## 2 Theory of Method  
The application of multiple sensors brings the problem of massive data, using compressed sensing technology to collect information at a sampling frequency far lower than Nyquist sampling frequency. Compared with the data before processing, the nonlinear reconstruction algorithm perfectly reconstructs the signal. The processed signal has both sparsity and robustness, avoiding the redundancy of the collected information. The algorithm is as follows:  
Suppose a one-dimensional discrete time signal \(X\)[50]:  
\[X\;=\;(x_{1},x_{2}\cdots x_{N})^{T} \tag{1}\]  
The sparse basis matrix \(\psi\) is used to calculate the sparse time signal \(X\):  
\[\psi\;=\;\left[\varphi_{1},\varphi_{2}\cdots\varphi_{N}\right]^{T} \tag{2}\]  
\[X\;=\;\sum\varphi_{i}s_{i} \tag{3}\]  
where \(\varphi_{i}\) is the \(i\)-th of \(N\) orthogonal basis vectors in the space, and the corresponding \(s_{i}\) is the coefficient of the corresponding orthogonal basis vector. Select the measurement matrix \(\Phi\) and calculate the observation value \(y\):  
\[\Phi\;=\;\left[\phi_{1},\phi_{2}\ldots\phi_{N}\right] \tag{4}\]  
\[y\;=\;\Phi X \tag{5}\]  
where \(\phi_{1},\phi_{2}\ldots\phi_{N}\) is the column vector of the measurement matrix \(\Phi\). The sensing matrix \(\Theta\) is as follows:  
\[\Theta\;=\;\Phi\psi \tag{6}\]  
If there is a constant \(\delta\), so that for any \(K\) sparse vector \(x(\|x\|_{0}\leq K)\), it satisfies the restricted equidistant characteristic, and then the sparse signal can be effectively and completely recovered [57]:  
\[(1-\delta)\|x\|_{2}^{2}\leq\|\Phi x\|_{2}^{2}\leq(1+\delta)\|x\|_{2}^{2} \tag{7}\]  
In each iteration, the column vector having the largest projection coefficient in the sensing matrix is selected by the residual to form the recovery matrix \(\tilde{\Phi}\)[58], and the column vector having the largest projection coefficient in the sensing matrix must be zeroed each time it is updated:  
\[\omega_{j}\;=\;\left|\left\langle\gamma_{i-1},\theta_{i}\right\rangle\right| \tag{8}\]  
\[\lambda_{i}\;=\;\text{argmax}\omega_{j} \tag{9}\]  
where \(\gamma_{i-1}\) is the residual of the \(i\)-1 th iteration, \(\theta_{i}\) is the column vector of the sensing matrix \(\Theta\), \(\lambda_{i}\) is the index of the maximum projection coefficient b, \(\lambda_{i}\) is the indexes of the maximum value of the projection coefficient \(\alpha_{j}\), and \(\widetilde{\Phi}\) is the set of column vectors having the largest projection coefficient in each iterative. Solve the Moore-Penrose pseudo-inverse of the recovery matrix, and then based on Modified Cholesky decomposition, obtain the inverse \(Q^{-1}\) of the symmetric matrix \(Q\):  
\[Q^{-1}\ =\ (L^{-1})^{T}D^{-1}L^{-1} \tag{10}\]  
The reconstructed signal \(\widetilde{x}_{i}\) obtained after each iteration is:  
\[\psi\;=\;\left[\varphi_{1},\varphi_{2}\cdots\varphi_{N}\right]^{T} \tag{2}\]0  
The residuals that need to be iterated next time are updated according to the recovery matrix \(\widetilde{\Phi}\) and the reconstruction vector \(\widetilde{x}_{i}\) in each iteration until the final requirement is reached, and the final reconstructed signal \(\widetilde{x}\) is obtained:  
\[\psi\;=\;\left[\varphi_{1},\varphi_{2}\cdots\varphi_{N}\right]^{T} \tag{2}\]1  
We hope to find a method that can perform a frequency division operation on signals with multiple degrees of freedom at the same time, in order to achieve the filtering effect, and can maintain the coupling relationship between the original signals to the greatest extent. Therefore, we use the recently proposed 2D-VMD technique to adaptively decompose signals with three degrees of freedom into multiple sets of feature scale components. Then, Renyi entropy is used as the optimization index of 2D-VMD analysis performance to measure the correlation and coupling between the characteristic scale components of various degrees of freedom in different states. The signal is reconstructed using the extracted effective target IMF component to improve the signal resolution and signal-to-noise ratio.