mitigating the impact of sinusoidal noise while maintaining spectral clarity. Additionally, the improved algorithm enhances resolution in the low-frequency region below 1 mHz by setting a fixed, lower taper number, ensuring higher precision in this critical frequency range. A further quantitative assessment through comparative analysis of multiple algorithms confirms the improved algorithm's enhanced performance in handling sinusoidal noise and its superior resolution across both the low-frequency and 1 Hz regions. The mean PSD derived from this setup serves as the true value, as illustrated in Figure 11a. Near 1 mHz, the adaptive sine multi-taper algorithm exhibits increased variance but reduced bias, resulting in a lower MSE. Its MSE is approximately 54% of that of the original algorithm, reflecting a reduction of 46%, and 45% of that of the LPSD algorithm, indicating a significant performance enhancement with a reduction of 55%.