# Sparse Envelope Spectra for Feature Extraction of Bearing Faults Based on NMF
## 3 Results  
### Numerical Experiment  
To evaluate the performance of the proposed method, according to the proposed vibration model of a rolling bearing [35; 36], the synthetic signal \(x(t)\) is generated by:  
\[x(t)=\sum_{k}A_{k}e^{-B_{k}t}\sin(2\pi f_{k}t-kT+\varphi_{k})+\sum_{m}P_{m}\sin (m2\pi f_{o}t)+n(t) \tag{16}\]  
The exponentially decaying component usually indicates a localized fault in rolling bearings or external excitation, such as electromagnetic interference. \(A_{k}\) is the amplitude of impulse. \(T\) is the period of impulse, \(f_{h}=1/T\) is the fault feature frequency, \(f_{k}\) is the natural frequency and \(B_{k}\) and \(\varphi_{k}\) are the damping ratio and phase respectively. It is obvious that the vibration response excited by a bearing fault is periodic. The second item of Equation (16) corresponds to the rotor fault, which is represented by the frequency harmonic component \(f_{o}\). \(P_{m}\) is the amplitude of the harmonic, and the last item \(n(t)\) is Gaussian white noise. Considering the low frequency position for the rotor harmonic fault, the impact excitations are only computed in an experiment. The outer ring fault-related frequency \(f_{h}\) is set to 80 Hz, and the excited natural frequency is set to 2200 Hz. Meanwhile, three non-periodic transient oscillations are added, where the carrier frequencies are 1000 Hz, 2600 Hz and 3000 Hz, and the fractional bandwidths are set to 0.1, 0.3 and 0.1. The phases are all zero. Finally, the Gaussian white noise with signal-to-noise ratio (SNR) = 0 is added, and the vibration response signal is sampled at 10,240 Hz. The components of the simulated signal are described in Figure 3, and its waveform and spectrum are shown in Figure 4, where the red line denotes the envelope of the valid fault component. In Figure 4, because of the strong background noise and the random transient interference, it is difficult to directly observe the impact response caused by the local fault and its frequency component. The Hanning window with 33 samples and 97% overlap rate is used for the STFT of the signals. Then, NMF is performed on the TFD of the simulated signal, where the low rank dimension \(r=5\). Figure 5 illustrates the decomposed spectral basis and the time weight vectors. For the distribution of spectral basis \(\mathbf{W}\), the basis \(\mathbf{w_{5}}\) has a wide frequency band of about 2000 Hz, which can contain the periodic transient oscillations of 2200 Hz and the transient impact excitation component of 2600 Hz. The frequency basis \(\mathbf{w_{2}}\) focuses on the frequency band around 3000 Hz and corresponds to the transient oscillation components of natural frequency 3000 Hz. The frequency basis \(\mathbf{w_{3}}\) is concentrated in the frequency band of 1000 Hz, which is the frequency feature of the other transient impact excitation components. In addition, the bases \(\mathbf{w_{1}}\) and \(\mathbf{w_{4}}\) correspond to the noise components. As for the distribution of the time weight matrix \(\mathbf{H}\), the two peaks \(\mathbf{h_{2}}\) at approximately 0.14 s and 0.22 s correspond to the two impulses whose natural frequency is 3000 Hz; the peak \(\mathbf{h_{3}}\) at approximately 0.35 s corresponds to the impulse whose natural frequency is 1000 Hz. Useful feature components of other time weight vectors are difficult to observe. Figure 4: **(a)** Waveform of the simulated signal; (b) Spectrum of the simulated signal. Figure 5: NMF on the TFD of the simulated signal. Figure 3: The model of simulated signal. The spectral bases \(\mathbf{w_{2}}\), \(\mathbf{w_{3}}\), and \(\mathbf{w_{5}}\), which contain the impact excitations, are selected to extract impulse responses. The sparse envelope spectra constructed with three frequency bases and weight vectors are given in Figure 6, in which most of the feature components are significantly enhanced and noise is almost eliminated.