# Sparse Low-Rank Based Signal Analysis Method for Bearing Fault Feature Extraction
## 3 The Proposed Fault Information-Based Sparse Low-Rank Algorithm  
The sparsity and low-rank characteristics of the fault features are explored in Section 2. To abstract the STFT coefficient matrix \(X\) corresponding to the fault information from the noisy matrix \(Y\), the TSLRM can be defined as:  
\[\arg\min_{X\in\mathbb{R}^{\text{norm}}}\left\{\|Y-X\|_{F}^{2}+\lambda_{0}\|X \|_{+}+\lambda_{1}\|X\|_{1}\right\} \tag{1}\]  
where \(\|X\|_{*}\) and \(\|X\|_{1}\) are the trace norm and \(l_{1}\) norm, respectively. \(\lambda_{i}\geq 0\) represents the regularization parameter. In Equation (1) trace norm \(\|X\|_{*}\) and \(\|X\|_{1}\) promote sparsity of singular values (SVs) and elements of matrix \(X\), respectively. Essentially, the trace norm is the \(l_{1}\) norm of the SVs of the matrix \(X\). The signal analysis method based on the TSLRM has been widely applied in graph denoising. However, TSLRM is insufficient in the bearing fault diagnosis field, since the bearing fault information is usually submerged by external interference. In particular, TSLRM is susceptible to the interference  
Figure 1: (a) Periodic impulsive features; (b) a partially enlarged view of (a).  
Figure 2: Time-frequency diagram: (a) periodic impulsive features; (b) noisy signal.  
from the HCs generated by the parts such as gears or rotors. To overcome this difficulty and to improve the capability of fault feature extraction, a FISLRA is proposed. In the FISLRA, the corresponding sparse and low-rank model can be designed as follows:  
\[X^{*}=\arg\min_{X\in R^{\text{min}}}\left\{F(X):=\frac{1}{2}\|Y-X\|_{\text{F}}^{ 2}+\lambda_{0}\sum_{i=1}^{k}\sigma_{i}(X)+\lambda_{1}\sum_{m_{1}=1}^{M_{1}} \sum_{m_{2}=1}^{M_{2}}\phi(X_{m_{1},m_{2}};a)\right\} \tag{2}\]  
where \(\text{Xand}Y\in C^{M_{1}\times M_{2}}\) are the STFT coefficient matrix; \(k=\min\) (\(M_{1}\), \(M_{2}\)). Note that the FISLRA is a signal processing technique based on matrix estimation, so the raw signal needs to be converted into a matrix \(Y\) through the STFT operator \(A^{T}\). The transformation relation is indicated as \(Y=A^{T}\ y\). The parameter settings of STFT in this paper are as follows: window size R = 64 with 50% window overlapping, and the length of Fourier transform is L = 256. Moreover, the STFT coefficient matrix \(X^{*}\) is the solution of optimization problem (2). Finally, obtaining the estimated fault feature \(x\) through inverse STFT, that is \(x=AX\). Moreover, it is worth noting that the \(l_{0}\) norm can promote the sparsity of the solution to the greatest extent, but since the sparsity regularized least squares cost function based on the \(l_{0}\) norm is proved to be a Non-deterministic Polynomial (NP)-hard problem, it is difficult to obtain the optimal solution. Therefore, the \(l_{1}\) norm instead of the \(l_{0}\) norm is widely used as a regularizer, since it can induce sparsity effectively while the optimal solution can be obtained easily. However, \(l_{1}\) norm tends to underestimate non-zero amplitude of the signal, which causes the underestimation of the signal of interest. Consequently, to further induce sparsity and to get over the shortcomings of the \(l_{1}\) norm which tends to underestimate non-zero amplitude of the signal [37], various NPFs \(\varphi\) are utilized. The detailed instructions about NPFs can be found in [37]. In this section, the \(l_{1}\) norm and several typical NPFs that can easily ensure the convexity of the regularized least squares cost function and effectively promote the sparsity of the solution are plotted in Figure 3. In the NPFs shown in Figure 3, 'atan' is more similar to the \(l_{0}\) norm than the other NPFs, which means the 'atan' function induces the sparsity more strongly than others [23]. Furthermore, it can effectively maintain the amplitude of the signal of interest. Therefore, to induce the sparsity of STFT coefficient matrix, the 'atan' penalty function is utilized in this paper.