# 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
\[f(t)=\cos(2\pi f_{1}t)+\frac{1}{2}\cos(2\pi f_{2}t)+e(t) \tag{17}\]  
where \(f_{1}\) and \(f_{2}\) represent the primary frequencies of the simulated signals, \(f_{1}=2\) Hz, \(f_{2}=5\) Hz, and \(e(t)\) is the irregular noise signal. To simulate testing conditions, a random noise of 18 dB is added to the signal \(f(t)\), resulting in the simulated signal for testing. Subsequently, the signal undergoes CEEMDAN decomposition, and the decomposed IMF components are illustrated in Figure 5. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Classifications** & **SNR/dB** & **RMSE** \\ \hline Soft threshold denoising & 23.07 & 0.126 \\ Hard threshold denoising & 23.25 & 0.123 \\ Thresholding denoising in literature [25] & 23.84 & 0.115 \\ Thresholding denoising in literature [26] & 24.32 & 0.109 \\ Threshold denoising in this article & 25.15 & 0.099 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: SNR and RMSE values after noise reduction. Figure 4: Comparison of noise reduction with various threshold functions. (**a**) Noisy signal. (**b**) Signal after improved threshold denoising. (**c**) Signal after soft threshold denoising. (**d**) Signal after hard threshold denoising. (**e**) Signal after improved threshold denoising in the literature [25]. (**f**) Signal after improved threshold denoising in the literature [26]. Figure 5 illustrates the outcome of decomposing the simulated signal using the CEEMDAN method. In this representation, the intrinsic mode function (IMF) components are arranged in order from high frequency to low frequency, representing the frequency components inherent in the original signal. The IMF components derived from the analog signal are subjected to correlation coefficient analysis and organized based on Equation (18). Table 3 presents the correlation coefficient values, aiding in the selection of IMF components that retain more original information for subsequent noise reduction and reconstruction. The correlation coefficient formula is as follows:  
\[\rho_{X,Y}=\frac{\cos(X,Y)}{\sigma_{X}\sigma_{Y}^{Y}}=\frac{E((X-\mu_{X})(Y- \mu_{Y}))}{\sigma_{X}\sigma_{Y}}= \tag{18}\]  
where \(X\) and \(Y\) represent two signals, \(E(\cdot)\) is the mathematical expectation of the signal, _cov_ is the covariance, and \(\sigma\) is the variance. Based on the given equation, the IMF components with correlation coefficients greater than 0.1 are selected for the sym8 wavelet basis and five-layer wavelet decomposition. Other correlation coefficients of IMF are disregarded, and the remaining components are reconstructed to yield the final noise-reduced signal. Figure 6 displays the noise reduction signal. \begin{table}
\begin{tabular}{c c c c c c c} \hline
**IMF Component** & **IMF1** & **IMF2** & **IMF3** & **IMF4** & **IMF5** & **IMF6** \\ \hline Correlation coefficient & 0.105 & 0.083 & 0.018 & 0.019 & 0.019 & 0.305 \\ IMF component & IMF7 & IMF8 & IMF9 & IMF10 & IMF11 & IMF12 \\ Correlation coefficient & 0.434 & 0.564 & 0.875 & 0.573 & 0.081 & 0.002 \\ \hline \end{tabular}
\end{table} 
Table 3: IMF correlation coefficient value of the simulated signal. Figure 5: Decomposition result of the simulated signal using the CEEMDAN method. The noise reduction signal images in Figure 6 appear relatively close overall, which is attributed to the simplicity of the initial signal. However, when dealing with noise reduction in complex signal images, comparing the noise reduction effect requires computing signal-to-noise ratio (SNR) and root mean square error (RMSE) for several signals after noise reduction. Table 4 provides the SNR and RMSE values of several signals after noise reduction. According to the comprehensive analysis of Figure 6 and Table 4, SNR after adopting a hard threshold lies between that of a soft threshold and that of an improved threshold, indicating a general noise reduction effect. The soft threshold demonstrates superior smoothness.