## 4 Materials and Methods
### Signal Decomposition Based on EEMD  
EMD decomposes the signal from high frequency to low frequency into a finite number of Intrinsic Mode Functions (IMFs) and the sum of the remaining terms. For the signal \(x(t)\), the implementation process of the EMD method is as follows:  
_Step 1_ : The acquisition of all local maxima and minima of the signal.  
_Step 2_ : Interpolation operations are performed on all maximum and minimum points, respectively, based on cubic splines.  
_Step 3_ : Fitting the upper and lower envelope and its average \(m_{1}\) and then calculaion of \(h_{1}=x(t)-m_{1}\).  
_Step 4_ : If \(h_{1}\) is an IMF, \(h_{1}\) is recorded as the first IMF component. If \(h_{1}\) is not an IMF, it is taken as the original data and repeated the above three steps \(k\) times, and finally the following is obtained:  
\[h_{1k}=h_{1(k-1)}-m_{1k}. \tag{5}\]  
Formula (5) meets the conditions of the IMF and can be regarded as the first-order IMF, denoted as \(c_{1}\). Among them, \(m_{1k}\) refers to the average value of the upper and lower envelope lines of \(h_{1(k-1)}\).  
_Step 5_ : Subtraction of the component \(c_{1}\) from the signal \(x(t)\) to obtain the residual signal \(r_{1}=x(t)-c_{1}\). Taking \(r_{1}\) as the original data, Steps 1, 2, 3 are repeated to extract the second component \(c_{2}\) of \(x(t)\). By analogy, when \(r_{n}\) is a monotonic function and the IMF can no longer be extracted, the loop ends, and the \(n\) IMF components are finally obtained. Therefore, the signal \(x(t)\) can be expressed as  
\[x(t)=\sum_{i=1}^{n}c_{i}+r_{n} \tag{6}\]  
where \(r_{n}\) is the residual function used to characterize the average trend of the signal.  
However, in the presence of abnormal signals such as intermittent signals, pulse interference, and noise, the identification and distribution of extreme points are compromised. This leads to the obtained envelope line becoming a superposition of the local envelope line of the abnormal signal and the envelope line of the true signal. Moreover, the estimation error in the envelope line propagates continuously throughout the decomposition process. Consequently, the components of the decomposed IMFs not only capture the inherent patterns of the true signal but also incorporate the characteristics of the abnormal signal and its adjacent feature time scales, thereby causing modal aliasing. [PERSON] and [PERSON] proposed the EEMD method in 2009 [27]. Based on the characteristic of uniform spectral distribution in white noise, the EEMD method introduces white noise into the signal. This addition ensures the continuity of the signal across different time scales. Subsequently, multiple Empirical Mode Decompositions are performed to effectively address the issue of mode mixing.  
Nevertheless, in practical applications, EEMD requires the setting of the amplitude coefficient k to add white noise and the overall average times M. If these parameters are improperly configured, it may result in an increased decomposition error for EEMD. Specifically, the amplitude coefficient k for adding white noise is inversely proportional to the accuracy of the EEMD decomposition result. However, if the amplitude of the added white noise is too small, the original signal may not be sufficiently enhanced, thereby failing to adequately address the issue of pattern confusion. Conversely, if the amplitude of theadded white noise is excessively large, ensuring that the decomposition error satisfies the required accuracy necessitates a sufficiently high number of overall averaging iterations M. Nevertheless, as M increases, the computational cost rises linearly. To strike an optimal balance between decomposition accuracy and computational efficiency, it is essential to determine the appropriate white noise amplitude coefficient k and the number of overall averaging iterations M.