# Weak Multiple Fault Detection Based on Weighted
## 4 Simulation Signal Analysis
The sparse optimization model established in TQWT can be written as:  
\[\underset{w_{1},w_{2}}{\arg\min}\left\|y-\Phi_{1}w_{1}-\Phi_{2}w_{2}\right\|+ \sum_{j=1}^{I_{1}+1}\lambda_{1,j}\left\|w_{1,j}\right\|_{1}+\sum_{j=1}^{I_{2}+1 }\lambda_{2,j}\left\|w_{2,j}\right\|_{1} \tag{26}\]  
\[x_{1}=\Phi_{1}w_{1}x_{2}=\Phi_{2}w_{2} \tag{27}\]  
where \(\Phi_{1}\) and \(\Phi_{2}\) denote the inverse TQWT having high and low Q-factors respectively, and \(\lambda_{1}\) and \(\lambda_{2}\) are the regularization parameters. The sparse coefficients \(w_{1}\) and \(w_{2}\) are obtained by solving the optimization problem. Then, through the signal reconstruction of Equation (25), the high-factors component and low-factors component of the signal can be obtained, which represent the harmonic component and the impact component, respectively. Therefore, TQWT and WOGS have similar ideas for extracting fault features. But based on different prior knowledge, the two algorithms build different models. In order to highlight the weak fault characteristics, we weighted the penalty terms in the WOGS model. The resonance characteristic of the signal is used to construct the wavelet basis in TQWT. However, we use the Morlet wavelet basis function in WOGS. Here, the parameters of the high oscillatory component are \(Q_{1}=7\)  
Figure 7: Results of the proposed WOGS algorithm: (**a**) Time domain waveform. (**b**) Frequency spectrum. (**c**) Envelope spectrum. \(r_{1}=5,J_{1}=10\), and the parameters of the low oscillatory component are \(Q_{2}=2,r_{2}=3.5,J_{2}=7\). Figure 8 illustrates the decomposition result of the TQWT, and Figure 8b shows that the impulse feature can be observed in the low oscillatory component. Furthermore, Figure 9a-c exhibits the envelope spectra of the high and low oscillatory and residual components. Figure 9b depicts some obvious peaks at 20 and 70 Hz, corresponding to the feature frequencies of the two different impulse components. The methods in [33; 34] use the OGS algorithm to extract multiple faults directly. Figure 10 displays the denoised result of the OGS algorithm with group size \(K=3\). Figure 10 shows that no evident periodic impulse features are available. The same signal was also processed by using soft threshold denoising. Figure 11 shows the denoised results of soft threshold. In addition to the impulse fault frequency, a large amount of noise also emerges. Compared with other methods that extract multiple fault information when rolling bearing signals are masked by strong noise, the WOGS algorithm can obtain good diagnostic results. The energy ratio at the characteristic frequency is used as an evaluation index to compare the fault feature extraction effects of the above algorithms. The energy ratio can be defined as:  
\[R_{f}=\frac{\sum_{t=1}^{4}S(t\times f_{i})+\sum_{j=1}^{4}S(j\times f_{o})}{S(f)} \tag{28}\]  
Figure 8: Decomposition results by TQWT: (**a**) High oscillatory component. (**b**) Low oscillatory component. (**c**) Residual component. where \(f_{i}\) and \(f_{o}\) correspond to the inner and outer fault characteristic frequencies, respectively. \(S(f)\) represents the energy of the entire envelope spectrum. The computer configuration is as follows: the processor--Intel Core i7-8550U; the CPU frequency--1.8 GHz; the memory--16 GB; the graphics card--NVIDIA GeForce MX150; operating system--Win10 (64 bit); the program--implemented on MATLAB R2015b. Table 1 shows the energy ratio \(R_{f}\) and CPU running time \(T_{f}\) of the above algorithms. The results show that WOGS algorithm has the largest energy ratio, which indicates that it has a stronger ability to extract weak multiple fault features. However, it is time-consuming to calculate the weight coefficients and Morlet wavelet transform in WOGS algorithm, so the proposed algorithm not only improves the extraction accuracy of the algorithm, but also sacrifices the algorithm efficiency.