## 3 Simulation and Experimental Verification  
### Case 1: A Numerical Experiment with Multiple Interferences  
In order to demonstrate the noise immunity and anti-periodic interference of the Mkurtogram, a simulated out race bearing fault signal with the is formed with Gaussian white noise, abnormal impulsive noise and gear meshing harmonic interferences. In this case, the above mentioned multiple interferences are added to the simulated signal referring to [33; 34]. Then, the simulated bearing signal can be modeled as:  
\[x(t)=s(t)+\delta(t)+m(t)+n(t) \tag{13}\]  
The first part \(s(t)\) is a sequence of repetitive impulses, which is generated as follows:  
\[s(t)=A(t){\sum_{n}}\mathrm{h}(t-n/f_{\mathrm{o}}-\tau) \tag{14}\]  
where \(A(t)\) signifies the amplitudes of the fault-induced repetitive impulses, and it is randomly selected from the uniform distribution U (1.2,1.5) in this section. Additionally, \(n\) is the impulse number,\(f_{\mathrm{o}}\) = 54 Hz is the FCF and \(\tau\) represents the random fluctuation generated by the rolling elements, which generally enjoys a percentage of 2-3% of \(T_{\mathrm{d}}\). \(T_{\mathrm{d}}\) is the time lag between two successive impulses. The impulse expression function \(\mathrm{h}(t)\) can be modeled as:  
\[\mathrm{h}(t)=e^{-\alpha t}\sin(2\pi ft) \tag{15}\]  
where \(\alpha\) = 860 represents the damping ratio, and \(f\) = 3200 Hz signifies the structure resonance frequency. The second part \(\delta(t)\) simulates the impulse interference, which shares the same form with Equation (15). The parameters are different from Equation (15), which match a large amplitude of 3.6, a damping ratio of 480, and the resonant frequency of 7200 Hz. The disturbance impulse is set at the time of 0.17 s. The third part \(m(t)\) is a harmonic interference generated by gear meshing, which can be modeled as:  
\[m(t)=\sum_{i=1}^{11}[(1.3-0.01i)\cos(i2\pi\times 12f_{\mathrm{r}}t)]+0.1\times \cos(2\pi f_{\mathrm{r}}) \tag{16}\]  
where the rotating frequency\(f_{\mathrm{r}}\) is set as 7 Hz and the meshing frequency \(f_{m}\) = 84 Hz can be calculated. The last part \(n(t)\) is the Gaussian white noise with the SNR of \(-12\) dB. The sampling frequency \(F_{\mathrm{s}}\) = 25.6 kHz with a sampling time length of 1 s. The time domain waveform of the first three parts and synthesized signal \(x(t)\) are depicted in Figure 3.  
Before applying the Mkurtogram to detect the optimal IFB, the input period \(T\) should be determined at first. In this case, the outer race fault period \(T_{\text{o}}=F_{\text{s}}/f_{\text{o}}=474\) can be calculated. Then, the Mkurtogram is applied to the simulated data and its corresponding results are displayed in Figure 4. From the Mkurtogram plotted in Figure 4a, the optimal IFB is found at level 5, which represents the decomposition scale of MODWPT. Meanwhile, the optimal center frequency \(f_{\text{c}}=3000\) Hz with a bandwidth \(B_{\text{w}}=400\) Hz is obtained, which is adjacent to the set resonant frequency of 3200 Hz. The optimal IFB is then taken to filter the simulated signal. From the SES shown in Figure 4c, the outer race \(\text{FCF}f_{\text{o}}\) and its multiple harmonics can be found obviously. Thus, the result indicates that the proposed Mkurtogram is valid to identify the bearing fault under the multiple interferences. As an additional detection in this case, the Mkurtogram is also applied to detect the gear meshing component based on the gear meshing period \(T_{m}=F_{\text{s}}/f_{\text{d}}=304.7\). The results are presented in Figure 5. From Figure 5a, the Mkurtogram finds a low-frequency band with the optimal center frequency \(f_{\text{c}}=800\) Hz and bandwidth \(B_{\text{w}}=1600\) Hz. From the filtered signal plotted in Figure 5b, the gear meshing repetitive transients are clearly revealed and the meshing frequency \(f_{m}\) along with its multiple harmonics are also dominated in the SES as displayed in the Figure 5c. Figure 4: Case 1: The results of Mkurtogram based on bearing fault period \(T_{\text{o}}\).