# Wavelet-Based Estimation of Damping from Multi-Sensor, Multi-Impact Data
## 5 Results and Discussion
### Numerical Simulation
To investigate this, we conducted an ANOVA test to analyze the effect of the number of recordings on damping ratio estimates using a non-aligned dataset. The results are presented in Table 2.  
Figure 3: Impact of sample shift on the accuracy of damping ratio extraction methods at 10 dB SNR (**a**) and 0 dB SNR (**b**), illustrating the degradation in performance with increasing misalignment from the true impact time. (**a**) 10 SNR. (**b**) 0 SNR. From the table, we observe the following:  
* The PP method shows a highly significant effect of the number of samples on damping ratio estimates, with an extremely low \(p\)-value (\(2.21\times 10^{-194}\)), indicating a strong influence of the number of recordings on its performance. * The pLSCF method also shows a significant effect with a \(p\)-value of \(1.27\times 10^{-15}\), although the influence is less pronounced compared to the PP method. * The CTWT-1, CTWT-2, LSRF, and Yoshida methods do not show significant effects of the number of samples on damping ratio estimates, as indicated by their high \(p\)-values. These results suggest that, while the PP and pLSCF methods are significantly affected by the number of recordings, the CTWT-1, CTWT-2, LSRF, and Yoshida methods are less influenced by the number of samples. This robustness further underscores the suitability of the LSRF and CTWT-2 methods for practical applications where the number of recordings may vary, alongside their previously noted accuracy and reliability under non-aligned conditions. In summary, the LSRF and CTWT-2 methods are not only accurate and reliable under varying SNR levels but also robust to the effect of the number of recordings, making them highly suitable for real-world applications where both impact time and the number of recordings are uncertain. The ANOVA test results for both the aligned (Table 1) and \begin{table}
\begin{tabular}{l c c} \hline \hline
**Method** & **F** & \(p\)**-Value** \\ \hline CTWT-1 & 2.49 & 0.114 \\ CTWT-2 & 1.52 & 0.217 \\ LSRF & 1.16 & 0.282 \\ pLSCF & 64.0 & \(1.27\times 10^{-15}\) \\ PP & 889.0 & \(2.21\times 10^{-194}\) \\ Yoshida & 0.002 & 0.969 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: ANOVA test results showing the effect of the number of recordings on damping ratio estimates using non-aligned dataset. Figure 4: (**a**,**b**) Damping ratio distribution and mean squared error of damping ratio estimates, respectively, for various methods using wavelet analysis to determine the time of impact across different signal-to-noise ratio (SNR) levels. These figures highlight the robustness and reliability of the methods under non-aligned conditions, when the time of impact is unknown. (**a**) Damping Ratio vs. SNR. (**b**) Mean Squared Error vs. SNR. non-aligned (Table 2) datasets reveal key differences in the sensitivity of damping ratio estimation methods to the number of recordings. For the CTWT-1 method, a moderate effect is observed in the aligned dataset with a \(p\)-value of 0.079, indicating some sensitivity to the number of recordings, while this effect diminishes in the non-aligned dataset with a \(p\)-value of 0.114, suggesting reduced sensitivity when the impact time is variable. The CTWT-2 method shows a significant effect in the aligned dataset with a \(p\)-value of 0.015, indicating notable sensitivity to the number of recordings, but this effect lessens in the non-aligned dataset with a \(p\)-value of 0.217. The LSRF method exhibits minimal sensitivity to the number of recordings in both scenarios, with \(p\)-values of 0.377 and 0.282 for aligned and non-aligned datasets, respectively. In contrast, the pLSCF method displays negligible sensitivity in the aligned dataset with a high \(p\)-value of 0.831, but becomes highly sensitive in the non-aligned scenario, with a very low \(p\)-value of \(1.27\times 10^{-15}\).