*2.2. Variational Mode Decomposition*

Variational mode decomposition (VMD) was put forward by Dragomiretskiy, K. and Zosso, D.in 2014 as a new method of time–frequency analysis based on traditional EMD [23]. The basic concept of VMD is to determine the center frequency and bandwidth of each IMF by searching for the optimal solution of the optimal function iteratively and decompose each IMF of frequency self-adaptively [24]. Research has shown that VMD could overcome the uncertainty of EMD in the center frequency and bandwidth and fundamentally reduce the endpoint effect and mode mixing. The basic steps of constructing the VMD objective function are as follows:

Analytic signals of each IMF are obtained by Hilbert transform to obtain a onesided spectrum.

$$
\left(\delta(t) + \frac{\dot{j}}{\pi t}\right) \cdot \mu\_k(t) \tag{7}
$$

where *μk*(*t*) is the kth IMF, and *δ*(*t*) stands for the pulse signal.

To modulate the spectrum of each mode to baseband, the exponential term *e*<sup>−</sup>*j<sup>ω</sup><sup>k</sup> t* is added to the analytical signal of each IMF.

$$
\left[ \left( \delta(t) + \frac{\dot{j}}{\pi t} \right) \cdot \mu\_k(t) \right] e^{-\dot{j}\omega\_k t} \tag{8}
$$

where *ωk* denotes the estimated center frequency of the kth IMF.

Calculate the gradient of the 2-norm of the above functions and construct the following constraint model:

$$\left\{ \begin{array}{ll} \min\_{\{u\_{k}\}, \{\omega\_{k}\}} \left\{ \sum\limits\_{k} \left\| \partial\_{t} \left[ \left( \delta(t) + \frac{j}{\pi t} \right) \mu\_{k}(t) \right] e^{-j\omega\_{k}t} \right\|\_{2}^{2} \right\} \\ \text{s.t.} & \sum\_{k} \mu\_{k} = f \end{array} \tag{9}$$

where {*μk*} is the set of each IMF, and {*<sup>ω</sup>k*} indicates the center frequency of each IMF. Continuous iteration is carried out to optimize the above constraint model, and each center frequency and bandwidth are searched to self-adaptively decompose signal frequency.

To solve the optimal solution of the constraint model, the constrained variational problem can be transformed into an unconstrained variational problem by introducing quadratic penalty factor *α* and Lagrange multiplier *<sup>λ</sup>*(*t*), the extended Lagrange expression is as follows:

$$\begin{aligned} L(\{\mu\_k\}\_\prime \{w\_k\}, \lambda) &= \\ \mu \sum\_k \left\| \partial\_t \left[ (\delta(t) + \frac{j}{\pi t}) \cdot \mu\_k(t) \right] e^{-j\omega\_k t} \right\|\_2^2 + \left\| f(t) - \sum\_k \mu\_k(t) \right\|\_2^2 + \left< \lambda(t), f(t) - \sum\_k \mu\_k(t) \right> \end{aligned} \tag{10}$$

where *α* is the quadratic penalty factor, which can ensure signal accuracy in the presence of Gaussian noise. *λ*(*t*) is the Lagrange multiplier, which can maintain strict constraint conditions. An alternate direction multiplier algorithm is used to solve the extended Lagrange function by alternating updates *uk*, *wk*, *λ* to find "saddle point" of extended Lagrange expression [23].
