## 3 Results
### Numerical Simulation  
To evaluate the performance of the proposed method, the ITD, complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN) [41], and proposed ICEITDAN method are compared using synthetic signals. CEEMDAN can reduce the mode mixing of EMD using white noise and has been widely used in rolling bearing fault diagnosis [41]. When local damage occurs on the surface of the bearing element, the periodic shock vibration signal caused by the local damage of the bearing can be expressed as a convolution form between a cyclic shock signal and an impulse response signal. InFormula (38), \(A_{i}\) is the cyclic impulse signal and \(h(t)\) is the impulse response signal, because the actual project is also accompanied by noise interference, so \(n(t)\) is Gaussian white noise. The analog signal is expressed in Formula (38):  
\[\left\{ \begin{array}{c}x(t)=s(t)+n(t)=\sum\limits_{i}A_{i}h(t-iT-\tau_{i})+n(t )\\ A_{i}=A_{0}\cos(2\pi f_{i}t+\varphi_{A})+C_{A}\\ h(t)=e^{-Ct}\sin(2\pi f_{n}t+\varphi_{m})\end{array} \right., \tag{38}\]  
where \(s(t)\) is the periodic fault shock, \(f_{i}\) is the frequency conversion, \(f_{s}=3380\) Hz, \(C\) is the attenuation coefficient (\(C=840\)), \(C_{A}\) is 1, \(f_{n}\) is the resonance frequency (\(f_{n}=100\) Hz), and \(f_{i}\) is the characteristic frequency of the inner ring fault (\(f_{i}=1/T=230\) Hz). \(\tau_{i}\) is the slight sliding of the \(i-th\) shock with respect to period \(T\), after which the sliding follows a normal distribution with a mean of 0 (the standard deviation is 0.5% of the frequency conversion). \(n(t)\) is Gaussian white noise with an SNR of \(-8\) dB and \(f_{s}\) is the sampling frequency (\(f_{s}=8192\) Hz). The number of analysis data points is 4096. A synthetic signal is used to evaluate the fault feature extraction performance of the proposed ICEITDAN-CQMA method because the analog signal in Formula (38) can reflect the inherent fault characteristics of rolling bearings and clearly and intuitively reflect the performance of the method that is proposed in this paper. Figure 2 shows a time domain diagram of the analog signal that is used in this study. Figure 3 presents the decomposition results of three types of signal decomposition methods. CEEMDAN decomposes signals into 13 modal components, which is far more than the ITD and ICEITDAN methods that are proposed in this paper. The modal-aliasing problem occurs on IMF 5 and IMF 6, while ICEITDAN alleviates this problem effectively. To evaluate the performance of the proposed method, the orthogonality index is used to compare the three methods. If one of two or more event species changes, other aspects will not be affected. Therefore, the orthogonality index can be selected to judge the independence between the decomposition components and determine whether the high- and low-frequency signals are accurately decomposed in the decomposed signal. Reference [42] also shows that the smaller the orthogonality index is, the higher the decomposition accuracy is, and the better the effect of endpoint effect suppression is via many experiments. Therefore, it is feasible to select an orthogonality index to evaluate the effect of endpoint effect suppression. In the comparison process, three performance indicators are considered--namely, the orthogonality indicator, the root mean square error (RMSE), and the run time--to quantitatively evaluate the decomposition performances of the methods [43; 44; 45; 46]. Theoretically, the mode components that are decomposed by ICEITDAN, ITD,  
Figure 2: Time domain diagram of the simulation signal. and CEEMDAN should be completely orthogonal--namely, the orthogonality index should be equal to zero. However, due to errors and environmental interference, the orthogonality index cannot be zero [42]. Therefore, the smaller the orthogonality index in the analysis is, the more accurate the decomposition is.