## 4 Experiments on Feature Extraction of Machinery Faults
### Feature Extraction of Impulse Fault on Bearing Dataset--Case 1
#### 4.1.2 Feature Extraction Based on SNTF  
Based on the feature extraction method proposed above, we first reconstructed the phase space for the selected data. The length of the signal \(x(n)\) was 20,480. With the speed at 2000 RPM, we obtained the embedding dimension \(m\) at \(2^{11}\) (2048), and the time delay \(\tau\) at 600. The number of the phase point vectors was 31 (rounded down for the consistency in vector length). Accordingly, the reconstructed phase space \(\mathbf{X}\in\mathbb{R}^{2048\times 31}\) was derived.  
When applying STFT to the phase point vectors of \(\mathbf{X}\), the width of tye Hanning window was set to 53. Thereupon, we acquired the three-way time-frequency data (\(2048\times 2048\times 31\)), on which the modulus operation was performed to obtain a non-negative time-frequency tensor \(\underline{\mathbf{Y}}_{tf}\in\mathbb{R}^{2048\times 2048\times 31}\).  
Then, we set the reduced-dimension index \(J\) at 3, knowing that the two high-frequency resonance bands in the spectrum (see Figure 5b) could be separated from the low-frequency component using SNTF. We let the sparseness regulation parameter \(\alpha\) be 0.20, and performed the SNTF-HALS method to factorize the tensor \(\underline{\mathbf{Y}}_{tf}\) into three sparse nonnegative matrices: \(\mathbf{A}\in\mathbb{R}^{2048\times 3}\), \(\mathbf{B}\in\mathbb{R}^{2048\times 3}\), and \(\mathbf{C}\in\mathbb{R}^{31\times 3}\). \(\mathbf{A}\) and \(\mathbf{B}\) respectively, indicate the data distributions in frequency domain and time domain, as presented in Figure 6, where we separately plotted them as curves.  
Figure 5: The vibration signal of bearing 1 for further analysis: (**a**) In the waveform only a faint trace of impulse is found; (**b**) the spectrum contains multiple frequency components, with resonance features located at around 1000 Hz, and between 3000 Hz and 5000 Hz.  
Figure 6: Decomposed factors of SNTF: (**a**) Frequency matrix \(A\); (**b**) time matrix \(B\).  
Based on the two factors, three TCDs (see Figure 7) were reconstructed via \(\mathbf{a_{s}b_{s}}^{T},(s=1,2,\dots,J)\), mentioned previously in Section 3.4. As a result, evident correlations exist between the factors and the TFDs. For example, the frequency-wise component at around 1000 Hz in reconstructed TFD 1 (Figure 7a) matches with the peak in the first graph in Figure 6a; both reconstructed TFD 2 and TFD 3 incorporate the prominent component at around 4200 Hz, which correspond with the trends of the 2 nd and 3 rd curves in Figure 6a.  
From the figures above we can see that the periodic impulse can be found in each TFD along the time axis and that this feature is more evidently suggested in the third TFD. In order to extract such a frequency component, we perform the inverse STFT on the third TFD to recover the latent component signal. Then the envelope demodulation is applied to generate the corresponding envelope waveform and spectrum, as presented in Figure 8.  
According to the envelope spectrum, the frequency component at 234.4 Hz is prominent, which can be identified as the characteristic frequency of the outer race fault of the bearing. Specifically, as the theoretical fault frequency is calculated with the hypothesis of rollers' pure rolling, the deviation is partly due to the sliding of the rollers. With this result, the efficacy of the feature extraction method based on Sparse NTF is revealed.