# A Deep-Learning-Based Bearing Fault Diagnosis Using Defect Signature Wavelet Image Visualization
## 3 Fault Diagnosis Methodology Using the Defect Signature Wavelet Image
### 2-D Data Representation with Defect Signature Wavelet Image Generation
For instance, the continuous wavelet transform has a similar skeleton with the Fourier transform. While the Fourier transform yields correlation coefficients between the original signal and a sinusoidal signal, continuous wavelet transform obtains correlation coefficients resulting from an inner product of the mother wavelet and the signal. Unlike the Fourier transform though, where the signal is converted into the frequency-domain, continuous wavelet transform transfers the signal to the time-frequency-domain by managing the shape of the mother wavelet. Here, the research controls the shape of the mother wavelet by adjusting the scaling and shifting parameters. Continuous wavelet transform, using a  
Figure 4: Generating the DSWI representation and process for fault diagnosis. smooth analytical mother wavelet, is able to identify the dynamic frequency characteristics of the signal at different scales. By employing various dilates and translations to the mother wavelet function, continuous wavelet transform coefficients reflect the resemblance of the signal to the wavelet at the current scale. The bump wavelet is a good choice for the continuous wavelet transform when signals are oscillatory, and when the researchers are more interested in time-frequency analysis than in the localization of transients. Moreover, bump wavelet has the best time resolution permitting separation of the start and the end times for each component of the signal with impressive precision for each of the performed tests. The bump is the symmetric wavelet in frequency and has an immediate relationship between the scale and the center frequency. The bump wavelet is defined by:  
\[\hat{\psi}_{bump}(\zeta)=e^{1-\frac{1}{1-\sigma^{2}(\zeta-\
u)^{2}}}\chi_{(\
u -1/\sigma,\
u+1/\sigma)}(\zeta) \tag{4}\]  
where \(\sigma>0\), \(\
u>0\) with \(\sigma\
u>1\), are the commonly used with continuous wavelets. The parameter \(\sigma\) controls the window widths of the time-frequency localization of the wavelets (plays a role in trans-shaping the mother wavelet \(\psi_{bump}\)) and has effects on the representation of the transformed signal. In the literature, the wavelet parameter \(\sigma\) is usually treated as a fixed constant. The bump wavelet \(\psi_{bump}\) is bandlimited and hence it has better frequency localization than other wavelet families. \(\zeta_{\psi}=\
u\) admits the peak frequency which is defined by \(\zeta_{\psi}:=\underset{\zeta}{\text{argmax}}\big{|}\psi_{bump}(\zeta)\big{|}\) and \(\chi\) denotes the indicator function. The translation parameter \(\
u\) is involved in the mother wavelet's location and specifies the properties of the resulting child wavelets. Therefore, this research also determines the characteristic signature of faults at various locations of the mother wavelet by controlling the translation. The high-resolution in the frequency of large-scale wavelet permits us to capture the harmonic of the slow-variation elements whenever the fine-resolution of time in small-scale wavelet allows us to catch the fast-variation elements in the AE data. The wavelet decomposition enables detection of the hidden details of transient impulse waveforms, which is significant for inspecting a signal which contains both high frequency and low frequency components. In the case of the bump wavelet, the wavelet representation is almost symmetric with respect to the scale associated with the peak frequency. Since most defect characteristic harmonics stay in the low frequency range, fine-resolution frequency band analysis is essential to exactly interpret the properties of the abnormal indications in the bearing. As mentioned previously, the mother wavelet works in the role of a bandpass filter that permits a special frequency band to pass across a range existing between two limiting frequencies. This paper scrutinized the multiple faults which occur in bearings by changing the cutoff frequencies of the bandpass filter with the frequency range which contains defect characteristics.