## 4 Simulation Analysis
### The Performance Analysis of the Proposed Method  
In order to analyze the decomposition performance of SVD method, we studied the relationship between the singular values of the SVD method and different kinds of signals first. Firstly, the Hankel matrix \(\mathbf{A}_{m}\) is constructed when applying the SVD denoise method, and there is a high degree of correlation between the adjacent rows [28]. As a result, the matrix \(\mathbf{A}_{m}\) must be a morphid matrix, and its rank \(r\) is far less than \(l\). Therefore, for an SVD denoising approach, the useful information should be determined by some singular components. In order to separate the effective periodic impulse component, the interference component, and the noise component of the fault signal, it is important to verify whether there were dislocations filtering the characteristics of the singular value distributions of the different signals. Hence, the periodic impulse fault signal, the interference signal, and the white noise signal are set as reference signals to build three groups of mixed signals. Mixed signal 1 is comprised of the interference signal and the noise signal, and mixed signal 2 is composed of the periodic impulse fault signal and the noise signal, while mixed signal 3 is comprised of the periodic impulse fault signal, the interference signal, and the noise signal. The three reference signals and the three mixed signals are analyzed based on the SVD denoise method. The delay step is set as \(m=20\). The singular values of these six signals are obtained, as shown in Figures 6 and 7. Interestingly, it can be found that the singular values of mixed signals are added by the reference signals, as shown in Figures 6 and 7. For example, the singular values of mixed signal 1 are added by the interference signal and the white noise signal. This showed that different singular values correspond to different kinds of signals. Therefore, the sensitive impulse feature signal can be reconstructed with the appropriate singular components, selected based on certain principles of the SVD method, thereby realizing the fault features extraction of the rolling bearings. Figure 5: (**a**) The stimulated fault vibration signal; (**b**) The envelope spectrum of (**a**). Figure 6: The singular components of these six signals. In order to verify the performance of proposed ACFHNR-SVD-TEO method, the simulated fault vibration signal of the rolling bearing is analyzed with the feature extraction method. The delay step \(m\) is set as \(m=20\). For comparison purposes, the kurtosis value and the ACFHNR index value corresponding to the 20 singular components of SVD are computed, as shown in Figure 8. Figure 8 exhibited that the kurtosis and the ACFHNR values of the second and the third singular components are larger than the other values, which tells us that the second and the third singular components are the effective signal features. However, apparently, it can be found that the ACFHNR index is very sensitive to the fault components, as the ACFHNR indexes corresponding to the second and the third singular components are much larger than the others, while the kurtosis did not have such good performance. As a consequence, the ACFHNR index has much better resolution than the kurtosis for extracting fault features. From the analysis above, the decomposition result of the SVD method is shown in Figure 9. Ten singular components are plotted, as shown in Figure 9a. Moreover, the singular components and their Teager energy operator spectra are obtained. Figure 9b,c correspond to the second and the third singular components, respectively. One can easily see that the fault frequency and its harmonics are clearly detected in the second and the third singular components. Based on the above analysis, it can be seen the ACFHNR index is very sensitive to the fault information, which has much better performance than the kurtosis for extracting the weak periodic impulse fault feature.