# A Deep-Learning-Based Bearing Fault Diagnosis Using Defect Signature Wavelet Image Visualization
## 3 Fault Diagnosis Methodology Using the Defect Signature Wavelet Image
### Bearing Fault Signature and Wavelet Analysis
The factor \(\
u\) has the role of shifting in time such that if the \(\
u\) gets a positive value, the mother wavelet is shifted to the right, and if the \(\
u\) gets a negative value, the mother wavelet is shifted to the left. To comprehend the role of the dilation \(\sigma\) in wavelet analysis, let us use [PERSON]'s theorem to transfer the Equation (1) to the frequency domain:  
\[W_{s}(\sigma,\
u)=\frac{1}{2\pi}\int\limits_{-\infty}^{+\infty}\hat{s}(w)\hat{ \psi}_{\sigma,\
u}^{*}(w)dw, \tag{2}\]  
where \(\hat{s}(w)\) represents the Fourier transforms of \(s(t)\) and \(\hat{\psi}_{\sigma,\
u}^{*}(w)\) are Fourier transforms of \(\psi_{\sigma,\
u}^{*}(t)\). Since \(\hat{\psi}(0)=0\), the transfer function of a bandpass filter is represented by \(\hat{\psi}(w)\), which means the  
Figure 3: 2-D representation of the signals corresponding to the inner and roller faults: (**a**) DSWI for inner fault, (**b**) envelope signal for inner fault, (**c**) envelope spectrum with FI defect frequency for inner fault, (**d**) inner defect frequency FI, (**e**) DSWI for roller fault, (**f**) envelope signal for roller fault, (**g**) envelope spectrum with SF defect frequency for roller fault, and (**h**) roller (ball spin) defect frequency SF, respectively. function \(s(t)\) is particularized by the decomposition with wavelet family in the form of a series of different frequency bandwidths. Furthermore, the energy bandwidth can be expressed by:  
\[\epsilon_{w}^{2}=\frac{1}{2\pi E}\int\limits_{0}^{+\infty}{(w-w_{c})^{2}\big{|} \hat{\psi}(w)\big{|}^{2}dw}, \tag{3}\]  
where \(w_{c}\) corresponds to the center frequency of \(\hat{\psi}(w)\), and \(E=(1/2\pi)\int_{0}^{+\infty}\big{|}\hat{\psi}(w)\big{|}^{2}dw\). Hence, the center frequency of the wavelet and the energy bandwidth of the wavelet are \((w_{c}/\sigma)\) and \((\epsilon_{w}/\sigma)\), respectively. Thus, since the scaling parameter \(\sigma\) changes, both the energy bandwidth and the center frequency of the wavelet vary. That means if the value of factor \(\sigma\) is large, the mother wavelet has the role of a zoom-in function and vice versa. Moreover, when the value of parameter \(\sigma\) is large, the bandpass width becomes diminutive, which yields an increase of resolution in frequency analysis. In this paper, the 2-D representation DSWI with the CWT spectra from the envelope signal of AE signal for the outer, inner, and roller faults are shown in Figures 2a and 3a,e, respectively. These figures depict the pattern considering both the frequency domain described by the defect envelope spectrum and the information from the time domain of the envelope signal which appears in the form of a periodic impulse. These figures also illustrate that depending on the amplitude of impulse and the attenuation process, these impulses cannot always be seen in the frequency spectrum. At some point, these defect frequencies can be diminished and not be seen even if this is the 1X harmonic which usually has higher energy than the others. This characteristic represents the non-stationarity of the system. Moreover, if the segment signal has a length of less than 0.1 s, the information about the bearing defects is missed. Hence, setting up the sampling rate and segment length appropriately is important to not lose the information.