## 3 The Proposed Hybrid Demodulation Technique
### A Hybrid Demodulation Technique via MID and VMD  
Since the acquired vibration signals of a rotating machine are usually multi-components and complex modulation signals, VMD can be used as a pre-processing method to decompose a complex signal into a series of mono-harmonic components so as to further improve the demodulation performance of the MID. Thus, the developed hybrid demodulation technique has complementary merits in the demodulation of a complex signal, whose effectiveness will be first demonstrated using a simulated signal. A numerical signal \(x_{2}(t)\) is simulated with an impulsive signal \(s(t)\), a multi-harmonic signal \(u(t)\), and Gaussian white noise \(n(t)\), which is written by [29]  
\[x_{2}(t)=s(t+T)+u(t)+n(t) \tag{17}\]where the impulsive signal \(s(t)\) and the multi-harmonic signal \(u(t)\) are separately given by  
\[s(t)=e^{-1000t}\cos(2\pi\cdot 3000t) \tag{18}\]  
\[u(t)=5\sin(40\pi t)+1.5\sin(80\pi t)+0.5\sin(120\pi t). \tag{19}\]  
The sampling frequency and number of sampling points of the signal in Equation (17) are set to 40,960 Hz and 4096, respectively. The carrier frequency of the impulsive component is 3000 Hz in Equation (18). The damping period of \(s(t)\) is \(T=0.01\) s, so its modulation frequency is 100 Hz. Two noise variances of the simulated signal are considered in order to verify the noise robustness of the proposed technique. The original temporal waveform of the simulated signal and its spectrum are shown in Figure 5. It can be seen in Figure 5a that the simulated impulsive signal is almost buried in the multi-harmonic signal by the added noise. MID is then used with the meaningful BLIMF2 for the demodulation. The impulsive components usually represent transient signatures generated by a local defect in a rotating machine. Sub-band components of the simulated signal are obtained by the VMD and EMD techniques, which are found in Figure 6a,b, respectively. VMD can not only effectively detect the impulsive signatures (see BLIMF2 in in Figure 6a), but also successfully identify the multi-harmonic component (see BLIMF1 in Figure 6a). However, EMD can only detect the impulsive signal (IMF3 in Figure 6b). Figure 5: Simulated vibration signal (**a**) the time domain waveform (**b**) the FFT spectrum. Figure 6: _Cont._  
\(\text{MID}_{\Delta f}^{\text{PSCoh}}\) is then used to analyze the raw simulated signal, and the result is shown in Figure 7. We can find that the cyclic frequency and the carrier frequency cannot be obviously detected due to the added strong noise as well as the influences of the multi-harmonic components. Since the impulsive signal has been successfully identified with VMD, MID is then used to demodulate the meaningful BLIMF2. \(\text{MID}_{\Delta f}^{\text{SCoh}}\) of the BLIMF2 is shown in Figure 8a, where it can be found that the carrier frequency is 3000 Hz and the cycle frequency (modulation frequency) as well as its multipliers are about 100 Hz. The validation of the proposed hybrid technique is thus well-demonstrated. Moreover, MID is also used to analyze the IMF3 achieved by EMD. \(\text{MID}_{\Delta f}^{\text{SCoh}}\) of the IMF3 is illustrated in Figure 8b. It can be clearly seen in Figure 8a,b that the result of the BLIMF2 is much better than that of IMF3. Figure 6: The decomposed BLIMFs of the simulated vibration signal using VMD (**a**) and IMF3 using EMD (**b**). Figure 7: The \(\text{MID}_{\Delta f}^{\text{SCoh}}\) of the original simulated vibration signal (the noise variance is 0.3). A noise variance of 0.7 is set in the simulated signal in Figure 9. MID\({}_{\Delta f}^{\text{PSCoh}}\) is then used to analyze the simulated signal, and the result is found in Figure 7a. We can find that the cyclic frequency and the carrier frequency cannot be obviously detected due to the strong noise. Once more, MID is also used to demodulate the BLIMF2 component.