## 3 Proposed IMCKD  
### Mckd  
The MCKD technique is based on selecting a Finite Impulse Response filter (FIR filter) to maximize the correlated kurtosis (CK), which takes advantage of the periodicity of the faults and is used to highlight the continuous pulse covered by the strong noise in the signal. Consider a discrete signal \(x\)(n) that is the response of the bearing excited by the fault impulses signal \(y\)(n). MCKD searches for a FIR filter \(f(l)\) (\(l\) is a filter length) to maximize the correlation kurtosis of the signal \(y(n)\) and the general expression for the inverse is given by [43]:  
\[y(n)=\sum_{l=1}^{L}f(l)x(n-l+1). \tag{5}\]Correlation kurtosis is defined as an indicator in the MCKD algorithm to evaluate the impact characteristics of the resulting signal \(y(n)\) (the fault impulses signal), which can reflect the proportion of periodic influence in the signal. The correlation kurtosis is defined as [28]:  
\[\begin{split}\text{CK}_{M}(T_{*})=\frac{\sum\limits_{n=1}^{N}{( \prod\limits_{m=0}^{M}{y_{n-mT_{*}}})}^{2}}{(\sum\limits_{n=1}^{N}{y_{n}}^{2}) }\\ \end{split} \tag{6}\]  
where \(M\) is the order of shift, \(L\) and \(N\) denote the length of filter and the sampling number of the signal respectively, \(f\) is the filter vector \(f=\left[f_{1},f_{2}\cdots,f_{L}\right]^{T}\), \(T_{*}\) is the deconvolution period and can be calculate by the following equation:  
\[T_{*}=f_{s}/f_{*} \tag{7}\]  
where \(f_{s}\) and \(f_{*}\) denote sampling frequency and fault characteristic frequency respectively.  
The optimization function is equivalent to solving the equation:  
\[\frac{d}{df_{k}}CK_{M}(T)=0(k=1,2,\cdots,L). \tag{8}\]  
The final iterative expression of the filter coefficients is represented:  
\[f=\frac{||y||^{2}}{2||g||^{2}}(\text{X}_{0}\text{X}^{T}{}_{0})^{-1}\sum\limits _{m=0}^{M}{X_{mT}\alpha_{m}} \tag{9}\]  
where \(\text{X}_{0}\text{X}^{T}{}_{0}\) is the Toeplitz autocorrelation matrix of the original signal \(x\) and the inverse \(\left(\text{X}_{0}\text{X}^{T}{}_{0}\right)^{-1}\) is assumed to exist.  
\[X_{r}=\left[\begin{array}{cccc}x_{1-r}&x_{2-r}&x_{3-r}&\cdots&x_{N-r}\\ 0&x_{2-r}&x_{3-r}&\cdots&x_{N-1-r}\\ 0&0&x_{3-r}&\cdots&x_{N-2-r}\\ \vdots&\vdots&\vdots&&\vdots\\ 0&0&0&\cdots&x_{N-L-r+1}\end{array}\right]r=\left[0\;T\;2T\cdots mT\right] \tag{10}\]  
where  
\[\alpha_{m}=\left[\begin{array}{cccc}y^{-1}{}_{1-mT}\;(y^{2}{}_{1}y^{2}{}_{1 -T}\cdots{}y^{2}{}_{1-MT})\\ y^{-1}{}_{2-mT}\;(y^{2}{}_{2}y^{2}{}_{2-T}\cdots{}y^{2}{}_{2-MT})\\ \vdots\\ y^{-1}{}_{N-mT}(y^{2}{}_{N}y^{2}{}_{N-T}\cdots{}y^{2}{}_{N-MT})\end{array} \right]\beta=\left[\begin{array}{cccc}y_{1}y_{1}{}_{1-T}&\cdots&y_{1-MT}\\ y_{2}y_{2-T}&\cdots&y_{2-MT}\\ \vdots\\ y_{N}y_{N-T}\cdots{}y_{N-MT}\end{array}\right]. \tag{11}\]  
The performance of MCKD must be guaranteed by the accurate and appropriate parameters, the input parameters, including the length of filter \(L\), the order of shift \(M\), the maximum count of iteration \(N\) and the deconvolution period \(T_{*}\).  
(1) Generally, the maximum count of iteration \(N\) is empirically set. The larger count of iteration may bring a better result when the algorithm is executed to convergence but needs more calculation time. Therefore, to ensure the algorithms converge in this paper, \(N=30\) is set as the maximum count of iteration in MCKD [37].  
(2) The deconvolution period \(T_{*}\) is obtained from the theoretical fault characteristic frequency mentioned above.  
(3) The optimal M and L are different when MCKD is applied in different signal. Obviously, the main problem in the practical application of MCKD is how to select the optimal parameters including the order of shift \(M\) and the filter length \(L\).