## 3 Proposed SRSVD Method  
The basic flowchart of the proposed method is shown in Figure 2, and the steps are explained in detail below.  
SR from the perspective of probability interpretation, the fidelity term of the SR model is given by the maximum likelihood estimation of \(\mathbf{y}\,|\,\mathbf{x}\) to be inferred likelihood estimator, and the fidelity term follows white Gaussian distribution. Therefore, the SR model has a strong ability to remove white Gaussian noise. Because a lot of noise in nature is caused by the irregular movement of some particles. Especially in the circuit, the irregular motion of electrons, such as thermal motion, produces strong white Gaussian noise. Therefore, the first purpose of using SR is to remove Gaussian white noise.  
Figure 1: Visualization of objective function under different norm constraints.  
Next, the reconstructed signal is decomposed by SVD, and the decomposed signal components are given proper weights by appropriate evaluation indexes to extract the favorable parts of the signal components. The specific evaluation index and weight selection will be given in Section 4 combined with the actual situation.  
Therefore, the SRSVD noise reduction algorithm steps can be summarized as follows:  
Step 1. Construct a dictionary matrix \(\mathbf{D}\) by prior knowledge or dictionary learning. For dictionary learning, its objective function is:  
\[\underset{D,A}{\operatorname{argmin}}\frac{1}{2}\|\mathbf{Y}-\mathbf{D}\mathbf{A}\|_{F}^{2},\quad s.t.\quad\left\|\mathbf{\alpha}_{i}\right\|_{0}\leq T,\quad i=1,2,\ldots,m \tag{7}\]  
where \(\mathbf{Y}=\left(\mathbf{y}_{i}\right)_{i=1}^{m}\in\mathbb{R}^{m\times n}\) is the sample matrix; it may consist of signals of nonparametric signal observation. \(\mathbf{A}=\left(\mathbf{\alpha}_{i}\right)_{i=1}^{m}\) is the sparse coefficient matrix and \(T\) is sparsity. \(\left\|\mathbf{X}\right\|_{F}=\sqrt{\frac{\sum}{i}\frac{\sum}{j}\left|x_{ij} \right|^{2}}\) is the Frobenius norm of the matrix \(\mathbf{X}\). Dictionary learning algorithm can construct an appropriate dictionary without prior knowledge of the sample, so it has a wider range of application fields than an analytic dictionary.  
Step 2. The sparse coefficients are solved by ISTA [20]. The objective function is shown in Equation (3) and the ISTA procedure is shown in Algorithm 1.  
Step 3. Reconstructed signal \(\mathbf{y}=\mathbf{D}\mathbf{x}^{(i+1)}\).  
Step 4. Construct Hankel matrix \(\mathbf{H}\) of new \(\mathbf{y}\), the construction method is shown in Equation (5).  
Step 5. Decompose \(\mathbf{H}\) as shown in Equation (6). The decompose components are denoted as \(f_{1},f_{2},\ldots,f_{n}\).  
Step 6. Each component is given weight according to the evaluation index. In this paper, we choose the periodic modulation intensity (PMI) [25] as the component evaluation index, which can be used as an important indicator to measure the information in the signal. The weight assigned to each component by PMI is denoted as \(w_{1},w_{2},\ldots,w_{n}\), and finally, after denoising \(\hat{\mathbf{y}}=\sum\limits_{i}^{n}w_{i}f_{1}\).  
Figure 2: Flowchart of SRSVD.  
```
Initialization:\(\mathbf{x}^{(0)}\), \(\mu<\left\|\mathbf{D}\right\|_{2}^{-2}\), Iter; For\(i=0\) to \(Iter\)do \(\mathbf{z}^{(i)}=\mathbf{x}^{(i)}\)-\(\mu\mathbf{D}^{\mathsf{T}}\Big{(}\mathbf{D}\mathbf{x}^{(i)}-\mathbf{y}\Big{)}\); \(\mathbf{x}^{(i+1)}=\text{soft}(\mathbf{z}^{(i)};\mu\lambda)\); endfor return\(\mathbf{x}^{(i+1)}\)
```  
**Algorithm 1**Iterative shrinkage-thresholding algorithm.