## 3 The Proposed Multi-Scale Demodulation Method
### Results and Discussion
Figure 14d; the fault characteristic frequency and its harmonics were not as clear as the proposed method, and there were many noise components. Under the 300 rpm working condition, the optimal demodulation scale chosen by the fast Kurtogram method was at level 3.6 and the frequency ranged from 4266.5 to 5119.5 Hz, as shown in Figure 14b. The corresponding demodulation spectrum is shown in Figure 14b; it is also difficult to observe the fault characteristic frequency and its harmonics due to the noise components. Under the 500 rpm working condition, the optimal demodulation scale chosen by the fast Kurtogram method was at level 4 and the frequency ranged from 1920 to 2560 Hz, as shown in Figure 14c. The corresponding demodulation spectrum is shown in Figure 14d; it is observed that the fault characteristic frequency of 28.13 Hz and its harmonics could be found, although it contained more noise components compared to the time-frequency impulse envelope demodulation result in Figure 14f. The demodulation results based on the enhanced Kurtogram method are shown in Figure 15. The enhanced Kurtograms of the three datasets are shown in Figure 14a-c. The corresponding demodulation spectra of the three datasets are shown in Figure 15d-f. It can be found the optimal  
Figure 14: Fast Kurtograms under (**a**) 100 rpm, (**b**) 300 rpm and (**c**) 500 rpm working conditions and the corresponding demodulation spectra under (**d**) 100 rpm, (**e**) 300 rpm and (**f**) 500 rpm working conditions. Figure 13: Impulse envelope waveforms under (**a**) 100 rpm, (**b**) 300 rpm and (**c**) 500 rpm working conditions and the corresponding demodulation spectra under (**d**) 100 rpm, (**e**) 300 rpm and (**f**) 500 rpm working conditions. demodulation scales chosen by the enhanced Kurtogram method were all at wavelet packet node (4, 3) for the three working conditions; the frequency ranged from 1920 to 2560 Hz. Under the 100 rpm working condition, the fault characteristic frequency of 5.625 Hz and its harmonics could be found as shown in Figure 15d, but it contained more noise components compared to the time-frequency impulse envelope demodulation result in Figure 13d. Under the 300 rpm working condition, the fault characteristic frequency of 16.88 Hz and its harmonics could be found as shown in Figure 15e; it also contained more noise components compared to the time-frequency impulse envelope demodulation result in Figure 13e. Similarly, under the 500 rpm working condition, the fault characteristic frequency of 28.13 Hz and its harmonics could be found as shown in Figure 15f; it also contained more noise components than the time-frequency impulse envelope demodulation result in Figure 13f. In summary, the enhanced Kurtogram method yielded better results than the fast Kurtogram method but contained more noise components than the proposed method. The demodulation results based on the traditional envelope analysis method are shown in Figure 16; the demodulation sub-bands were chosen depending on experience. Under the 100 rpm working condition, the demodulation sub-band was chosen at around a 10\(\times\) meshing frequency to include the dominant frequency of the spectrum in Figure 11d--specifically, from 120 to 320 Hz--and it was observed that only the fault characteristic frequency of 5.625 Hz could be found, as shown in Figure 16a. Under the 300 rpm working condition, the demodulation sub-band was chosen around a 4\(\times\) meshing frequency to include the dominant frequency of the spectrum in Figure 11e--specifically, from 170 to 370 Hz--and it was observed that there were no fault-related components in the demodulation spectrum, as shown in Figure 16b. Under the 500 rpm working condition, the demodulation sub-band was also chosen at around a 4\(\times\) meshing frequency to include the dominant frequency of the spectrum in Figure 11f--specifically, from 350 to 550 Hz--and it was observed that there were also no fault-related components in the demodulation spectrum, as shown in Figure 16c.