## 3 The Method Proposed in This Paper  
### ADMM-1 DNet  
For the outdoor high-noise signal reconstruction problem, the compressed measured value \(y\) according to CS theory is shown in Equation (11). Where the original signal \(x\in\mathbb{R}^{n}\), \(e\in\mathbb{R}^{m},\ m<n\) corresponds to the noise in the acquired signal and \(\left\|e\right\|_{2}\leq\eta\). \[y=Ax+e. \tag{11}\]  
We assume that there exists a redundant sparse transformation \(\mathbf{\Phi}\in\mathbb{R}^{N\times n}(N>n)\) called the analysis operator that makes \(\mathbf{\Phi}x\) approximately sparse. To obtain \(x\), a common approach is to transform it into solving the \(l_{1}\)-minimization problem as follows:  
\[\min_{x\in\mathbb{R}^{n}}\left\|\mathbf{\Phi}x\right\|_{1}\text{s.t.}\left\|Ax-y \right\|_{2}\leq\eta. \tag{12}\]  
The ADMM algorithm is a classical convex optimization algorithm for solving the problem of Equation (12), which considers the equivalent generalized LASSO form of Equation (12), as follows:  
\[\min_{x\in\mathbb{R}^{n}}\frac{1}{2}\left\|Ax-y\right\|_{2}^{2}+\lambda\left\| \mathbf{\Phi}x\right\|_{1}. \tag{13}\]  
Introducing the dual variables \(z,u\in\mathbb{R}^{N}\), Equation (13) is transformed into the following:  
\[\min_{x\in\mathbb{R}^{n}}\frac{1}{2}\left\|Ax-y\right\|_{2}^{2}+\lambda\left\| z\right\|_{1},\text{ s.t.}\ \mathbf{\Phi}x-z=0. \tag{14}\]  
The optimization problem of Equation (14) can be solved using an iterative scheme of ADMM for the penalty parameters \(\rho>0\), \(k\in\mathbb{N}\) and the initial points \((x^{0},z^{0},u^{0})\):  
\[x^{(k+1)}=\left(A^{T}A+\rho\ \mathbf{\Phi}^{T}\mathbf{\Phi}\right)^{-1}\left(A^{T}y+ \rho\ \mathbf{\Phi}^{T}\Big{(}z^{(k)}-u^{(k)}\Big{)}\right), \tag{15}\]  
\[z^{(k+1)}=S_{\lambda/\rho}\Big{(}\mathbf{\Phi}x^{(k+1)}-u^{(k)}\Big{)}, \tag{16}\]  
\[u^{(k+1)}=u^{(k)}+\mathbf{\Phi}x^{(k+1)}-z^{(k+1)}. \tag{17}\]  
Iterative steps (15)-(17) converge to the solution of Equation (14) [54], i.e., \(\left\|Ax^{(k)}-y\right\|_{2}^{2}+\left\|z^{(k)}\right\|_{1}\to p^{*}\) and \(\mathbf{\Phi}x^{(k)}-z^{(k)}\to\ 0\) as \(k\to\infty\). The idea of ADMM-1 DNet is to map the aforementioned ADMM iterative scheme into a multilayer DNN, where each layer corresponds to an iteration of ADMM. The \begin{table}
\begin{tabular}{l} \hline \hline
**CS Based on ADMM + K-SVD** \\ \hline
**Step 1:** The monitoring end receives the observation \(y\), and the reference vibration signal is used as a training sample to obtain the initial dictionary \(\mathbf{D}_{0}\). \\
**Step 2:** The ADMM algorithm is used to solve the optimization problem in Equation (1), resulting in the initial sparse approximation coefficients \(\theta_{0}\). Solving for \(\tilde{y}_{0}=\mathbf{D}_{0}\theta_{0}\). gives the initial \\ reconstructed signal data. \\
**Step 3:** Update the sparse transformation dictionary \(\mathbf{D}\) with the K-SVD dictionary learning algorithm, and use the updated dictionary \(\mathbf{D}\) to make a sparse representation of the preliminary reconstructed data from the previous step. \\
**Step 4:** Solve the updated sparse coefficients \(\theta_{i}\) with the ADMM algorithm to update \(\tilde{y}_{i}=\mathbf{D}_{i}\theta_{i}\). \\
**Step 5:** Take the data obtained in step 2 as the input data for the next iteration; repeat Steps 3 and 4 until the algorithm converges. Output the reconstructed data \(\tilde{y}\) to realize the vibration signal reconstruction. \\ \hline \hline \end{tabular}
\end{table} 
Table 1: CS based on ADMM + K-SVD. first Equation (15) is transformed into the update rules (16) and (17), and the second Equation (16) into (17). \[ \begin{split} u^{(k+1)}&=(I-W)u^{(k)}+Wz^{(k)}+b-S_{ \lambda/\rho}\Big{(}(-I-W)u^{(k)}+Wz^{(k)}+b\Big{)},\\ z^{(k+1)}&=S_{\lambda/\rho}\Big{(}(-I-W)u^{(k)}+Wz^{( k)}+b\Big{)}\end{split}