## Theory
### The Basic Theory of Blind Source Separation  
Assume that there are \(R\) source signals being linearly mixed into \(J\) observed signals. For each signal, \(N\) samples are available. The following BSS model is considered:  
\[\mathbf{T}=\mathbf{D}\mathbf{S}+\mathbf{N} \tag{11}\]  
where \(\mathbf{T}=[\mathbf{t}_{1},\mathbf{t}_{2},\ldots,\mathbf{t}_{J}]\in R^{N\times J}\) contains the observed data, \(\mathbf{S}=[\mathbf{s}_{1},\mathbf{s}_{2},\ldots,\mathbf{s}_{R}]\in R^{N\times R}\) with the \(R\) unknown source signals. \(\mathbf{D}\in R^{R\times J}\) represents the unknown composite matrix and \(\mathbf{N}\in R^{N\times J}\) denotes additive noise. Commonly, the problem of BSS can be divided into three situations: the underdetermined BSS with the condition of \(R>J\); the positive definite BSS if \(R=J\); and the over-determined BSS when \(R<J\). A clearly model of BSS is as shown as Figure 3.  
The general goal in BSS is to recover the unknown source in \(S\) and the unknown composite vectors in \(D\), given only the observed data \(T\), as shown in Figure 3. The research of this paper is concerned about the single channel underdetermined BSS, namely SCBSS, which means the number of observed signals is less than that of the source signals and the number \(J\) is equal to 1. Hence, the observed data \(T\) will be a vector with length \(n\). Therefore, the target of SCBSS is to obtain the unknown source \(\mathbf{U}\in[u_{1},u_{2},\ldots,u_{R}]\) and the unknown composite vectors \(\mathbf{P}\) from the observed vector \(T\). Additionally, the most important part aims to recover source \(\mathbf{U}\), which should be as close to the unknown source \(S\) as possible. Obviously, if the composite matrix \(\mathbf{D}\) is known, the SCBSS will be a very simple problem of linear equations to obtain the source signals. However, in the practical engineering condition, the composite \(\mathbf{D}\) is ordinarily unknown, so recovering the source signals has become a significant problem, especially in the underdetermined condition. The general solution to this problem is not unique and various approaches have been proposed, ranging from applying independence assumptions to non-negativity and sparsity constraints. Then, the independent component analysis (ICA), which assumes the sources to be statistically independent, is introduced. However, the ICA is a typical matrix separation method, which demands the strictly statistical independence. Conversely, in actual situations, the sources may not always be statistically independent; therefore, the result provided by ICA method was not satisfactory as expected. The improved TSSA based on CP-WOPT is proposed in this paper, which is mentioned in the Section 2.2. The observed data can be converted into a tensor firstly. Then, using the CP-WOPT method to solve the obtained tensor, we can get \(R\) rank-1 sub-tensor and divide them into several parts. Therefore, we can choose some parts sub-tensors to reconstruct as vector data, which can be regarded as the expected source signals. The number of the divided tensors is equal to the number of the source signals, which can be determined by the method of EMD-SVD-BIC [30]. EMD-SVD-BIC algorithm can be performed by three steps. Firstly, the IMFs of single-channel observation signal \(x(t)\in R^{N}\) is obtained by EMD, thus we can get a multi-dimensional \(\text{data}_{imf}(t)=(x(t),c_{1}(t),c_{2}(t),\ldots,c_{l}(t),r_{l}(t))^{\text{T}}\), where \(c_{i}(t)\), \((i=1,2,\ldots,l)\) is the IMFs and \(r_{l}(t)\) is remainder. Then, we can solve the correlation matrix as \(R_{x}=E\Big{[}s_{inf}(t)s_{imf}{}^{\text{H}}(t)\Big{]}+\sigma^{2}I_{M-n}\), where \(s_{imf}(t)\) represents the source signal component, \(M=l+2\), \(I_{M-n}\) is unit matrix, and \(\sigma^{2}\) denotes the noise power.