## 4 Experiments on Feature Extraction of Machinery Faults
### Feature Extraction of Impulse Fault on Bearing Dataset--Case 2
#### 4.2.3 Comparison and Analysis  
For verification, we also conducted the comparisons with MCKD, SVD, and MSK for case 2.  
In the MCKD model, the deconvolution period is given at 84 (the sampling rate is 12,000 Hz and the _fc_ is 143.6 Hz) and the length of MCKD filter is 300. The periodic train of impulse is extracted, prominent in the filtered waveform, as can be seen in Figure 17. Still, the frequency spectrum does contain the characteristic frequency of an inner ring fault, _fc_ and its harmonics, but it is relatively more interfered by irrelevant components compared to the SNTF results.  
Figure 16: Results of feature extraction of reconstructed TFD 2 (dominant frequency band—around 3200 Hz) and TFD 3 (dominant frequency band—within 1000 Hz): (**a**) Envelope waveform of TFD 2; (**c**) envelope spectrum of TFD 2; (**b**) envelope waveform of TFD 3; (**d**) envelope spectrum of TFD 3.  
Figure 17: Results based on MCKD in case 2: (**a**) In the deconvolution signal, impulses are revealed; (**b**) the frequency spectrum also captures the characteristic fault, but it is interfered.  
In the SVD model, a Hankel matrix was built, based on the one-dimesional vibration signal for decomposition. Then, the components with large singular values were restored from the right singular vector and the left singular vector, which were further processed via envelope analysis, as displayed in Figure 18. We can see that the spectrum exposes the running speed frequency _fr_, the characteristic frequency _fc_, and other interference. The fault feature was extracted but not so prominently as by the SNTF model.  
In the MSK model, the central frequency \(f_{\textit{ctr}}\) appears at 3375 Hz, level five, with a maximum kurtosis as presented in Figure 19a, and the frequency bandwidth \(\Delta f\) is thereby obtained at 750 Hz. Thus, the signal within the band of from 2526 Hz to 4125 Hz is filtered and selected for envelope demodulation. From the envelope spectrum in Figure 19b, we can see that multiple peaks stand out, including the characteristic frequency of inner ring fault, _fc_. As this impulse component is affected by a few other insignificant components, we hold that the MSK method is also less effective in this case.  
Based on the comparative studies in Sections 4.1 and 4.2, we can see that the SNTF-based method has evidently better performance of fault feature extraction, the results of whi  
Figure 19: Results based on MSK in case 2: (**a**) Kurtogram of the signal; (**b**) envelope spectrum of the filtered signal.  
Figure 18: Results based on SVD in case 2: (**a**) The frequency spectrum of the recovered signal (2800 Hz to 3800 Hz); (**b**) envelope spectrum.  
impulses can be extracted in one of the feature subspaces. An important advantage rooted in our SNTF-based framework is the utilization of multiple TFDs, which allows comprehensive observations of a single vibration signal, despite the varying degrees of impulse fault and the changing forms of noise in different time intervals. Conventional approaches to feature extraction consider a single time-frequency spectrogram [8; 30], which might receive better results after improvement.