## 3 The Preliminaries
### Wavelet-Based Vibration Imaging (WVI)  
To obtain discriminant features from the preprocessed vibration signal, intrinsic information of the vibration signal should be utilized, such that it can provide enough information about MDTF types of defects. For this reason, a proper method that can highlight the key representative elements of MDTF-type defects in gearbox vibration signal is needed. Accordingly, the optimized output sub-bands from the LADT, which contains condensed defect-related useful information, are converted into two-dimensional time-frequency representation images by employing the CWT method; these images are called WVIs. These WVIs, which carry enough fault-related information, are referred to as the enriched feature pool in this paper. The enriched feature pool of the WVIs can be utilized for identifying each defect type of MDTF states (i.e., PC, DT1, DT2, DT3, DT4, DT5, DT6) of the gearbox under variable speed conditions. The process of WVI formation can be explained in detail as follows:  
To overcome the limitation of Fourier transform in processing non-linear and non-stationary signals, and the limitation of STFT with fixed timing-window transforming observation, a wavelet approach has been developed. The wavelet transform uses a mother wavelet for decomposing a signal into the spatiotemporal domain. The mother wavelet can be adjusted by expanding or compressing during the transforming process [30]. We denote the wavelet function as \(\varphi(t)\), with \(\varphi(\omega)\) as Fourier transform. To apply the wavelet approach in terms of reversible transform, the admissibility condition must be satisfied:  
\[C_{\phi}=\int_{-\infty}^{\infty}\frac{\left|\phi(\omega)\right|^{2}}{\left| \omega\right|}d\omega<\infty, \tag{4}\]  
where \(C_{\phi}\) is the admissibility constant. This (inequality 4) approximates \(\phi(\omega)=0\), which can be presented as:  
\[\int_{-\infty}^{\infty}\varphi(t)dt=0, \tag{5}\]  
and this requirement also makes clear that the mother function is a band-pass filter. The term \"wavelet\" implies a small oscillation wave with the finite length of the window function, and \"mother function\" can be understood as a prototype function, such as Morlet  
Figure 3: Block diagram of the LADT. wavelet or Daubechies wavelet, whose variants are the wavelet window functions. The actual wavelets are generated from a mother wavelet by the following equation:  
\[\varphi_{s,\tau}(t)\ =|s|^{-\frac{1}{2}}\ \varphi\bigg{(}\frac{t-\tau}{s}\bigg{)}, \tag{6}\]  
where \(\tau\) is the translation parameter and \(s\) represents dilation in Equation (6). The translation parameter represents time in the wavelet domain. The dilation is the inversion of frequency. This scale of wavelet technique is analogous to the scale of map architecture. A large scale in mapping indicates the globalized scenery, and a smaller scale indicates more detail. Similar principles can be applied to the wavelet approach; the high scale (i.e., \(s\gg 1\), low frequency) is used for observing the global features of a signal because the wavelets are expanded for extracting the low-frequency components, such as the large time window of STFT. In contrast, the low scale (i.e., high frequencies, \(s\ll 1\)) is used for observing more details of a signal, called local features. Consider the vibration sub-band \(x(t)\) and the given wavelet family \(\varphi_{s,\tau}(t)\), the continuous wavelet transform of \(x(t)\ \in L^{2}(\mathbb{R})\) is calculated [31] by following inner products equation:  
\[CWT_{x}^{\varphi}(s,\tau)\ =<x,\varphi_{s,\tau}>=|s|^{-\frac{1}{2}}\int_{- \infty}^{\infty}x(t)\varphi^{*}\bigg{(}\frac{t-\tau}{s}\bigg{)}dt. \tag{7}\]  
Equation (7) represents the coefficients of _CWT_. _CWT_ coefficients are the combination of translation series (time series) and scale (1/frequency) series, which can be utilized for constructing the vibration imaging feature spaces (scalograms).