## 4 Experiments on Feature Extraction of Machinery Faults
### Feature Extraction of Impulse Fault on Bearing Dataset--Case 1
#### 4.1.3 Comparisons with Other Methods  
In order to verify the effectiveness of feature extraction, some representative models were adopted for comparisons. Figure 8: Results of the envelope demodulation: (**a**) The envelope waveform of latent component three; and (**b**) the corresponding envelope spectrum. Figure 7: Reconstructed time-frequency distributions based on the vector product: (**a**) Reconstructed time-frequency distribution (TFD) 1; (**b**) reconstructed TFD 2; (**c**) reconstructed TFD 3. Maximum correlated kurtosis deconvolution (MCKD) is one of the mainstream algorithms for revealing specific impulse series from a signal and is used for analyzing the same vibration signal. First, based on the sampling rate _fs_ of 20,000 Hz and the characteristic frequency, extracted previously at 234.4 Hz (0.0043 s), we derived the deconvolution period _Tp_ at 86 (rounded). Then, the length of the filter \(L\) also required determination. For finding a best \(L\), we set a loop interval in calculation, e.g., 100 to 500, to locate the one with a maximum correlated kurtosis. In this case, \(L\) was 400. The latent impulses extracted by MCKD are presented in Figure 9. The frequency spectrum shows the characteristic frequency and its harmonic. However, it also contains more irrelevant components as compared to the SNTF envelop spectrum. Another drawback of MCKD is that it requires an accurate period _Tp_ in advance. A deviation between the theoretical characteristic frequency and the authentic one might produce undisirable results. Next, we performed the feature extraction based on singular value decomposition (SVD), which has extensive applications in the field of signal processing. In essence, SVD is a generalization of eigenvalue decomposition and pertains to matrix analysis methods. To employ this method, we first consecutively selected 2048 points of data from the vibration signal, and then converted the data into a Hankel matrix, \(\mathbf{H}\in\mathbb{R}^{2019\times 30}\), with 30 row vectors (similar to PSR but with a time delay of 1). \[\mathbf{H}=\left[ \begin{array}{c}\mathbf{s}_{1}^{m}\\ \mathbf{s}_{2}^{m}\\ \vdots\\ \mathbf{s}_{j}^{m}\\ \vdots\\ \mathbf{s}_{30}^{m}\end{array} \right]^{T}=\left[ \begin{array}{cccc}x_{1}&x_ {2}&\cdots&x_{2019}\\ x_{2}&x_{3}&\cdots&x_{2020}\\ \vdots&\vdots&\ddots&\vdots\\ x_{j}&x_{j+1}&\cdots&x_{j+2018}\\ \vdots&\vdots&\ddots&\vdots\\ x_{30}&x_{31}&\cdots&x_{2048}\end{array} \right]^{T}. \tag{28}\]  
With the help of SVD, we obtained the following singular values: \(\sigma_{1}\geq\sigma_{2}\geq\sigma_{3}\ldots\geq\sigma_{30}\), which reveal the distribution of relevant signal and irrelevant interference. After employing the greatest three singular values for signal reconstruction, the frequency band of 4000 to 5000 Hz was restored, as shown in Figure 9(a). Eventually, we performed the Hilbert transform to derive the  
Figure 9: Results based on maximum correlated kurtosis deconvolution (MCKD): (**a**) The deconvolution signal after filtering; (**b**) the frequency spectrum of the deconvolution. envelope spectrum, as presented in Figure 10b. Similarly, the fault characteristic frequency was also extracted in the envelope spectrum, while the feature is less evident than the previous methods. Finally, the maximum spectrum kurtosis (MSK) method was also adopted to extract the fault feature. Based on STFT, the central frequency and the resolution of the optimal bandpass filter can be obtained. The central frequency \(f_{\mathit{ctr}}\) is selected when the value of kurtosis reaches a maximum, and the resolution \(\Delta f\) is determined by the sampling rate and the window width; i.e., \(\Delta f=2f_{s}/N_{w}\). We used a segment of 2048 points of the original signal and set the width of Hanning window at 45, from which we derived \(f_{\mathit{ctr}}\) at 3750 Hz and \(\Delta f\) at 663 Hz, as shown in Figure 11a. Thus the characteristic frequency band of (3087 Hz, 4413 Hz) was accessed and thereupon employed for envelope demodulation.