## 5 Application of the Proposed Method
### Simulation Analysis
#### 5.1.2 Compound Faults Feature Extraction for Simulation Signal  
Firstly, the sparse decomposition of the parallel dual-Q-factor bases is applied to extract and sparsely represent the compound fault components according to the proposed method. The decomposition parameters are shown in Table 3, and the analysis results are shown in Figure 8. Figure 8a,b are the low resonance components which is obtained by using the sparse decomposition method and the Hilbert envelope spectra of decomposed components respectively. It can be seen from Figure 8b that peaks are obvious at \(f_{r}\) (24.38 Hz), \(f_{o}\) (61.88 Hz), \(f_{i}\) (300 Hz) and their harmonics. Hence, the results indicate that the method can successfully identify and extract features of the simulated signal composed of impact components of the inner race and outer race simultaneously. However, the two fault impact signals are overlapped with each other, and the two fault characteristic frequencies and their harmonics are interlaced with each other. Therefore, it is difficult to accurately extract the two-fault characteristic information from the envelope spectrum. To further separate the inner and outer race fault features from the composite fault signals, according to the different fault periods of the inner and outer race of the rolling bearing, the adaptable deconvolution techniques was applied to extract the periodic features from the low resonance component. The deconvolution period \(T_{*}\) (* is \(i\), \(o\)) of the signal is shown in Table 4.  
Figure 8: The analysis results of case 1 with the parallel dual-Q-factor bases sparse decomposition: (**a**) Low resonance component; (**b**) Hilbert envelope spectrum. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Attribute** & \(\mathbf{Q_{1}}\), \(\mathbf{Q_{2}}\) & **Redundancy (\(r_{1}\), \(r_{2}\))** & **Number of Layers (\(J_{1}\), \(J_{2}\))** \\ \hline Low resonance component & 1 & 4 & 27 \\ High resonance component & 6 & 4 & 12 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The sparse decomposition parameters of case 1. Figure 7: The time-domain waveform of the compound fault simulation signal: (**a**) Outer race and inner race fault; (**b**) compound signal with gauss noise. Then, we selected the optimal parameter (\(M\), \(L\)) as mentioned in Section 3.2.2  
(1) When T\({}_{o}\) = 248, Figure 9a,b show the relationship between \(M\) and MTEK, and the relationship between the filter length \(L\) and the TEK by iterative calculation respectively. It was found that when \(M\) = 3, the MTEK value was the largest from Figure 9a, and when \(L\) = 20, the TEK value was the largest from Figure 9b, so the optimal parameters \(T_{o}\), \(M_{o}\), and \(L_{o}\) were determined [248; 3; 20]. (2) When T\({}_{i}\) = 51, Figure 9c,d show the relationship between \(M\) and MTEK, and the relationship between the filter length \(L\) and the TEK by iterative calculation respectively. It was found that when \(M\) = 5, the MTEK value was the largest from Figure 9c, and when \(L\) = 55, the TEK value was the largest from Figure 9d, so the optimal parameters \(T_{i}\), \(M_{i}\), and \(L_{i}\) were determined [51; 5; 55]. The low resonance component was deconvoluted and filtered with the obtained optimal parameters [\(T\)-, \(M\)-, \(L\)-] (* is \(i\), \(o\)) using MCKD. Finally, the analysis result is presented in Figure 10. It is noticeable that the impulse components in the time-domain of the outer race fault and the inner race fault can be found clearly in Figure 10a,c, respectively. From Figure 10b,d, the harmonic components of the characteristic frequencies of 61.88 Hz and 300 Hz are revealed noticeably. It is consistent with the theoretical fault characteristic frequency (\(f_{o}\) (62 Hz),\(f_{i}\) (300 Hz)). The results indicate that the proposed method can successfully separate the features of the simulated signal composed of the outer race fault and the inner race fault.