## 3 Algorithm Simulation Comparison
### Simulation Comparison of ASD Estimation Algorithms
When assessing computational speed over a data processing duration of five hours, the execution time of the refined adaptive sine multi-taper algorithm is significantly reduced to just 24% of that demanded by the original algorithm, reflecting a reduction of 76%. Additionally, it requires only 12% of the time taken by the LPSD algorithm, indicating a reduction of 88%, and 75% of the time used by the LISA-LPSD algorithm, corresponding to a reduction of 25%, as shown in Table 1 in 'Sinusoidal Simulation: Adaptive Sine Multi-Taper vs. Others'. The refined adaptive sine multi-taper algorithm shows clear advantages in resolution and computational efficiency in ASD estimation. To further corroborate the system's overarching performance acceptance criteria, supplementary simulations of noise density ASD estimation processes were undertaken. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Algorithm**} & \multicolumn{2}{c}{**In Sinusoidal Simulation Adaptive Sine**} & \multicolumn{2}{c}{**Under Noise Metric Simulation Adaptive**} \\  & \multicolumn{2}{c}{**Multi-Taper vs. Others**} & \multicolumn{2}{c}{**Sine Multi-Taper vs. Others**} \\  & \multicolumn{2}{c}{**MSE Reduction(\%)**} & \multicolumn{2}{c}{**Runtime Reduction(\%)**} & \multicolumn{2}{c}{**MSE Reduction(\%)**} & \multicolumn{2}{c}{**Runtime Reduction(\%)**} \\ \hline Original & & & & \\ Adaptive Sine & 45 & 76 & 85 & 68 \\ Multi-taper & & & & \\ LPSD & 34 & 88 & 64 & 86 \\ LISA-LPSD & 90 & 25 & 64 & 31 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Performance: Adaptive Sine Multi-Taper Algorithm vs. others in sinusoidal stimulation and under noise metric simulation. Figure 3: Performance Comparison of LPSD, LISA-LPSD, and Adaptive Sine Multi-Taper Algorithms in sine excitation simulation. (**a**) Squared Bias, Variance, and MSE for LPSD, LISA-LPSD, and Adaptive Sine Multi-Taper Algorithms before and after improvement; (**b**) Computation time comparison using LPSD, Adaptive Sine Multi-Taper, Original Adaptive Sine Multi-Taper, and LISA-LPSD Algorithms. As illustrated in Figure 4a, the noise spectral density function is formulated as follows: \(\rm n(f)=\sqrt{1+\left(\frac{3 mHz}{f}\right)^{4}}\), with f ranging from 0.1 mHz to 1 Hz and featuring a corner frequency at 3 mHz. Initially, random phases were simulated, followed by the creation of both positive and negative frequency spectra to form a complete spectrum. Subsequently, an inverse Fourier transform was applied, yielding the time series illustrated in Figure 4b. This time series serves as the foundational input for the ensuing spectral density estimation analysis. As illustrated in Figure 5a, during the noise spectral density estimation process, the input signal exhibits a notably steep spectral slope characteristic in the low-frequency region, necessitating higher resolution demands. To address this issue, the improved algorithm enhances low-frequency resolution by setting a fixed, lower taper number for frequencies below 1 mHz, as shown in Figure 5b. A further quantitative assessment through comparative analysis of multiple algorithms confirms the improved algorithm's performance in low-frequency resolution. Similarly, as illustrated in Figure 6a, it is evident that in the vicinity of 1 mHz, the improved adaptive sine multi-taper algorithm exhibits an increase in variance relative to the original algorithm; however, both bias and MSE are notably reduced. Furthermore, when compared to the LPSD algorithm, the overall MSE remains comparatively lower. At the 1 mHz threshold, the MSE of the adaptive sine multi-taper algor  
Figure 4: Evaluation of noise spectrum spectral density. (**a**) Metrics for assessing noise spectrum spectral density; (**b**) Temporal sequences for noise spectrum spectral density analysis. Figure 5: Comparison of algorithm effectiveness in estimating noise spectral density under noise metric simulation. (**a**) Noise spectrum spectral density comparison; (**b**) Taper number comparison before and after improvements for adaptive sine multi-taper algorithm.