## 3 Results
### Numerical Simulation
The standard is defined as the ratio of periodic modulation components to the generalized interference (PMGI). The PMGI standard can be expressed as follows:  
\[PMGI=\frac{R_{a}^{\prime}(T)-R_{a}^{\prime}(\alpha)}{R_{a}^{\prime}(T)}=1- \frac{R_{a}^{\prime}(\alpha)}{R_{a}^{\prime}(T)}=1-\frac{E_{g}}{E_{p}+E_{g}} -\frac{2E_{pg}}{E_{p}+E_{g}}, \tag{40}\]  
\[ \begin{array}{l}R_{a}^{\prime}(\alpha)=\underset{0<\alpha<T}{\mathrm{argmax}} \left[\int(p(t)+g(t)+n^{\prime}(t))\left(\begin{array}{c}p(t+\alpha)+\\ g(t+\alpha)+n^{\prime}(t+\alpha)\end{array}\right)dt\right]\\ =0+\int p(t)g(t+\alpha)dt+\int g(t)p(t+\alpha)dt+0+\int g(t)g(t)dt+\\ 0+0+0+0=E_{g}+2E_{pg}\end{array} . \tag{41}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Decomposition Method** & **IO** & **RMSE** & **Time (s)** \\ \hline ICEITDAN & 0.0216 & \(7.1305\times 10^{-17}\) & 24.59 \\ CEEMDAN & 0.1241 & \(8.8015\times 10^{-16}\) & 171.54 \\ ITD & 0.0262 & 0.0186 & 0.14 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison indicators for the three methods. According to the proposed PMGI guidelines, only the growth of \(E_{p}\) can induce the growth of PMGI. The higher the energy of the periodic modulation component is, the closer the PMGI is to 1. Therefore, in this paper, we select the mode components for which the PMGI value exceeds the average value for reconstruction and perform envelope spectrum transformation to evaluate the performance of the method that is proposed in this paper. The PMGI values of the three methods are presented in Table 2. The PMGI values of the first two modes exceed the average values in the CEEMDAN decomposition results, whereas only one mode component exceeds the average values in ITD and ICEITDAN. The envelope analysis method mainly forms the envelope spectrum by adopting the form of envelope detection of the signal. Its main strategy is to use the envelope spectrum peak of the signal to judge the type of fault. This method is also referred to as an envelope solution. Its main advantage is that it can demodulate the fault vibration signal in high-frequency wells and provide effective interference cancellation for subsequent studies [45]. Therefore, we must reconstruct the first two mode components for CEEMDAN, select the first mode component for ITD and ICEITDAN, and obtain envelope spectra for the three signals after reconstruction. The results are presented in Figure 5. Only six times can the frequency fault feature frequency be clearly identified using the CEEMDAN and ITD methods. However, the method that is proposed in this paper identifies more than ten fault frequency peaks. Moreover, the amplitude of the envelope spectrum that is obtained via the CEEMDAN method is much smaller than that obtained by the method that is proposed in this paper; thus, the method that is proposed in this paper outperforms the others in fault feature extraction. Figure 4: Comparison of the reconstruction errors of three methods: CEEMDAN, ICEITDAN, and ITD. (**a**) CEEMDAN, (**b**) ICEITDAN, and (**c**) ITD. According to the comparison that is presented above, the ICEITDAN decomposition method that is proposed in this paper outperforms CEEMDAN and ITD. Therefore, in the next step, we will focus on the demodulation and analysis of ICEITDAN-CQMA for signals. Based on the principles and results of the reconstruction of the sensitive mode components that are selected above, only the mode component GPR1 in this analog signal exceeds