## 2 Variational Mode Decomposition
If \(K\) is too small, multiple components of the signal may be contained in one mode simultaneously, or one component may not be able to be estimated. If K is too large, one component in the signal could be included in multiple modes, and the mode central frequencies obtained by iteration will eventually overlap. To address this problem, a mode number fluctuation method is proposed in this paper to determine the mode number K. The detailed procedure is as follows, and the flowchart is shown in Figure 1.  
1. The initial value of the mode number is \(K=K_{0}\);
2. When the mode number is \(K_{0}\), determine whether the mode central frequencies overlap;
3. If the central frequencies overlap, decrease the mode number and perform VMD until the central frequencies do not overlap. Return \(K\);
4. If the central frequencies do not overlap, increase the mode number and perform VMD until the central frequencies overlap. Return \(K-1\). A simulation test is designed to verify the effectiveness of VMD. Simulation signal \(y_{(t)}\) consists of interval signal \(x_{1(t)}\), linear frequency modulation signal \(x_{2(t)}\), and cosine signal \(x_{3(t)}\); the time domain waveform is shown in Figure 2. The simulation signal is decomposed via the VMD, EMD, and LMD methods, as shown in Figures 3-5. The decomposition result shows that the VMD method decomposes various signal components effectively and the decomposed signal and original signal have a high degree of coincidence. In comparison, the EMD- and LMD-processed intrinsic modes have various degrees of mode aliasing and signal distortion. The main reasons are considered as follows: both the EMD and LMD algorithms define local mean functions and local envelope functions based on extreme points (maxima and minima), and their envelope estimation errors will be amplified after repeated recursive decomposition. Therefore, the EMD and LMD algorithms are very sensitive to noise. When the decomposed signal contains interval or adjacent frequency components, modal aliasing often occurs in the EMD and LMD algorithms, and the adjacent frequency components are hard to decompose. In addition, due to the effect of end effects, the signal contains some false components, resulting in demodulation errors. In the simulation signal \(y_{(t)}\), the frequencies of \(x_{1(t)}\) and \(x_{2(t)}\) are 5 Hz and 10 Hz, respectively, with adjacent frequency components. In addition, \(y_{(t)}\) also contains interval signal components. Therefore, the simulation signal \(y_{(t)}\) cannot be effectively decomposed by using the EMD and LMD algorithms. VMD assumes that each intrinsic mode function has limited bandwidth and a different central frequency. To ensure that the sum of the estimated bandwidths for the intrinsic mode functions is minimal, a variation problem is solved via a transformation. Each intrinsic mode function is demodulated to the corresponding base frequency band. Finally, each intrinsic mode function and its corresponding central frequency are extracted. VMD is an adaptive Wiener filter group, which shows better noise robustness. Therefore, VMD effectively solves the mode aliasing problem and has significantly superior anti-noise performance compared with EMD and LMD. Additionally, because the VMD algorithm is applied in the frequency domain and belongs to a complete non-recursive algorithm, it has higher computational efficiency than EMD and LMD. \[x_{1}(t)=\left\{ \begin{array}{l}\cos(20\pi t),0.2<t<0.25,0.4<t<0.45,0.6<t<0.6 5,0.8<t<0.85\\ 0,\text{other}\end{array} \right. \tag{9}\]  
\[x_{2}(t)=0.6\cos(10\pi t+10\pi t^{2}) \tag{10}\]  
\[x_{3}(t)=\cos(100\pi t) \tag{11}\]  
\[y(t)=x_{1}(t)+x_{2}(t)+x_{3}(t) \tag{12}\]  
Figure 1: Method for determining the number of variational mode decomposition (VMD) modes. Figure 3: Simulation signal \(y_{(l)}\) VMD results. (**a**) IMF1; (**b**) IMF2; (**c**) IMF3.