*2.1. VMD*

Variational modal decomposition (VMD) is developed by University of California scholars Dragomiretskiy et al., in 2014 [18]. Based on Wiener filtering, this method searches for the optimal solution of the input signal within the framework of variational model. It can adaptively update the center frequency, bandwidth, and corresponding sub signals, and decompose the independent components of the signal from the frequency domain. As a non-recursive signal analysis method, the core idea of the VMD method is to determine the intrinsic mode function (IMF) by solving the variational problem. Therefore, in the VMD algorithm, the IMF component obtained by signal decomposition is different from that in EMD and LMD algorithms. The original signal is non recursively decomposed into several IMF components with limited bandwidth:

$$
\mu\_k(t) = A\_k(t) \cos[\phi\_k(t)] \tag{1}
$$

In the above formula, *Ak*(*t*) is the instantaneous amplitude of *μk*(*t*), and *Ak*(*t*) ≥ 0. *φk*(*t*) is phase and *φk*(*t*) ≥ 0. *<sup>ω</sup>k*(*t*) is the instantaneous phase of *μk*(*t*).

$$
\omega\_k(t) = \phi\_k \iota(t) = \frac{d\phi\_k(t)}{dt} \tag{2}
$$

In the above formula, *μk*(*t*) can be regarded as a harmonic signal, its amplitude is *Ak*(*t*) and its frequency is *<sup>ω</sup>k*(*t*).

It is assumed that each mode has a limited bandwidth with central frequency, and the central frequency and bandwidth will be updated continuously in the decomposition process. Then the variational problem can be expressed as finding *r* modal functions *μk*(*t*) and minimizing the estimated bandwidth forthe sum of all modal functions. The sum of modes is the input signal.VMD algorithm can obtain *k* discrete modes *μk*(*t*) (*r* ∈ 1, 2, ··· , *R*) by decomposing signal *X*(*t*). Then, the frequency bandwidth of each modal signal is estimated in the following manner.

(1) Hilbert transform is extended to the modal function, and the marginal spectrum is obtained. 

$$\left(\delta(t) + \frac{\dot{j}}{\pi t}\right) \* \mu\_k(t) \tag{3}$$

(2) Each estimated center band is modulated to the corresponding fundamental band.

$$\left[ (\delta(t) + \frac{\dot{j}}{\tau \mathbf{t} t}) \* \mu\_k(t) \right] e^{-j\omega\_k t} \tag{4}$$

(3) The square *L*<sup>2</sup> norm of the demodulated signal gradient is obtained.

$$\left\|d\_{\mathbf{f}}[(\delta(t) + \frac{\dot{f}}{\tau \mathbf{t} t}) \* \mu\_{\mathbf{k}}(t)]e^{-j\omega\_{\mathbf{k}}t}\right\|\_{2}^{2} \tag{5}$$

By constructing the VMD variational constraint model based on the above formula, the following formula can be formed.

$$\begin{cases} \min\_{\{\mu\_k\}, \{\omega\_k\}} \left\{ \sum\_{k=1}^k \|d\_l[(\delta(t) + \frac{j}{\pi t}) \* \mu\_k(t)]e^{-j\omega\_k t}\|\_2^2 \right\} \\ \qquad \text{s.t. } \sum\_{k=1}^k \mu\_k(t) = \mathbf{x}(t) \end{cases} \tag{6}$$

In the above formula, {*μk*(*t*)} = {*μ*1, *μ*2, *μ*3, ···*μk*} is the function set of each mode. {*<sup>ω</sup>k*} = {*<sup>ω</sup>*1, *ω*2, *ω*3, ···*ωk*} is the central frequency set. *δ*(*t*) is the unit pulse function. *dt* is the derivative of the function over time t. s.t. is the constraint. *x*(*t*) is the original input signal, where *k* is the number of decompositions.

By introducing penalty factor and Lagrange multiplication operator b, the constrained model problem in Equation (5) can be transformed into a non-constrained model problem, as in Equation (6).

$$L(\{\mu\_k\}, \{\omega\_k\}, \lambda) = a \sum\_{k=1}^k \left\| d\_l[(\delta(t) + \frac{j}{\pi t}) \* \mu\_k(t)]e^{-j\omega\_k t} \right\|\_2^2 + \left\| \mathbf{x}(t) - \sum\_{k=1}^k \mu\_k(t) \right\|\_2^2 + \left< \lambda(t), \mathbf{x}(t) - \sum\_{k=1}^k \mu\_k(t) \right> \tag{7}$$

By constantly iteratively searching for the minimum point of Lagrange function *L*, the original input signal a will be decomposed into *k* modal functions *μk*(*t*).

When using VMD decomposition algorithm to adaptively decompose the signal, the decomposition parameters need to be set in advance. Theoretical research shows that the parameters that have a grea<sup>t</sup> impact on the decomposition effect mainly include the number of decomposition *k* and the penalty parameter α. Therefore, setting these two parameters only by experience will bring grea<sup>t</sup> errors to the decomposition results of VMD. Among them, the size of *k* value is directly related to the decomposition effect of VMD. If the value of *k* is too small, it will lead to under decomposition of the signal, and the resulting signal is not completely decomposed into components. If the value of *k* is too large, the signal will be decomposed too much, resulting in over decomposition. At the same time, the value of a has a certain impact on the bandwidth of the decomposed component. If the value of a is too small, the bandwidth of the decomposed component will be too large, and some components will include other components. If the value of a is too large, the bandwidth of the decomposed component will be too small, and some components in the decomposed signal may be lost.
