# Sparse Envelope Spectra for Feature Extraction of Bearing Faults Based on NMF
## 3 Results
### Experimental Verification  
#### 3.2.1 Case 1  
To further explore the effectiveness of the proposed method, the data from the Case Western Reserve University Bearing Data Center [37] is used for the verification. The test stand consists of a two-horsepower motor with a torque transducer and a dynamometer to apply different loads. Vibration data was collected using accelerometers, which were attached to the housing with magnetic  
Figure 8: Envelope spectrum from weight vector: (**a**) \(h_{2}\); (**b**) \(h_{3}\); (**c**) \(h_{5}\). Figure 9: Fault characteristics extracted by the fast spectral kurtosis method: (**a**) Kurtogram of the simulated signal, the global maximum is achieved for \(f_{c}=960\) Hz and Level = 6; (**b**) Waveform of the filtered signal; (**c**) Spectrum of the filtered signal; (**d**) Envelope spectrum of the filtered signal. bases. [PERSON] [38] conducted a comprehensive analysis of each type of fault in the data set and summarized all the data into three types of faults, namely, obvious (Y), weak (P), and cannot be diagnosed (N). Faults marked as \"P\" or \"N\" are recommended for the performance analysis of the newly proposed algorithm. The fault of the inner ring bearing numbered 277 DE, marked as \"P\", is adopted for identification. The sensor for 277 DE data is located at the motor driving end, and far from the fan end of the fault bearing, which means the test signal is coupled with a large amount of interference components. The analyzed data set consists of 4096 sampling data with a sampling frequency of 12 kHz, and its waveform and frequency spectrum are shown in Figure 10. According to the structural parameters of the bearing and the rotation speed, the characteristic frequency of the fault bearing is \(f_{i}=142.9\) Hz. In the frequency spectrum of Figure 10b, distinct frequency bands can be observed except for the harmonic frequency of the rotation speed, and the high frequency region is located in the range of 1900-4000 Hz. After STFT is computed with the same parameters in the simulation experiment, the TFD matrix is decomposed by NMF with low rank \(r=5\); the basis and weight vectors are shown in Figure 11. Because the high frequency bands usually have small noise interference, for the five decomposed frequency regions in Figure 11, the spectral bases \(\mathbf{w_{3}}\), \(\mathbf{w_{4}}\), \(\mathbf{w_{5}}\) are located at the frequency band of 1900-4000 Hz and selected to extract the impulse response. The three selected frequency regions and the sub-matrix of corresponding weight are adopted to calculate the inner product of the squared envelope, and the sparse envelope spectra are displayed in Figure 12.  
In Figure 12a, constructed by frequency basis \(\mathbf{w_{3}}\) and weight vector \(\mathbf{h_{3}}\), the fault characteristic frequency \(f_{i}\) can be observed clearly. The sparse envelope spectrum constructed by the spectral bases \(\mathbf{w_{4}}\) and \(\mathbf{w_{5}}\) are shown in Figure 12b,c. Thereinto, an obvious rotational frequency of 29.2 Hz and double harmonic components are found, while the frequency of the inner ring fault is smaller. Although the frequency band regions of the three frequency bases are all located in the 1900-4000 Hz region, the three sparse envelope spectra are different. It can be supposed that \(\mathbf{w_{3}}\) corresponds to the vibration response excited by fault, while \(\mathbf{w_{5}}\) can be generated by other components, such as motor vibration excitation or bearing friction. Due to the overlapping of multi-frequency bands, the bases \(\mathbf{w_{3}}\) and \(\mathbf{w_{5}}\) are very close, as shown in Figure 13. Therefore, except for \(\mathbf{w_{3}}\), the sparse envelope spectrum constructed by the basis \(\mathbf{w_{5}}\) and the weight \(\mathbf{h_{5}}\) also contains part of the features of the periodic impact response. Figure 10: Inner ring fault bearing: (**a**) Waveform of the fault signal; (**b**) Frequency spectrum of the fault signal. Figure 11: NMF on the TFD of the inner ring fault bearing.