# An Optimal Resonant Frequency Band Feature Extraction Method Based on Empirical Wavelet Transform
## 4 Simulations and Comparative Analysis  
A simulation experiment is made to compare the effectiveness of the proposed method with other methods. The simulation signal is composed of a periodic impulse component and strong background noise to simulate the vibration signal with impulse characteristics when defects occur in rotating machinery:  
\[x(t)=y_{0}e^{-2\pi f_{n}\zeta t}\times\sin(2\pi f_{n}\sqrt{1-\xi^{2}t}) \tag{17}\]  
where the displacement constant of the bearing is \(y_{0}\) = 2.5, the relative damping coefficient is \(\zeta=0.1\), the resonance frequency \(f_{n}\) = 900 Hz, \(t\) is the sampling moment, the impact fault period is \(T\) = 0.00625 s (the characteristic frequency is \(f_{c}\) = 160 Hz), the sampling frequency is \(f_{s}\) = 12 kHz and simulated data length is 4096. The periodic vibration pulse signal is generated as shown in Figure 2a. Meanwhile, adding SNR = \(-4\) dB white Gaussian noise to the original signal, the waveform of the mixed signal is illustrated in Figure 2b. It can be seen that due to the influence of noise interference, it is difficult to find pulses in the time domain. To test the effectiveness of the proposed method, the AEWT is applied to analyze the analog signal. The parameterless clustering scale-space histogram method is applied to the detected signal, and the boundaries of each mode are shown in Figure 3. The ASN of all modes, the correlation coefficient with the original signal, and the results of the AASN are provided in Table 1. In these modes, the highest value of the AASN appears in mode 3. Among them, the highest value of the ASN appears in mode 3, and the highest correlation coefficient appears in the mode 15, so the AASN is consistent with the results of the ASN. Figure 3: Analyzed result of the boundaries detected using the parameterless scale-space method. Figure 2: (a) Time-domain Waveform of Original signal; (b) Time-domain Waveform of SNR = \(-4\) dB mixing Signal. For simplicity, only modes 1-6 near the determination conditions are shown in Figure 4a, as they may best meet the requirements of the characteristic resonance band. The corresponding envelope spectrum is presented in Figure 4b. Due to the influence of a strong background noise, the parameterless scale-space segments the frequency band too narrowly. It can be seen that in these modes, fault-related modulation information is not identified. As shown in the dotted circle, the envelope spectrum of mode 3 shows no feature information. Therefore, we should consider merging the adjacent modes to construct the optimal resonance demodulation frequency band. According to the correlation coefficient merging strategy, it is determined whether the mode adjacent to the central mode satisfies the merge condition. As shown in Table 1, IMF1, IMF2, and IMF4 meet the requirements. The envelope spectrum of the combined mode is shown in Figure 5, which can clearly identify the modulation characteristics of the fault correlation frequency and multiple harmonic waves. Under the influence of a strong noise, due to the large number of interference components, there is a large deviation when using a correlation analysis as the basis for the merger. However, the ASN can reflect the real results more accurately, and the product can play the role of an adaptive correction. The simulation results show that the proposed fault center mode selection principle and the correlation coefficient combination principle are both reliable and effective. To verify the superiority of the proposed method, we also use the EMD method for the test signal. The EMD decomposition results are presented in Figure 6a. For simplicity, we only show the top five IMFs. It can be seen that the EMD is sensitive to noise, and no periodic pulse information can be seen in the decomposed IMF. The corresponding envelope spectrum is shown in Figure 6b. Although the fault frequency can be extracted by IMF1 and IMF2, it is not obvious and the relevant harmonic components cannot be extracted.