# A Review on Data-Driven Condition Monitoring of Industrial Equipment
## 2 Data-Driven Condition Monitoring of Industrial Equipment
### Data Processing Methods
#### 2.1.2 The Wavelet Transform Approach  
The WT approach can be used to decompose the signal at multiple scales and obtain local features in the time-frequency domain through a series of wavelet basis functions and is one of the most important tools in the time-frequency analysis of non-stationary signals. Let \(\Psi(t)\in L^{2}(R)\) (where \(L^{2}(R)\) is the square integrable real number space), then the Fourier transform of \(\Psi(t)\) is when the following admissibility condition is met:  
\[C_{\Psi}=\int\limits_{-\infty}^{+\infty}\frac{|\hat{\Psi}(\omega)|^{2}}{| \omega|}d\omega<+\infty \tag{2}\]  
where \(\hat{\Psi}(\omega=0)=0\). \(\Psi(t)\) represents the mother function, which can be shifted, stretched, or compressed to give a wavelet sequence (sub wavelet) after the transformation as follow. \[\Psi_{a,b}(t)=\frac{1}{\sqrt{a}}\Psi\left(\frac{t-b}{a}\right)a,b\in R,\ a\
eq 0 \tag{3}\]  
where the role of a is to scale the mother function \(\Psi(t)\), known as the scale factor, and the role of b is to temporally localize the mother wavelet, known as the shift factor. The WT inherits and expands upon the concept of STFT localization, resolving the issue that the window size in STFT does not change with frequency by providing a time-frequency window that fluctuates with frequency. Therefore, the WT can be utilized for condition monitoring by analyzing changes in the signal's mutation value and frequency structure, as well as by utilizing the WT to denoise and extract waveform information from the signal [30; 43; 44]. In [45], [PERSON] and [PERSON] used WT to decompose the stator current signal of an induction motor and extract low frequency features to represent the fault information. All the useful information of the fault is obtained by superimposing the waveforms, distinguishing the healthy and faulty states of the motor, and achieving the detection of broken rotor bar faults. In addition, many variants of the WT have also been successfully applied to equipment condition monitoring. These include continuous wavelet transform (CWT) [46; 47], discrete wavelet transforms (DWT) [48; 49; 50; 51], and empirical wavelet transform (EWT) [52]. CWT is often used to extract and analyse the wavelet features of a signal. [PERSON] and [PERSON] [53] used CWT to analyze the Monoblock centrifugal pump vibration signals and obtained feature sets consisting of different wavelets and used them as inputs to a classifier to obtain the maximum fault identification capability for the system. Finally, the diagnosis of single and hybrid faults of the pump bearing was achieved by combining the decision tree approach. In addition, CWT is also used to convert time series signals into time-frequency images. For example, [PERSON] et al. [54] converted the bearing vibration signal into a time-frequency image by a series of CWT and used it as the input to a convolutional neural network. After training the network with multilayer pooling, unique features corresponding to faults can be captured from the original CWT images, which are then used as inputs to the gcForest model for fault classification. The results show that the hybrid model achieves high accuracy in detecting and diagnosing multiple bearing faults for datasets of different sizes. [PERSON] et al. [55] used CWT to convert time series of vibration, pressure, and sound signals of hydraulic axial piston pumps under different operating conditions into two-dimensional time-frequency images and developed a convolutional neural network fault diagnosis model based on an adaptive learning rate strategy for each signal. The results showed that the average accuracy of fault diagnosis using different signals was 97.33%, 99.48% and 98.77%. In [56], [PERSON] et al. used the DWT with multi-resolution analysis (MRA) to perform a five-level wavelet decomposition of the original pump vibration signal to obtain the low and high frequency features of the signal.