# Sparse Envelope Spectra for Feature Extraction of Bearing Faults Based on NMF
## 2 Materials and Methods
### Feature Extraction Based on Sparse Envelope Spectrum  
As a spectral basis, each column vector of \(\mathbf{W}\) represents different frequency regions. Thus, the original signal \(x(t)\) can be filtered with the base \(\mathbf{w_{i}}\) containing the fault characteristic information, and the filtered spectrum is given by:  
\[X_{f}(f)=X(f)\mathbf{w_{i}} \tag{10}\]  
This can be followed by performing the inverse Fourier transform (IFT) on the filtered spectrum; the filtered waveform is calculated as follows:  
\[x_{f}(t)=\textbf{IFT}(X_{f}(f)) \tag{11}\]  
Next, the envelope signal is obtained from the filtered signal \(x_{f}(t)\), which is defined as:  
\[x_{e}(t)=\sqrt{x_{f}^{2}(t)+x_{f}^{\prime 2}(t)} \tag{12}\]  
where \(x_{f}^{\prime}(t)\) is the Hilbert transform of the signal \(x_{f}(t)\), and \(x_{f}^{\prime}(t)=\frac{1}{\pi}\int_{-\infty}^{+\infty}\frac{x(t)}{t-t}d\tau\). Finally, removing the zero-frequency component, the spectrum of the envelope signal is computed by Fourier transform (FT):  
\[H_{W}(f)=\textbf{FT}(x_{e}(t)). \tag{13}\]  
After computing the envelope spectrum, the feature information of the fault frequency can be observed. However, the acquired vibration signal includes not only the impact excitation caused by bearing defects, but also the external impacts. The envelope spectrum usually contains a lot of noise and other interference components. Therefore, to improve performance, it is necessary to further analyze the selected filter band. Generally speaking, each spectral basis \(\mathbf{w_{i}}\) describes the frequency region of the transient oscillation, while the weight vector \(\mathbf{h_{i}}\) indicates its time envelope. Thus, performing Fourier transform on \(\mathbf{h_{i}}\) can display the frequency components of the time envelope:  
\[H_{H}(f)=\textbf{FT}(\mathbf{h_{i}}). \tag{14}\]  
Obviously, the band-pass filtering is realized by the spectral base \(\mathbf{w_{i}}\) for the transform of the whole signal. The weight vector \(\mathbf{h_{i}}\) is obtained from the TFD directly, where the frequency component of each slip moment is calculated by the local signal. This difference leads to the different amplitude distributions of the two envelope spectra. However, the periodic impact component, as the main characteristic, has a larger amplitude in both envelope spectra. Therefore, to enhance the fault-related frequency, the product of the two squared spectra is defined as:  
\[H(f)=H_{W}^{2}(f)\bullet H_{H}^{2}(f). \tag{15}\]  
The squares of the components represent the energy of the frequency component, which can highlight a larger component than that of the amplitude spectrum [3]. The inner product, denoted by \"\(\bullet\)\", can strengthen the common frequency of the envelope spectra and reduce other frequencies. As a result, the envelope spectrum becomes sparse, and the feature frequency of bearing faults can be clearly observed naturally. The algorithm of the sparse envelope spectrum extracted by combining \(\mathbf{W}\) and \(\mathbf{H}\) is described as follows. **Step 1:** Calculating the STFT coefficients of the vibration signal with a low side-lobes window function and obtaining its TFD matrix. To better extract the impact response waveform, it is recommended to use a smaller window length;  
**Step 2:** Applying NMF on the TFD matrix with low rank \(r\). Considering that the frequency spectrum often contains multiple frequency regions, \(r\) is set to \(k\leq r\leq k+3\), where \(k\) is the estimatednumber of resonance regions.