# 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  
Bearing faults can occur with many types of damage presenting such as spalling, pitting, misaligned races, waviness that happens due to improper installation, abrasive wearing, manufacturing error, material fatigue, and so on. In general, the fault in each bearing element has a specific representative frequency. When the fault appears on a bearing component, the interaction of defects with other surfaces generates pulses with small duration which lead to an increasing vibrational energy at that specific frequency. These particular frequencies depend on the geometry characteristics of the bearing such as the number of rolling elements (or balls) \(N_{roll}\), the rolling element's diameter \(D_{R}\), the cage diameter or pitch diameter \(D_{p}\), the contact angle of the balls \(\alpha\), and the rotational frequency \(Sp\). This phenomenon will generate a high peak at a particular position in the spectrum from the FFT analysis. However, the damage frequency is amplitude-modulated to the high-frequency region that causes indiscriminate visualization when we observe the spectrum with the conventional FFT method. To overcome this drawback, the demodulation method is used with the Hilbert transform and envelope analysis. By these methods, a signal is filtered by the bandpass filter in a frequency band in which the fault impulse is amplified by structure resonances and is applied to remove the carrier signal. The envelope signals of bearing outer, inner, and roller faults are illustrated in Figure 2b, Figure 3b, and Figure 3f, respectively. The obtained envelope signal contains richer diagnostic information both in terms of the repetition frequency of ball-bass and ball-spin frequency about bearing fault. The envelope spectra, obtained by applying the FFT to the envelope signal with specific defect frequencies FO, FI, and SF, for respective cases of outer, inner, and roller faults, are illustrated in Figures 2c and 3c,g, \begin{table}
\begin{tabular}{c c l} \hline \hline
**Devices** & \multicolumn{2}{c}{**Detailed Specification**} \\ \hline \multirow{3}{*}{AE sensor R15I-AST} & - & Resonant frequency: 150 kHz (Ref in V/ubar) \\  & - & Operating range: 50–400 kHz \\  & - & Peak sensitivity: \(-\)22 dB (Ref in V/ubar) \\ \hline \multirow{3}{*}{Vibration sensor PCB-622B01} & - & Frequency range: from 0.2 to 15,000 Hz \\  & - & Measurement range: \(\pm\)490 m/s\({}^{2}\) \\  & - & Sensor sensitivity: 100 mV/g \\ \hline \multirow{3}{*}{DAQ type NI 9234} & - & Operating condition: \(-\)40 \({}^{\circ}\)C to 70 \({}^{\circ}\)C operating \\  & - & Dynamic range: 102 dB \\ \cline{1-1}  & - & Resolution: 24-bit \\ \cline{1-1}  & - & IPEE signal conditioning with AC coupling (2 mA) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Specifications of measuring sensors and data acquisition card. Among the time-frequency decomposition methods, the short-time Fourier transform is constrained by the time-frequency resolution. To obtain an exact time resolution requires an analysis window to be short, whereas a long analysis window involves an accurate frequency resolution. The wavelet analysis is a recommended methodology to process the nonstationary AE signals, and it is acceptable to detect the temporary changes in the signal. In wavelet methods, the AE signals are decomposed in terms of a zero-mean function of a family of wavelets that keep an invariable shape but are able to be dilated and shifted in time. The continuous wavelet transform (or an admissible wavelet) projects an AE signal \(s(t)\) onto a family of zero-mean functions \(\psi_{\sigma,\
u}(t)\) (family of wavelets):  
\[W_{s}(\sigma,\
u)=\int\limits_{-\infty}^{+\infty}s(t)\psi_{\sigma,\
u}^{*}(t)dt, \tag{1}\]  
where \(\psi_{\sigma,\
u}^{*}(t)\) represents the complex conjugate, \(\sigma\) stands for a dilation factor, and \(\
u\) is a translation factor. The wavelets remain normalized, such that \(\|\psi_{\sigma,\
u}\|=1\), as the mother wavelet is normalized.