## 2 Theory Background  
### The Fundamental of SVD Denoising Method  
SVD is an orthogonal decomposition method. For any real matrix \(\mathbf{A}\in\mathbf{R}_{m\times n}\) with rank \(L\), it can be decomposed into the product of three matrices [5]: the orthogonal matrix \(\mathbf{U}\in\mathbf{R}_{m\times n}\), the diagonal matrix \(\mathbf{D}=\left[diag(\sigma_{1},\sigma_{2}\ldots,\sigma_{l}),\ 0\right]\), and the orthogonal matrix \(\mathbf{V}\in\mathbf{R}_{m\times n}\), as follows:  
\[\mathbf{A}=\mathbf{U}\mathbf{D}\mathbf{V}^{\mathbf{T}} \tag{1}\]  
where \(L=\min(m,n),\sigma_{1}\geq\sigma_{2}\ldots\geq\sigma_{l}\). \(\sigma_{i}(i=1,\ 2\ldots L)\) are the singular values of the matrix \(\mathbf{A}\). There are several steps to denoise the measurement signal with the SVD method. Firstly, the Hankel matrix should be constructed [24; 25; 26; 27] as follows:  
\[\mathbf{A}_{m}=\left[ \begin{array}{cccc}x(1)&x(2)&\cdots&x(n)\\ x(2)&x(3)&\cdots&x(n+1)\\ \vdots&\vdots&\ddots&\vdots\\ x(m)&x(m+1)&\cdots&x(N)\\ \end{array} \right] \tag{2}\]  
where \(1<n<N,\ m=N-n+1\). The parameter \(m\) is used to determine the number of decomposed components in SVD, which is usually selected less than \(n\)[15]. Secondly, the dynamic propriety of the matrix \(\mathbf{A}_{m}\) can be revealed based on the reconstructed attractor in the reconstructed phase space. As a result, the matrix \(\mathbf{A}_{m}\) can be expressed as [28]:  
\[\mathbf{A}_{m}=\mathbf{D}+\mathbf{W} \tag{3}\]  
where \(\mathbf{D}\) is the smooth signal in the reconstructed phase space and \(\mathbf{W}\) is the noise and interference signal in the reconstructed phase space. To denoise the original signal, it is necessary to find the best approximation matrix for matrix \(\mathbf{D}\). According to the singular value decomposition principle, the matrix \(\mathbf{A}_{m}\) can be decomposed as follows:  
\[\mathbf{A}_{m}=\mathbf{U}\mathbf{E}\mathbf{V}^{\mathbf{T}} \tag{4}\]  
where \(\mathbf{E}=\left[diag(\sigma_{1},\sigma_{2}\ldots,\sigma_{m}),\ \mathbf{0}\right]\) and \(L=\min(m,n),\sigma_{1}\geq\sigma_{2}\ldots\geq\sigma_{l}\). \(\mathbf{U}\) and \(\mathbf{V}^{\mathbf{T}}\) are the left and right singular matrixes, respectively, and they satisfied \(\mathbf{U}=\left[u_{1},u_{2}\ldots,u_{m}\right],\ \mathbf{V}=\left[v_{1},v_{2}\ldots,v_{n}\right]^{ \mathbf{T}}\), where \(u_{i}\) is the \(i\)th column vector of singular matrix \(\mathbf{U}\) and \(v_{i}\) is the \(i\)th column vector of singular matrix \(\mathbf{V}\). As a result, \(\mathbf{A}_{m}\) can be computed as:  
\[\mathbf{A}_{m}=\sigma_{1}u_{1}v_{1}+\sigma_{2}u_{2}v_{2}+\ldots+\sigma_{m}u_{m }v_{m} \tag{5}\]  
Finally, according to the singular value decomposition theory and the best approximation theorem of a matrix in the sense of Frobeious norm, it can be obtained that the different kinds of signals are characterized by different singular values [28]. Therefore, by removing the singular values corresponding to the noise components and interference harmonics, we can get the new diagonal matrix \(\mathbf{E}^{\prime}=\left[diag(\sigma_{\alpha},\sigma_{j}\ldots,\sigma_{ \beta}),\ 0\right]\). The new Hankel matrix \(\mathbf{A}_{m}{}^{\prime}\) can be computed as:  
\[\mathbf{A}_{m}{}^{\prime}=\sigma_{\alpha}u_{\alpha}v_{\alpha}+\sigma_{j}u_{j} v_{j}+\ldots+\sigma_{\beta}u_{\beta}v_{\beta} \tag{6}\]  
where the matrix \(\mathbf{A}_{m}{}^{\prime}\) is supposed to be the best approximation of matrix \(\mathbf{A}_{m}\). Compared to the original matrix \(\mathbf{A}_{m}\), its noise components and interference harmonics are compressed significantly.