# Feature Extraction of Bearing Weak Fault Based on Sparse Coding Theory and Adaptive EWT
## 3 AEWT Based on Optimized Component Screening
### Spectrum Segmentation Based on Scale Space Method  
After proposing the EWT, [PERSON] proposed a spectrum division method based on scale-space theory [22]. This method can quickly extract the mode of the median value  
Figure 1: Fourier spectrum segmentation.  
of the spectrum and has the advantages of self-adaptability and no parameterization. In EWT, the boundary and number of band-pass filters greatly affect the results of signal decomposition and fault feature extraction.  
Assuming that \(\varphi(x)\) is a function defined in the range of \([0,x_{\max}]\), the continuous scale-space representation \(\psi(x,t)\) is determined by the convolution of the kernel function \(\lambda(x;t)\) and \(\varphi(x)\):  
\[\psi(x,t)=T_{t}[\varphi](x)=\lambda(x;t)*\varphi(x) \tag{10}\]  
where \(\lambda(x;t)\) is the Gaussian kernel, given as, \(\lambda(x;t)=\frac{1}{\sqrt{2\pi t}}e^{-x^{2}/(2t)}\), and \(t\) represents the scale parameter.  
The scale-space representation of function \(\varphi(x)\), which can be understood as: \(\psi(x,t)\), should be smoother when \(t\) increases, and all modes with characteristic lengths less than \(\sqrt{t}\) will be eliminated. In practical applications, truncation is often used to obtain a filter, and the discrete scale space is expressed as follows:  
\[\psi(m,t)=\sum_{n=-\mathrm{K}}^{+\mathrm{K}}\varphi(m-n)\lambda(n;t) \tag{11}\]  
The discrete Gaussian kernel in Equation (11) is expressed as follows:  
\[\lambda(n;t)=\frac{1}{\sqrt{2\pi t}}e^{-n^{2}/(2t)} \tag{12}\]  
where K is as large as possible so that the approximation error can be ignored. Usually \(\mathrm{K}=\alpha\sqrt{t}+1(3\leq\alpha\leq 6)\). When \(\alpha=6\) the approximation error is less than \(10^{-9}\).  
In order to achieve discretization, the following discretization of scale parameters is also required:  
\[\sqrt{t}=s\sqrt{t_{0}},s=1,\cdots,S_{\max} \tag{13}\]  
where \(S_{\max}\) is an integer. Usually \(\sqrt{t_{0}}=0.5\), and \(S_{\max}=2x_{\max}\).  
The size of the scale parameter \(t\) can change the spectrum division boundary. In other words, when the scale parameter \(t\) decreases, the more minimum points that are obtained, and the more spectral boundary points there are. When the scale parameter \(t\) increases, the spectral lines are smoother, fewer minimum points are obtained, and there are fewer spectral boundary points. Therefore, when the scale parameter is too small, for broadband signals such as a periodic transient impulse signal, the modulated frequency band disassembled easily, resulting in the destruction of the integrity of the periodic shock characteristics, which reduces the recognition accuracy of fault features. Because narrowband filtering is more effective, the original scale parameter \(\sqrt{t_{0}}=0.5\) is modified to \(\sqrt{t_{0}}=2.0\).  
In \(\psi(m,t)\), the number of minimum points about the variable m is a decreasing function of the scale parameter \(t\). If we suppose N0 is the initial number of minimum points, then each minimum point can generate a curve \(\mathrm{X}_{i}(i\in[1,N_{0}])\) with length \(L_{i}\).  
The adaptive searching of Fourier spectrum segmentation points can be transformed into the searching of meaningful modes in the scale space. If the curve \(\mathrm{X}_{i}\) is long enough, that is, under a certain threshold \(T\), when \(L_{i}>T\) is satisfied, the minimum point corresponding to the curve \(\mathrm{X}_{i}\) is the boundary point of the modal in the scale space, which is precisely the Fourier spectrum segmentation boundary point. Then modes are quantified as determined by a threshold \(T\) so that the curve \(\mathrm{X}_{i}\) with a length greater than \(T\) is the boundary of spectrum segmentation, and meaningful modes are obtained. At present, there are five methods to determine automatically the threshold \(T\): Mean, K-Means, Half-Normal, Otsu and Empirical-law. By analyzing and comparing the application effect of these five methods on the vibration signal of a bearing, this paper selects the Empirical-Law method to determine the threshold \(T\).