# 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  
The overall process of the proposed methodology to construct the DSWI and bearing fault diagnosis is presented in Figure 4. Fundamentally, by virtue of the Hilbert transform, the signal envelope can be computed. The one-second AE signal \(s(t)\) in the time domain is converted to the Hilbert domain \(\widetilde{s}(t)\) using the Hilbert transform [27; 28]. The Hilbert transform applies the convolution of \(s(t)\) with the signal of \(1/\pi t\) that produces \(\widetilde{s}(t)=s(t)*(1/\pi t)\). Then the method calculates the analytic signal in a complex number form with both \(s(t)\) in the real part and \(\widetilde{s}(t)\) in the imaginary part as \(s_{a}(t)=s(t)+\widetilde{s}(t)\) in quadrature, where \(j\) represents the imaginary unit. Immediately, an advantage is detailed in that the demodulating of the extraction of the spectrum section is effectively executed by an ideal filter, which helps to distinguish it from adjacent components which will be considerably stronger such as the gear mesh frequencies. Following that, the absolute value of \(\textit{env}(t)=\big{|}s_{a}(t)\big{|}=\big{|}s(t)+\widetilde{s}(t)\big{|}\) is computed to yield the signal envelope. Then, the square root of FFT with signal \(\textit{env}(t)\) performs the envelope spectrum. In fact, it is more desirable to analyze the square of the envelope signal instead of the envelope itself. A simple argument for that is by comparison of the spectra of a squared signal with that of a rectified signal. In mathematical terms, it should be considered that a rectified signal is the same as the square root of the squared signal. Likewise, the envelope of the signal is calculated as the square root of the squared envelope. When the square root operator is applied, it launches extraneous components which do not appear in the original squared signal, and this is the reason to create the masking of the desired information. Because the entire operation is calculated digitally, it is impossible to erase the high harmonics by using lowpass filtration, and they generate the alias to the measurement range, which causes masking. In addition, when applying the one-sided spectrum, by considering the analytic signals whose squared envelope is constructed by multiplication with its complex conjugate, the spectra of the squared envelope is the convolution of the respective spectra. When this convolution is carried out, the result only yields different frequencies, e.g., sideband spacings. These different frequencies will contain the desired modulation information. Then the envelope signal is supplied to the continuous wavelet transform. The continuous wavelet transform with damage frequency filter band is applied after obtaining the envelope signal to generate the DSWI representation. The use of wavelet transforms to detect local faults in bearings has been described by many authors. However, most of the literature on utilization of wavelet decomposition for fault diagnostics makes the error of considering the performance only in the time-domain (mostly for denoising) and on a short recorded segment of signal, frequently shorter than the longest modulation period. Usually, the assertion is that the wavelet transform is more advanced than envelope analysis. Nevertheless, many authors fail to realize that the wavelet coefficients' squared modulus is helpful for a squared envelope signal, and much diagnostic information can be derived by analyzing the frequency domain of such squared envelope signals. As discussed, frequency-domain analysis of the envelope signal often makes evident fault repetition (in the form of transition peaks) and modulation patterns which are often difficult to recognize in time-domain signals, especially when the modulation phenomenon is so strong that the transient impulse is only stimulated when the fault point is inside the load-zone. These impulses are thus created with greatly varied amplitude.