## 3 Theory Analysis
### MPA Optimized MVMD  
The MVMD algorithm is an extension of the VMD algorithm from one-dimensional to multi-dimensional. By establishing a constrained variational model expression, the consistency of multi-channel data component frequency in the signal decomposition process is guaranteed. Figure 2: Comparison of transformation results of different time–frequency analysis methods for noisy PD signals. (**a**) FSWT; (**b**) STFI; (**c**) CWT. The multi-channel signal is decomposed into \(k\) components with the same center frequency and minimized total bandwidth. The corresponding constrained variational optimization problem is expressed as:  
\[ \begin{array}{l}\min\limits_{\{u_{k,c}\},\{\omega_{k}\}}\left\{\sum\limits_{k }\sum\limits_{c}\left\|\partial_{t}\left[u_{+}^{k,c}(t)\mathrm{e}^{-\mathrm{i} \omega_{k}t}\right]\right\|_{2}^{2}\right\}\\ s.t.\frac{\sum\limits_{k}u_{k,c}}(t)=x_{c}(t),c=1,2,\ldots,C\end{array}  \tag{12}\]  
In (12), \(t\) is time, \(x_{c}(t)\) is the original signal, \(k\) is the number of intrinsic modes, \(u_{k,c}(t)\) is the \(k\)th mode of the \(c\) channel and \(\omega_{k}\) is the center frequency of \(k\) intrinsic modes. The Lagrange multiplication operator is introduced to transform (12) from a constrained problem to an unconstrained problem. The associated augmented Lagrange function is illustrated in (13). \[ \begin{array}{l}\mathcal{L}\big{(}\{u_{k,c}\},\{\omega_{k}\},\lambda_{c} \big{)}=\alpha\underset{k}{\sum}\sum\limits_{c}\left\|\partial_{t}\left[u_{+} ^{k,c}(t)\mathrm{e}^{-\mathrm{i}\omega_{k}t}\right]\right\|_{2}^{2}+\\ \underset{c}{\sum}\left\|x_{c}(t)-\underset{k}{\sum}u_{k,c}(t)\right\|_{2}^{ 2}+\underset{c}{\sum}\left\langle\lambda_{c}(t),x_{c}(t)-\underset{k}{\sum}u _{k,c}(t)\right\rangle\end{array}  \tag{13}\]  
The unconstrained problem is solved using the alternating direction multiplier algorithm, with the center frequency update given by (14), completing the adaptive decomposition of multivariate signals. \[ \begin{array}{l}\omega_{k}^{n+1}=\frac{\sum\limits_{c}\int_{0}^{\infty} \omega\left|\partial_{k,c}(\omega)\right|^{2}d\omega}{\sum\limits_{c}\int_{0} ^{\infty}\left|\partial_{k,c}(\omega)\right|^{2}d\omega}\end{array}  \tag{14}\]  
The modal number \(k\) in the MVMD algorithm determines the precision of the decomposition. If the \(k\) value is too large, there will be pseudo components in the IMF component. If the \(k\) value is too small, it will lead to modal aliasing. The penalty factor \(\alpha\) in the MVMD algorithm determines the bandwidth of the IMF component. If the \(\alpha\) value is too large, it will affect the fidelity of the reconstructed data. If the \(\alpha\) value is too small, it will make some components contain other components. Therefore, the determination of \(k\) and \(\alpha\) is very important for the decomposition effect of MVMD. In order to solve the above problems, this paper proposes to use the MPA algorithm to optimize MVMD decomposition parameters \(k\) and \(\alpha\). MPA is a meta-heuristic optimization algorithm proposed in 2020 [26], which is derived from the optimal foraging strategy in marine organisms. It searches for the optimal solution by simulating the process of predator preying on prey. In the MPA framework, the sea represents the global scope of the optimization problem, the elite predator matrix represents the optimum solution in the global range and the prey matrix is the optimum solution obtained in the current iteration. On the basis of the complex interaction of predator and prey, the MPA iteration process is divided into four key links:  
(1) Population initialization: the MPA pass (x) randomly determines the initial position in the feasible region and starts the optimization process, as in (15). \[ \begin{array}{l}X_{0}=X_{\min}+rand(X_{\max}-X_{\min})\end{array}  \tag{15}\]  
In (15), \(X_{\max}\) represents the upper bound of the search space, while \(X_{\min}\) represents the lower bound. \(\mathrm{rand}(\cdot)\) is a random number that falls within the interval [0, 1].