# Incipient Fault Feature Enhancement of Rolling Bearings Based on CEEMDAN and MCKD
## 2 Theoretical Basis
### Maximum Correlated Kurtosis Deconvolution  
The MCKD method relies on selecting a FIR filter and using correlation kurtosis as the evaluation measure to maximize the kurtosis value of the filtered signal and highlight the periodic shocks obscured by noise. The signal \(y_{n}\) correlation kurtosis is expressed as follows:  
\[CK_{M}(T)=\frac{\sum\limits_{n=1}^{N}{(\prod\limits_{m=0}^{M}{y_{n-mT}})^{2} \over(\sum\limits_{n=1}^{N}{y_{n}^{2}})^{M+1}}}{M+1\over} \tag{7}\]  
where \(M\) is the number of shifts, \(m\in[0,M]\), \(T\) is the period of the shock signal, \(N\) is the number of samples of the input signal. If the effect of noise on the acquired vibration signalis ignored, MCKD is essentially the recovery of the actual signal \(x_{n}\) to \(y_{n}\) through a filter, namely:  
\[y=f*x=\sum\limits_{k=1}^{L}f_{k}x_{n-k+1} \tag{8}\]  
where \(L\) is the length of the filter, \(x_{n}\) is the original signal, \(T\) is the transpose.  
The objective function of MCKD is as follows:  
\[\max_{f}CK_{M}(T)=\max_{f}\frac{\sum\limits_{n=1}^{N}{(\prod\limits_{m=0}^{M} y_{n-mT})}^{2}}{{(\sum\limits_{n=1}^{N}y_{n}^{2})}^{M+1}} \tag{9}\]  
To find the optimal filter to maximize the correlation kurtosis value, the solution equation is as follows:  
\[\frac{d}{df_{k}}CK_{M}(T)=0,k=1,2,\cdots L \tag{10}\]  
The filtering coefficients \(f\) are obtained as:  
\[f=\frac{\parallel y^{2}\parallel}{2\parallel\beta\parallel^{2}}{\left(X_{0} X_{0}^{T}\right)}^{-1}\sum\limits_{m=0}^{M}{\left(X_{mT}\alpha_{m}\right)} \tag{11}\]  
\[\beta=\left[\begin{array}{cccc}y_{1}y_{1-T}&\cdots&y_{1-MT}\\ y_{2}y_{2-T}&\cdots&y_{2-MT}\\ &\vdots&\\ y_{N}y_{N-T}&\cdots&y_{N-MT}\end{array}\right] \tag{12}\]  
\[\alpha_{m}=\left[\begin{array}{c}y_{1-mT}^{-1}{\left(y_{1}^{2}y_{1-T}^{2} \cdots y_{1-MT}^{2}\right)}\\ y_{2-mT}^{-1}{\left(y_{2}^{2}y_{2-T}^{2}\cdots y_{2-MT}^{2}\right)}\\ &\vdots\\ y_{N-mT}^{-1}{\left(y_{N}^{2}y_{N-T}^{2}\cdots y_{N-MT}^{2}\right)}\end{array}\right] \tag{13}\]  
\[X_{r}=\left[\begin{array}{cccc}x_{1-r}&x_{2-r}&\cdots&x_{N-r}\\ 0&x_{1-r}&\cdots&x_{N-1-r}\\ \vdots&\vdots&\vdots&\vdots\\ 0&0&\cdots&x_{N-L-r+1}\end{array}\right]_{L\times N};r=\left[0,T,\cdots,mT\right] \tag{14}\]  
With the above description, the iterative solving process of the filter system \(f\) is as follows:  
1. Initialization parameters deconvolution period \(T\), filter length \(L\) and shift number \(M\), error convergence accuracy \(\varepsilon\), the maximum number of iterations \(M_{1}\), and filter coefficients \(f=\left[0,0,\cdots,1,-1\cdots,0,0\right]^{T}\);
2. Calculate \(X_{T}\), \(X_{0}\), and \(\left(X_{0}X_{0}^{T}\right)^{-1}\) of the signal \(x_{n}\);
3. Calculate the filtered signal \(y\);
4. Calculate \(\alpha_{m}\) and \(\beta\) from \(y\);
5. According to Equation (11), calculate the new filter coefficients \(f\);
6. Compare the correlation kurtosis difference \({}_{\triangle}CK_{M}(T)\) before and after signal filtering. If \({}_{\triangle}CK_{M}(T)\) is less than threshold \(\varepsilon\), the iteration stops; otherwise, return to step (3) to continue the cycle.