## 2 Theory Background  
### Singular Spectrum Decomposition  
SSD is a novel signal decomposition approach derived from SSA. While SSA has a powerful capability of recovering sub-component signal from the multi-component signals [31], how to properly choose the embedding dimension and select the principal components for reconstructing a specific sub-component are its two main obstacles. The use of SSD improves the process by allowing the SSA fundamental parameters to be selected adaptively. Given an original multi-component signal \(y(n)\), \(n=1,2,\ldots,N\), its sub-component signals, referred to as SSCs, can be iteratively extracted using the following steps:  
**(a)** Embedding  
If the embedding dimension is selected as \(K\), \(1<K<N\), a \(K\times L\) matrix \(\mathbf{Y}\) named Hankel matrix can be generated. The Hankel matrix consists of \(K\) lagged vectors \(y_{i}=[y(i),\)\(\ldots,y(N),\)\(y(1),\)\(\ldots,\)\(y(i-1)]\), \(i=1,\)\(2,\)\(\ldots,\)\(N-K+1\). Taking the one-dimensional signal \(y(n)=\{1,\)\(2,\)\(3,\)\(4,\)\(5\}\) as an example, if the embedding dimension \(K\) is set to \(3\), the corresponding Hankel matrix becomes:  
\[\mathbf{Y}=\left[\begin{array}{cccccc}1&2&3&4&5\\ 2&3&4&5&1\\ 3&4&5&1&2\end{array}\right]. \tag{1}\]  
In the SSD method, the embedding dimension is adaptively selected as \(K=f_{s}/f_{\max}\), where \(f_{\max}\) represents the dominant frequency in the power spectral density (PSD) of \(y(n)\) and \(f_{s}\) indicates the sampling frequency.  
**(b)** DecompositionThe Hankel matrix \(\mathbf{Y}\) is then decomposed using singular value decomposition (SVD):  
\[\mathbf{Y}=\mathbf{U}\mathbf{D}\mathbf{V}^{\mathrm{T}} \tag{2}\] \[=\left[\mathbf{u}_{1},\mathbf{u}_{2},\cdots,\mathbf{u}_{K}\right]\left[\begin{array} []{ccccc}\lambda_{1}&0&0&0&\mathbf{0}\\ 0&\lambda_{2}&\cdots&0&\mathbf{0}\\ \vdots&\vdots&\ddots&\vdots&\mathbf{0}\\ 0&0&\cdots&\lambda_{K}&\mathbf{0}\\ \end{array}\right]\left[\begin{array}{c}\mathbf{v}_{1}^{\mathrm{T}}\\ \mathbf{v}_{2}^{\mathrm{T}}\\ \vdots\\ \mathbf{v}_{N}^{\mathrm{T}}\end{array}\right]\] \[=\lambda_{1}\mathbf{u}_{1}\mathbf{v}_{1}^{\mathrm{T}}+\lambda_{2}\mathbf{u}_{ 2}\mathbf{v}_{2}^{\mathrm{T}}+\ldots+\lambda_{K}\mathbf{u}_{K}\mathbf{v}_{K}^{\mathrm{T}}\] \[=\mathbf{H}_{1}+\mathbf{H}_{2}+\cdots+\mathbf{H}_{K}\]  
where \(\mathbf{U}\in\mathbf{R}^{K\times K},\mathbf{D}\in\mathbf{R}^{L\times L}\) and \(\mathbf{V}\in\mathbf{R}^{N\times N}\). Both \(\mathbf{U}\) and \(\mathbf{V}\) are orthogonal matrices, which contains the left and right vectors, respectively. The main diagonal of \(\mathbf{D}\) carries the singular values.  
After the SVD process, the Hankel matrix \(\mathbf{Y}\) is decomposed into \(K\) principle components.  
**(c)** Grouping  
The \(R\) (\(R<K\)) principle components, whose left eigenvectors reflect a domain frequency in the range \(\left[f_{\max}-\Delta f,f_{\max}+\Delta f\right]\), are selected out to reconstruct the sub-component signal. \(\Delta f\) is determined by the Gaussian interpolation of the PSD of \(y(n)\).  
**(d)** Reconstruction  
The selected \(R\) principal components are employed to generate a rank-R approximation of \(\mathbf{Y}\). Then the corresponding sub-component can be recovered by applying diagonal averaging to the adjusted version of this matrix. For instance, the adjusted version of the above matrix shown in Equation (1) can be described as:  
\[\mathbf{X}=\left[\begin{array}{ccccc}&&1&&\\ &1&2&&\\ 1&2&3&4&5\\ 2&3&4&5&\\ 3&4&5&&\end{array}\right]. \tag{3}\]  
The detailed introduction to SSD can refer to [24].