# Rolling Bearing Incipient Fault Detection Based on a Multi-Resolution Singular Value Decomposition
## 4 Simulation Studies
To be specific, the two singular values obtained by the decomposition of the harmonic signals are quite different from each other, the two singular values obtained by the decomposition of noise signals are nearly equal, and the singular values obtained by the decomposition of periodic shock waves are different. However, the difference between them is neither too large nor close to zero. The simulated signal shown in Figure 3 was decomposed using the MRSVD method, and the number of decomposed layers was set to 15. The singular values and the kurtosis coefficients corresponding to the obtained detail signals of each layer are shown in Figure 4a,b. It can be seen from Figure 4a that the singular values of the first several detail signals decreased rapidly because the noise was removed relatively quickly at the beginning. Then the singular values of the detail signals decreased gradually with the increase of the number of layers. To be specific, the noise removal became slow with the increase of the number of decomposed layers. As shown in Figure 4b, because the first detail signal contained a large amount of noise, its kurtosis coefficient was nearly equal to 3. As the number of decomposed layers increased, the kurtosis value also increased, because the noise component contained in the detail signal decreased layer by layer and the periodic impact component increased layer by layer. When the number of decomposed layers was 5, the kurtosis value of the detail signal reached a maximum, and the kurtosis value decreased with the increase of the decomposed layer because the periodic shock component contained in the residual approximate signal decreased layer by layer. Therefore, the conclusions pertaining to the kurtosis value analysis of the detail signals in Section 3.1 are verified in Figure 4b. Figure 5 shows the time domain and frequency domain waveforms of the first detail signal obtained by the decomposition of the simulated signal via the MRSVD method. The main component of the signal in Figure 5 is the noise signal. Therefore, there is no highlighted frequency in the frequency spectrum of Figure 5. The time-domain and frequency-domain waveforms of the last layer of the approximate signal obtained by the decomposition of the simulated signal via the MRSVD method are plotted in Figure 6a,b. As stated in Section 3.1, most of the energy of the harmonic signal was \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(\sigma\)Hankerl Matrix & H & H\({}_{\text{s}}\) & H\({}_{\text{n}}\) & H\({}_{\text{h}}\) \\ \hline \(\sigma_{a}\) & 354.8617 & 44.5982 & 69.4838 & 346.2494 \\ \hline \(\sigma_{d}\) & 73.3264 & 14.2705 & 66.8183 & 17.5887 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Result singular values of the singular value decomposition SVD of matrix H, H\({}_{\text{s}}\), H\({}_{\text{n}}\), and H\({}_{\text{h}}\). Figure 4: The singular values and kurtosis coefficients corresponding to the obtained detail signals of each layer. (**a**) The singular values corresponding to the obtained detail signals of each layer. (**b**) The kurtosis coefficients corresponding to the obtained detail signals of each layer. retained in the last layer of the approximation signal, so the highlighted frequencies in Figure 6b are the harmonic frequencies \(f_{1}\),\(f_{2}\) and \(f_{3}\). The five signals with maximum kurtosis were selected (see the red box in Figure 4b), and these five signals were added together in order to form a synthetic signal. Figure 7 shows the time-domain waveform of the synthetic signal and its Hilbert envelope demodulation spectrum. Although some noise was also contained in the synthetic signal, the fault characteristic frequency and its multiples are highlighted in the Hilbert envelope demodulation spectrum shown in Figure 7b. In the above simulation analysis, the number of decomposed layers of the MRSVD method was set to 15 (Figure 4b). The corresponding sequence number of the detail signal with the maximum kurtosis value is 5.