## 2 Theoretical Background
### Review of SVD  
The singular value decomposition of a matrix \(\mathbf{A}\) can be formulated as  
\[\mathbf{A}=\mathbf{U}\mathbf{\sum V}^{\mathbf{\tau}} \tag{6}\]  
where \(\mathbf{U}=[u_{1},u_{2}\dots,u_{m}]\in\mathbf{R}^{m\times n}\), \(\mathbf{V}=[v_{1},v_{2}\dots,v_{m}]\in\mathbf{R}^{m\times n}\) are orthogonal matrices. Additionally, \(u_{m}\) is the left singular vector and \(v_{m}\) is the right singular vector. The column vectors of \(\mathbf{U}\) and \(\mathbf{V}\) are the orthonormal eigenvectors of \(\mathbf{A}\mathbf{A}^{\mathbf{\tau}}\) and \(\mathbf{A}^{\mathbf{\tau}}\mathbf{A}\). Additionally, \(\mathbf{\sum}\) is a diagonal matrix storing the singular values of \(\mathbf{A}\) in descending order  
Step 1: Construction of Hankel matrix.  
The vibration signal, which is measured by an accelerometer, is always expressed as a vector. In order to prepare the signal for the SVD, it was reshaped into a matrix. In this paper, a Hankel matrix is utilized.  
For a signal \(\mathbf{x}=[x(1),x(2),\dots x(N)]\), it can be reshaped into a Hankel matrix as follows:  
\[\left(\begin{array}{cccc}x(1)&x(2)&\dots\dots x(n)\\ x(2)&x(3)&\dots x(n+1)\\ \vdots&\vdots&\ddots\vdots\\ x(m)&x(m+1)&\dots\dots x(N)\end{array}\right) \tag{7}\]  
where \(m=N-n+1\).  
Step 2: Signal decomposition and reconstruction Hankel matrix \(\mathbf{A}\) can be expressed as the sub-matrix \(\mathbf{A}_{\mathbf{i}}\), as follows  
\[\mathbf{A}=[\mathbf{u}_{1},\mathbf{u}_{2},\ldots\mathbf{u}_{m}]\left(\begin{array}{cccc} \sigma_{1}&0&\cdots 0&0\\ 0&\sigma_{2}&\cdots 0&0\\ \vdots&\vdots&\ddots\vdots&\vdots\\ 0&0&\cdots\sigma_{m}&0\\ \end{array}\right)\left(\begin{array}{c}\mathbf{v}_{1}^{\intercal}\\ \mathbf{v}_{2}^{\intercal}\\ \vdots\\ \mathbf{v}_{n}^{\intercal}\\ \end{array}\right) \tag{8}\] \[=\sigma_{1}\mathbf{u}_{1}\mathbf{v}_{1}^{\intercal}+\sigma_{2}\mathbf{u}_{2} \mathbf{v}_{2}^{\intercal}+\ldots+\sigma_{m}\mathbf{u}_{m}\mathbf{v}_{m}^{\intercal}=\mathbf{ A}_{1}+\mathbf{A}_{2}+\ldots+\mathbf{A}_{m}\]  
where \(\mathbf{u}_{i}\in R^{m\times 1}\) is the \(i\)th column vector of left singular matrix \(\mathbf{U}\), \(\mathbf{v}_{i}\in R^{n\times 1}\) is the \(i\)th column vector of right singular matrix \(\mathbf{V}\). \(\mathbf{A}_{m}\) is sub-matrix. \(x_{i}\) can be obtained from \(\mathbf{A}_{i}\) by extracting the first row and last column of \(\mathbf{A}_{i}\), Equation (7) states that if all the SCs are added, the raw signal can be reconstructed.  
\[x=\sum\limits_{i=1}^{m}x_{i} \tag{9}\]  
Step 3: Signal denoising  
The commonly used method for selecting singular values is to delimit the singular values through a certain standard, and then select all the singular values before the limit for signal reconstruction. The first \(r\) largest singular values on the singular value difference spectrum for signal reconstruction are selected, i.e.:  
\[\widetilde{\mathbf{A}}=[\mathbf{u}_{1},\mathbf{u}_{2},\ldots\mathbf{u}_{n}] \left(\begin{array}{cccc}\sigma_{1}&0&\cdots 0&0\\ 0&\sigma_{2}&\cdots 0&0\\ \vdots&\vdots&\ddots\vdots&\vdots\\ 0&0&\cdots\sigma_{r}&0\\ 0&0&\cdots 0&0\\ \end{array}\right)\left(\begin{array}{c}\mathbf{v}_{1}^{\intercal}\\ \mathbf{v}_{2}^{\intercal}\\ \vdots\\ \mathbf{v}_{m}^{\intercal}\\ \end{array}\right) \tag{10}\] \[=\mathbf{A}_{1}+\mathbf{A}_{2}+\ldots+\mathbf{A}_{r}\Rightarrow \widetilde{x}=\sum\limits_{i=1}^{r}x_{i}\]  
where \(\widetilde{x}\) and \(\widetilde{\mathbf{A}}\) are the denoised signal and matrix, respectively.