## 2 Methodology and Principles
### CEEMDAN Principles and Improvements
Interpolate the mean extreme points \((\tau_{k},X_{k}),k=1,2,\ldots,K\) using a cubic spline curve to obtain the mean fit curve \(m(t)\). 4. Separate the mean curve from the extended signal \(x^{n}(t)\) under different weights to obtain the residual signal \(r(t)\). \[\overline{\text{Im}f_{1}(t)}=\frac{1}{K}\underset{i=1}{\overset{K}{\sum}} \,IMF_{1}^{i}(t)\]0 (11) Introducing the weight coefficient here, the mean curve \(m(t)\) with different weights can be separated from the signal. 5. If \(r(t)\) does not satisfy the condition of the IMF of the intrinsic modal function, let \(x^{n}(t)=r(t)\) and repeat steps (2) to (4) until the first IMF with different weights is obtained, from which the result with the smallest orthogonality is selected as the optimal IMF component, and after removing the extension portion, which is denoted as \(IMF_{im}\), the mean value of its signal amplitude is calculated as follows: \[\overline{\text{Im}f_{1}(t)}=\frac{1}{K}\underset{i=1}{\overset{K}{\sum}} \,IMF_{1}^{i}(t)\]1 (12) Separating \(\overline{imf_{i}}\) from the original signal \(X(t)\) yields the residual \(r_{i+1}(t)\) of the signal. \[\overline{\text{Im}f_{1}(t)}=\frac{1}{K}\underset{i=1}{\overset{K}{\sum}} \,IMF_{1}^{i}(t)\]2 (13)
6. Based on the residuals, repeat the iterative decomposition of the signal by the signal processing methods in steps (1) to (5), and compute the IMF and residuals for each order iteratively in turn. The original signal \(X(t)\) is finally decomposed into a sum of IIMFs and a residual term. At this point, \(X(t)\) can be expressed as: \[\overline{\text{Im}f_{1}(t)}=\frac{1}{K}\underset{i=1}{\overset{K}{\sum}} \,IMF_{1}^{i}(t)\]3 (14)  
In step (1), the added white noise signal is improved, and in steps (4) and (5), the weighted mean curves with different weights are separated from the original signal by introducing the weighting coefficients \(\beta\); the orthogonality index of the decomposed result is calculated. The smaller the orthogonality index, the lower the confusion degree of each IMF in the decomposition result, which is conducive to the further extraction of fault features. Therefore, the decomposition result with the smallest orthogonality index is preferred from the decomposition results with different weighting coefficients \(\beta\) as the final IMF of the improved CEEMDAN algorithm. When the weighting coefficient \(\beta\) is too large, the mean curve is separated by amplification, resulting in the useful information contained in the original signal \(X(t)\) also being eliminated, and the IMF obtained after decomposition is prone to distortion. When \(\beta\) is too small, it is difficult to determine the overall optimality of the resulting IMF. Therefore, the weighting factor \(\beta\) is made to take values between 0 and 2 by 0.1 steps. In the range of 0 to 2, the maximum value of the weighting factor \(\beta\) is 2, which is not prone to excessive separation. Weight values change continuously when taken at a certain step size. The overall optimality of the decomposition results is not compromised due to the lack of weight values. Of course, since the smaller the step size, the more the number of weights; theoretically, a better overall optimality can be obtained, but the increase in the amount of computation and the improvement of the overall optimality of the actual cost-effectiveness is not high, so the value is taken at a step size of 0.1.