# Wavelet-Based Estimation of Damping from Multi-Sensor, Multi-Impact Data
## 5 Results and Discussion
### Numerical Simulation
Moreover, the Yoshida method performs similarly to the CTWT-2 method, with almost the same error levels. The mean squared error (MSE) for Yoshida is closely aligned with those for CTWT-2, indicating comparable accuracy. Furthermore, the pLSCF method begins to exhibit reduced performance at an SNR of 5 dB, indicating a higher susceptibility to noise compared to other methods. While it is accurate at higher SNR levels, its performance deteriorates as the level of noise increases, falling behind other methods. Figure 1 highlights the superior performance of the LSRF and PP methods, which stand out with the lowest error and highest precision across most SNR levels in scenarios where the impact is known to be at the beginning of the recorded signal. The CTWT-1 method follows closely, outperforming CTWT-2 and Yoshida. The Yoshida method, with almost the same accuracy as CTWT-2, demonstrates strong performance as well. The \begin{table}
\begin{tabular}{l c c} \hline \hline
**Method** & **F** & \(p\)**-Value** \\ \hline CTWT-1 & 3.09 & 0.079 \\ CTWT-2 & 5.91 & 0.015 \\ LSRF & 0.78 & 0.377 \\ pLSCF & 0.045 & 0.831 \\ PP & 0.625 & 0.429 \\ Yoshida & 0.47 & 0.493 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The effect of the number of recordings in ANOVA test for aligned dataset. pLSCF method, though effective at higher SNR, becomes more sensitive to noise at lower SNR levels. While Figure 1 demonstrates the damping ratio extraction methods assuming the impact is known to be at the beginning of the signal, this scenario is not realistic. In many practical situations, the exact time of impact is not fixed at the beginning. To address this, Figure 2 presents the scenario where the time of the impact is random but known. This more realistic condition reveals that variations in method performance are generally not significant, except for specific observations at higher SNR levels. In this analysis, \"frequency normalized\" refers to the process in which we adjust for the frequency content of the impact. By knowing the time of impact, we can perform an FFT on the impact signal and then divide the FFT of the measurement signal by the FFT of the impact. This operation yields the system's frequency response, which we can then use for damping extraction methods. Comparing Figure 2 with Figure 1 suggests that the CTWT methods, LSRF, and pLSCF have maintained stable performance across varying conditions of impact timing, with minimal changes in error rates compared to the fixed impact scenario. This demonstrates robustness against timing variations, supporting their applicability in unpredictable environments. However, some differences were more pronounced under the random impact conditions at higher SNR levels. The PP method exhibited a slight worsening in performance, reflecting a decrease in effectiveness when faced with timing variations. The Yoshida method showed a significant decrease in performance at higher SNR levels, which emphasizes its vulnerability to timing variations. This decline suggests that, while the method can work well under ideal conditions, its reliability decreases significantly with more noise and timing variability. Figure 1: Comparison of damping ratio extraction methods across varying SNR levels when the impact is known to be at the beginning of the signal. The dashed line represents the true damping ratio value. (**a**) Shows the damping ratio performance, while (**b**) presents the mean squared error performance. LSRF consistently exhibits the lowest error, demonstrating superior performance. (**a**) Damping Ratio vs. SNR. (**b**) Mean Squared Error vs. SNR. Figures 1 and 2 demonstrate the accuracy of the damping ratio extraction methods under the assumption that the exact impact time is known. However, this assumption may not be valid in practical applications, especially when no sensor is available on the hammer to detect the impact time. In FFT-based methods, normalization in the frequency domain requires an estimate of the impact time.