## 3 Proposed Online Fault Diagnosis Model
### Heterogeneous Feature Pool Configuration  
To discriminate fault signatures, we use maximum possible fault features extraction [14; 15]. Different paradigms exist for the extraction of heterogeneous fault features, in which statistical features from time-domain and frequency-domain signals are used to uncover meaningful information for fault diagnosis. We use statistical features from the time-domain AE signal: \(f_{1}\), root-mean-square (\(RMS\));\(f_{2}\), square root of the amplitude (\(SRA\));\(f_{3}\), kurtosis (\(KV\));\(f_{4}\), skewness (\(KV\));\(f_{5}\), peak-to-peak (\(PPV\));\(f_{6}\), crest factor (\(CF\));\(f_{7}\), impulse factor (\(IF\));\(f_{8}\), margin factor (\(MF\));\(f_{9}\), shape factor (\(SF\));\(f_{10}\), kurtosis factor (\(KF\)); and frequency-domain features of the signal, including\(f_{11}\), frequency center (\(FC\)); \(f_{12}\), RMS frequency (\(RMSF\)); and \(f_{13}\), root variance frequency (\(RVF\)) [4]. Equations (1) and (2) present mathematical equations describing the extracted statistical features. Figure 5: Flow diagram of the proposed online bearing fault diagnosis model, where \(F_{dsk}\) is the number of fault modes in the diagnostic system knowledge (DSK). \[RMS =\left(\frac{1}{N}\sum\limits_{i=1}^{N}x_{i}^{2}\right)^{1/2},\] \[SRA =\left(\frac{1}{N}\sum\limits_{i=1}^{N}\sqrt{\left[x_{i}\right]} \right)^{2},\] \[KV =\frac{1}{N}\sum\limits_{i=1}^{N}\left(\frac{x_{i}-\overline{x}}{ \sigma}\right),\] \[SV =\frac{1}{N}\sum\limits_{i=1}^{N}\left(\frac{x_{i}-\overline{x}}{ \sigma}\right)^{3},\] \[PPV =\max(x_{i})-\min(x_{i}),\] \[CF =\frac{\max(\left[x_{i}\right])}{\left(\frac{1}{N}\sum\limits_{i= 1}^{N}x_{i}^{2}\right)^{1/2}}, \tag{1}\] \[IF =\frac{\max(\left[x_{i}\right])}{N\sum\limits_{i=1}^{N}\left[x_{i} \right]},\] \[MF =\frac{\max(\left[x_{i}\right])}{\left(\frac{1}{N}\sum\limits_{i= 1}^{N}\sqrt{\left[x_{i}\right]}\right)^{2}},\] \[SF =\frac{\left(\frac{1}{N}\sum\limits_{i=1}^{N}x_{i}^{2}\right)^{1/2} }{\frac{1}{N}\sum\limits_{i=1}^{N}\left[x_{i}\right]},\] \[RMS =\left(\frac{1}{N}\sum\limits_{i=1}^{N}x_{i}^{2}\right)^{1/2},\]0  
\[RMS =\left(\frac{1}{N}\sum\limits_{i=1}^{N}x_{i}^{2}\right)^{1/2},\]1 \[RMS =\left(\frac{1}{N}\sum\limits_{i=1}^{N}x_{i}^{2}\right)^{1/2},\]2 \[RMS =\left(\frac{1}{N}\sum\limits_{i=1}^{N}x_{i}^{2}\right)^{1/2},\]3  
To identify particular failure modes, bearing defect properties must be observed at different signal frequencies. Therefore, we extract \(RMSF\) values from frequency ranges that include harmonics of the following bearing defect frequencies [29]: Ball pass frequency of the outer raceway (\(BPFO\)), ball pass frequency of the inner raceway (\(BPFI\)), ball spin frequency (\(BSF\)), and fundamental train frequency (\(FTF\)). They are calculated as follows:  
\[ \begin{array}{l}BPFO=\frac{N_{rollers}\times F_{shuff}}{2}\left(1-\frac{B_{d }}{P_{d}}\cos\alpha_{angle}\right),\\ BPFI=\frac{N_{rollers}\times F_{shuff}}{2}\left(1+\frac{B_{d}}{P_{d}}\cos \alpha_{angle}\right),\\ BSF=\frac{P_{d}\times F_{shuff}}{2\times B_{d}}\left(1-\frac{B_{d}}{P_{d}} \cos\alpha_{angle}\right)^{2},\\ FTF=\frac{F_{shuff}}{2}\left(1-\frac{B_{d}}{P_{d}}\cos\alpha_{angle}\right),\end{array}  \tag{3}\]  
where \(N_{rollers}\) is the number of rollers, \(F_{shuff}\) is the shaft speed in hertz, \(\alpha_{angle}\) is the contact angle, \(B_{d}\) is the roller diameter, and \(P_{d}\) is the pitch diameter. RMSF features are calculated by first determining the power spectral envelope of the AE signal and then identifying the defect frequency ranges of \(BPFO\), \(BPFI\), and \(2\times BSF\) up to their third harmonics. Except for \(BPFO\), which is caused by the force of the roller on the stationary outer raceway fault, these frequencies can experience some modulation. \(BPFI\) and its harmonics are amplitude-modulated by \(F_{shuff}\), and \(2\times BSF\) is modulated by the rotational speed of the cage (\(FTF\)). Hence, amplitude modulation produces sidebands as an explicit symptom of inner-raceway and roller faults in bearings.