# Fault Diagnosis of Vehicle Gerboxes Based on Adaptive Wavelet Threshold and LT-PCA-NGO-SVM
## 2 Principle of the Method
### Principle of Wavelet Threshold Denoising
#### 2.1.3 Simulation Experiment and Result Analysis  
After the noise reduction process, signals containing noise can be denoised, and commonly selected evaluation indexes for denoising effect include the signal-to-noise ratio (SNR) and root mean square error (RMSE). The denoising effect is considered superior when SNR is larger and RMSE is smaller. SNR and RMSE are defined as follows:  
\[\text{SNR}=10\text{log}\left(\sum_{i=1}^{N}\frac{S_{i}^{2}}{\left(Z_{i}-S_{i} \right)^{2}}\right) \tag{15}\]  
\[\text{RMSE}=\frac{1}{N}\sum_{i=1}^{N}\left(Z_{i}-S_{i}\right) \tag{16}\]  
Here, \(N\) is the signal length, \(S_{i}\) is the original signal, and \(Z_{i}\) is the denoised signal. The simulated annealing experiment with an improved threshold function is conducted to obtain the optimal value. The simulation noise reduction experiments involve adding Gaussian white noise to the bump signals using MATLAB R2023b, employing a sym6 wavelet base, and setting five decomposition layers for wavelet decomposition. Test  
Figure 2: Noise reduction flowchart. ing involves the use of the soft threshold function, hard threshold function, two improved thresholds from the literature [25; 26], as well as the improved threshold function. The variable \(a\) in the threshold function is initially explored with three values (\(a=0.1\), \(a=0.5\), \(a=0.9\)) in Equation (8) for noise reduction. Given the randomness in noise addition, the subsequent results represent the average of five noise reduction attempts to mitigate errors. Table 1 summarizes the obtained signal-to-noise ratio (SNR) and root mean square error (RMSE) values. It is observed that different values lead to varied results, lacking clear regularity. To refine the selection of \(a\), the simulated annealing algorithm is introduced, determining the optimal value as 0.7124 for improved threshold noise reduction. Figure 3 displays the simulated annealing iteration curve. In Table 1, the \(a\) value obtained by simulated annealing exhibits the largest SNR and the most effective noise reduction compared to other \(a\) values. Noise reduction analysis is performed using soft thresholding, hard thresholding, and improved thresholding from the literature [25]. Figure 4 displays improved thresholding from the literature [26], the improved thresholding function, and the signal after noise reduction. Table 2 presents the evaluation index calculation results of various threshold functions for processing bump signals. The signal graph in Figure 4 highlights that, when compared with the conventional soft threshold, hard threshold, and improved thresholds from the literature [25; 26], the proposed improved threshold function preserves better continuity in the signal after noise reduction and retains local features. The computational results in Table 2 show that results achieved by the improved method surpass those of other threshold functions, while the RMSE is smaller. Consequently, the improved method exhibits superior noise reduction effectiveness. \begin{table}
\begin{tabular}{c c c} \hline \hline \(a\) & **SNR/dB** & **RMSE** \\ \hline \(a=0.1\) & 22.61 & 0.133 \\ \(a=0.5\) & 24.52 & 0.106 \\ \(a=0.7124\) & 25.15 & 0.099 \\ \(a=0.9\) & 23.78 & 0.116 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: SNR and RMSE values for various \(a\) parameters. Figure 3: Simulated annealing iteration curves. Having established the effectiveness of simulated annealing in the aforementioned analyses, practical tests were conducted using real extracted signals. When obtaining the actual signal in engineering training, the measurement sensor detects not only its own vibration signal but also considerable interference noise. Therefore, in simulated signal noise reduction tests, it becomes crucial to introduce irregular noise signals into the constructed multi-harmonic vibration simulation signal \(f(t)\).