# Gearbox Fault Diagnosis Based on ICEEMDAN-MPE-AWT and SE-ResNeXt50 Transfer Learning Model
## 2 ICEEMDAN-MPE-AWT Denoising
### AWT Denoising  
Wavelet threshold denoising decomposes signals into various spectral components through wavelet transform. Then the noise wavelet coefficients in high-frequency bands are filtered by setting thresholds, and the effective signal wavelet coefficients in low-frequency bands are extracted and retained to accomplish the objective of noise removal. Therefore, the core of wavelet threshold denoising is a process that combines the measured vibration signal filtering with effective feature extraction. In wavelet threshold denoising, threshold estimation and the threshold function play a key role in the wavelet threshold denoising effect. A threshold that is too large results in the loss of the effective signal, while one that is too small fails to achieve the denoising effect. Consequently, it is crucial to determine the appropriate threshold. The heuristic threshold selection method [24] relying on optimal selection can dynamically modify and choose thresholds throughout, considering noise characteristics and wavelet decomposition, which often achieve better practical results.  
This article improved upon the heuristic threshold method and proposed an adaptive threshold choice method. Since the wavelet transform divided the subbands into high and low frequencies based on thresholds at each wavelet decomposition level, using a globally consistent fixed threshold could potentially result in the original signal departing from its original shape when removing noise signals in different signal subspaces across different decomposition levels. By introducing a correction factor to dynamically adjust the thresholds, a threshold choice method based on the correction factor was proposed as shown in Equation (13) [25]:  
\[\lambda_{L,k}=\sigma^{\prime}\sqrt{2\lg N}/(S_{L,k}+b), \tag{13}\]  
where, \(\lambda_{L,k}\) is the adaptive threshold, \(\sigma^{\prime}\) is the estimate of the noise standard deviation \(\sigma\), \(S_{L,k}\) are sub-band level parameters, which are used to indicate the threshold modification across various layers of decomposition, and can be introduced as below.  
\[S_{L,k}=2^{(L-k/L)}, \tag{14}\]  
where \(L\) is the total number of layers of wavelet decomposition; \(k\) is the current number of decomposition layers; and \(b\) is the regulatory factor.  
This paper proposed an adaptive threshold calculation method based on the number of wavelet decomposition layers, as shown in Equation (15).  
\[\lambda_{L,k}=(k/L)\times\sigma^{\prime}\sqrt{2\lg N}/(S_{L,k}+b). \tag{15}\]  
During the procedure for threshold selection, the convergence factor \(\theta=k/L\) is used to further adjust the calculation result of the threshold, and the threshold amplitude is gradually decreased with the addition of the number of decomposition layers in order to upgrade the resolution of the low-frequency signal and improve the noise reduction effect. The selection of small- and medium-wave bases of wavelet transform should try to select the wavelet basis function with tight support for orthogonality and symmetry [26], and it is similar to the waveform of the signal to be processed so as to reduce the calculation amount and avoid waveform distortion in the course of signal decomposition and reconstruction. The dbN series wavelet bases show good performance in this respect, and the wavelet threshold denoising based on db4 is especially remarkable. Therefore, in this paper, the db4 wavelet basis was selected for signal denoising. Another important factor that affected the effectiveness of the wavelet transform is the number of wavelet decomposition levels. Too many decomposition layers will distort the reconstructed signal, and too few decomposition layers will not reduce the noise. Therefore, it is also very important to choose the appropriate number of wavelet decomposition layers. In the current paper, the number of wavelet decomposition layers can be obtained by Equation (16):  
\[J=\text{fix}(\text{log}_{2}(\frac{f_{s}}{f_{0}})\sqrt{\frac{1}{8}}+0.5)-1, \tag{16}\]  
where a fix is rounded in the direction of zero.