2.1 CEEMDAN
Suppose a signal is denoted as \(x\left(t\right)\), and\(b_{i}\left(t\right)\) is the white noise series with a mean value of
0 and a variance value of 1.\(\ D_{k}(.)\) is the k-th order modal
operator generated by EMD method.. To overcome the mode mixing caused by
the previous two modes, \(D_{k}(b_{i}\left(t\right))\) is used to
extract the k-th mode. \(E(.)\) is the operator that generates the local
mean of the signals to be decomposed. The procedures of CEEMDAN are
explained as follows:
(1) Add Gaussian white noise to the original series and the signal of
the i -th realization can be expressed
as:
\(x_{i}\left(t\right)=x\left(t\right)+\beta_{0}D_{1}(b_{i}\left(t\right))\)(1)
(2) Use the EMD method to calculate the mean value of the local mean of
the signal \(x_{i}\left(t\right)\) with the white noise. Then, the
first residual value is obtained as:
\(r_{1}(t)=\frac{1}{I}\sum_{i=1}^{I}{E(x\left(t\right)+\beta_{0}D_{1}\left(b_{i}\left(t\right)\right))}\)(2)
(3) With the first residual value, the first intrinsic mode function is
calculated as:
\(d_{1}\left(t\right)=x\left(t\right)-r_{1}(t)\) (3)
(4) Take the mean value of the local mean of\(r_{1}\left(t\right)+\beta_{1}D_{2}\left(b_{i}\left(t\right)\right)\)as the estimated value of the second residual value\(r_{2}\left(t\right)\), then the second intrinsic mode function is:
\(d_{2}\left(t\right)=r_{1}\left(t\right)-r_{2}\left(t\right)\)(4)
(5) For k=3, …, K, the k-th residual value is described as:
\(r_{k}(t)=\frac{1}{I}\sum_{i=1}^{I}{E(x\left(t\right)+\beta_{k-1}D_{k}\left(b_{i}\left(t\right)\right))}\)(5)
(6) From the k-th residual value, calculate the k-th intrinsic mode
function:
\(d_{k}\left(t\right)=r_{k-1}\left(t\right)-r_{k}\left(t\right)\)(6)
(7) Repeat steps (5) to (6) until the residual \(r_{k}\left(t\right)\)satisfies one of the following conditions (Adarsh and Reddy., 2018): ①
it cannot be further decomposed by EMD method; ② it satisfies theIMF condition; ③ the number of the local extrema is less than
three.
Thus, the original signal \(x\left(t\right)\) can be decomposed into
the k -number of IMF components and one trend term\(r_{K}\left(t\right)\):
\(x\left(t\right)=\sum_{k=1}^{K}{d_{k}\left(t\right)+r_{K}\left(t\right)}\)(7)