## 5 Verification Based on Leakage Signal in Check Valve
Figure 10, Figure 11, and Figure 12 present the signal processing results for three distinct fault conditions: (1) entire seal surface defect, (2) local seal surface defect, and (3) eccentric spool defect, respectively. Each figure is organized in a three-column format: the first column displays the original signal, the second column shows the noise-corrupted input signal, and the third column presents the enhanced output signal from the SaRT-CMSR system. Table 1 shows the number of pairs used in calculating the correlation between the input and output signals in the check valve leakage experiment. The amplitudes of the original signal and the output signal in the frequency domain are shown in Table 2. The experimental results demonstrated significant cross-correlation gains: 6.9569 for the globally damaged spool, 4.4128 for the locally damaged spool, and 5.7554 for the eccentric fault spool. Notably, the proposed SaRT-CMSR method successfully enhanced all three types of fault signals that were initially completely obscured by white noise, achieving both effective noise reduction and accurate fault frequency identification. Specifically, the method preserved the original signal's peak frequency while significantly amplifying its amplitude. These findings, derived from a comprehensive analysis of the leakage signals across the three distinct fault conditions, conclusively validate the feasibility and effectiveness of the SaRT-CMSR method for enhancing valve-leakage signals in practical applications. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Valve Core**} & **The Number of Relationships** & **The Number of Relationships** & \\  & **Between the Noisy Input** & **Between the Noise-Canceled** & **Cross-Correlation** \\  & **Signal and the Original** & **Output Signal and the** & **Gain** \\  & **Signal** & **Original Signal** & \\ \hline Entire seal surface defect & 0.0453 & 0.3151 & 6.9569 \\ \hline Local seal surface defect & 0.0436 & 0.1924 & 4.4128 \\ \hline Eccentric spool defect & 0.0691 & 0.3977 & 5.7554 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of cross-correlation between input and output signals of check valve leakage experiment. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Valve Core**} & **Original Signal Spectral** & **Output Signal** & **Is the Peak** & **Amplitude** \\  & **Amplitude** & **Spectral Amplitude** & **Frequency Equal** & **Ratio** \\ \hline Entire seal surface defect & 0.00108 & 0.01123 & Yes & 10.3981 \\ \hline Local seal surface defect & 0.00148 & 0.00356 & Yes & 2.4054 \\ \hline Eccentric spool defect & 0.00267 & 0.03124 & Yes & 11.7004 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Frequency domain amplitude ratio of original and output signals of check valve leakage experiment. Figure 12: Signal-processing diagram of eccentric spool defect check valve. To further investigate the influence of the controlled system parameters, the leakage signal from an eccentric spool defect check valve was employed as a case study. The controlled system parameters were set to \(a=b=1\). The noise reduction output signal processed by the SaRT-CMSR method is shown in Figure 13. After fixing the parameters of the controlled system, the output cross-correlation gain of the system was 3.3204. The spectral analysis indicated that while the processed signal maintained the same peak frequency as the original signal, its amplitude was significantly attenuated, failing to achieve the desired enhancement effect. A comparative analysis between Figures 12 and 13 showed a suboptimal performance for the output in Figure 13. These experimental findings provide empirical validation of the parametric analysis results presented in Section 4, confirming the critical role of parameter optimization in system performance.