## 2 Methodology and Principles  
### CEEMDAN Principles and Improvements  
CEEMDAN is an improved signal adaptive decomposition method based on EEMD and CEEMD, which has evolved from the EMD algorithm. EMD decomposes the original signal into a series of IMFs and a residual component, where each IMF represents a single-component signal with different physical meanings in the original signal. The principle of CEEMDAN is similar to EEMD, where two opposite white noise signals are added to the original signal, and then EMD is used to decompose it. Both EEMD and CEEMD effectively improve the mode mixing phenomenon of EMD, but they require multiple iterations to eliminate the influence of added white noise and reduce reconstruction errors. Increasing the number of ensemble averages leads to increased computational complexity, affecting efficiency. To address this issue, CEEMDAN adaptively adds white noise at each stage of the EMD decomposition, and regardless of the number of ensemble averages, the reconstruction error of CEEMDAN is almost zero. Therefore, CEEMDAN not only improves the mode mixing phenomenon of EMD but also solves the computational complexity issue of EEMD and CEEMD. CEEMDAN is widely applied in the field of signal denoising [23; 24; 25], such as in mechanical fault diagnosis and predictive maintenance [26; 27; 28]. It improves the quality and accuracy of signals by decomposing non-stationary vibration signals into multiple IMFs and noise and then removing noise through adaptive noise estimation. The specific steps of the CEEMDAN algorithm are as follows:  
1. Construct a time-domain signal sequence \(X^{n}(t)\) with noise. Create a white noise signal sequence \(\omega^{n}(t)\) with a standard deviation of \(\varepsilon_{0}\). As shown below, the white noise signal is combined with the original signal \(X(t)\) to generate a new signal: \[X^{n}(t)=X(t)+\varepsilon_{0}\omega^{n}(t),n=1,2,\ldots,K\] (1)
2. Apply EMD to decompose the time-domain signal sequence \(X^{n}(t)\), extract the first-order IMF component, and compute the average value of its signal amplitude. \[\overline{\text{Im}f_{1}(t)}=\frac{1}{K}\underset{i=1}{\overset{K}{\sum}} \,IMF_{1}^{i}(t)\] (2) After the first-level IMF decomposition, the residual of the signal is: \[r_{1}(t)=X(t)-\overline{\text{Im}f_{1}}(t)\] (3)
3. Based on the residuals, the signal \(r_{1}(t)+\varepsilon_{1}EMD_{1}(\omega^{n}(t))\) is reconstructed by adding white noise according to the signal processing method described in step (1). Extract the second-order IMF component through EMD decomposition. The calculation method for the average value of this component's signal amplitude and the new residual is obtained with the following equation: \[\overline{\text{Im}f_{2}}(t)=\frac{1}{K}\underset{n=1}{\overset{K}{\sum}}\,EMD _{1}(r_{1}(t)+\varepsilon_{1}EMD_{1}(\omega^{n}(t)))\] (4) \[r_{2}(t)=r_{1}(t)-\overline{\text{Im}f_{2}}(t)\] (5)
4. Repeat the decomposition of the signal following the same rules, as shown in the equation below. Compute the residual \(r_{k}(t)\) of the \(k\)-th order IMF and the average value \(\overline{\text{Im}f_{k+1}}(t)\) of the \((k+1)\)th order IMF. Repeat this process until the residual \(r_{k}(t)\) satisfies the conditions stated in Equation (8) or the signal cannot be further decomposed using EMD. Equations (6) through (8) are shown below:  
\[r_{k}(t)=r_{k-1}(t)-\overline{\text{Im}f_{k}}(t) \tag{6}\]  
\[\overline{\text{Im}f}_{k+1}(t)=\frac{1}{K}\sum_{n=1}^{K}EMD_{1}(r_{k}(t)+ \varepsilon_{k}EMD_{k}(\omega^{n}(t))) \tag{7}\]  
\[\sum_{t=0}^{T}\frac{|r_{k-1}(t)-r_{k}(t)|^{2}}{r_{k-1}^{2}(t)}\leq SD_{k} \tag{8}\]  
According to Equations (1)-(8), it can be observed that the signal decomposition process of CEEMDAN follows the same path as EMD. To obtain the mean curve, a cubic spline interpolation is fitted to the extreme points of the original signal, and this mean curve is then used for continuous iterative sieving.