## 2 Blind Source Separation Techniques
### Generalized Eigen Decomposition (GED)  
According to Equations (1) and (3), the BSS method aims to extract independent sources from their observed mixtures, without prior knowledge of the mixing matrix \(\mathbf{A}\) or the source signals \(\mathbf{s}\). In other words, given a set of mixed signals \(\mathbf{x}\), the BSS method seeksto find a separation matrix \(\mathbf{W}\) such that the estimated sources \(\hat{\mathfrak{s}}=\mathbf{W}\mathbf{x}\) are as independent as possible and approximate the true source \(\mathbf{s}\) as closely as possible. The solution process generally includes two steps [9; 32]. The initial step in the BSS method involves performing a whitening transformation on the observed mixtures. This transformation aims to linearly transform the measured data such that the correlation matrix of the output vector matches the identity matrix. At this stage, the dimensionality of the measured vector is reduced to match that of the source vector. Then, the next step in the BSS method is to estimate the separation matrix or the inverse of the mixing matrix \(\mathbf{A}\). Another method includes simultaneous diagonalization of matrix pencil \((\mathbf{R}_{x1},\mathbf{R}_{x2})\), namely, generalized eigen decomposition (GED). The calculation method of a matrix includes considering two signals with distinct energy [33], calculating one matrix in the filtering version of a mixed signal [34], and calculating the delay correlation matrix [28; 35; 36]. According to \(\hat{\mathfrak{s}}=\mathbf{W}\mathbf{x}\), the goal of GED is to seek the separation matrix \(\mathbf{W}\) which satisfies Equation (20). \[\mathbf{W}\mathbf{A}=\mathbf{P}. \tag{20}\]  
If the \(\mathbf{P}\) matrix is a diagonal matrix or a permutation matrix, then the estimated source signals \(\hat{\mathfrak{s}}\) are obtained by multiplying the separation matrix \(\mathbf{W}\) with the observed mixtures \(\mathbf{x}\). The generalized eigenvalue decomposition involves simultaneous diagonalization of two matrices, known as a matrix pencil. In the context of BSS, the matrix pencil is formed by the correlation matrices \(\mathbf{R}_{x1}\) and \(\mathbf{R}_{x2}\) of the observed mixtures and the estimated source signals, respectively. For a given set of observed mixtures, let \(\mathbf{X}\) be an \(m\times N\) matrix containing \(N\) samples of each of the \(m\) measured signals. The correlation matrices for \(X\) can be calculated as  
\[\mathbf{R}_{x1}=\frac{1}{N}\mathbf{X}\mathbf{X}^{\mathrm{T}},\mathbf{R}_{x2}=\frac{1}{N}\mathbf{X} \mathbf{H}^{\mathrm{T}}\mathbf{H}\mathbf{X}^{\mathrm{T}}. \tag{21}\]  
In [9], the correlation matrix \(\mathbf{R}_{x2}\) is a time-delayed version of the estimated source signals, while \(\mathbf{R}_{x1}\) is described by Equation (21) as the correlation matrix of the observed mixtures. In [36], both matrices in the matrix pencil can be time-delayed. In particular, \(\mathbf{R}_{x2}\) for a delay \(d\) can be written as  
\[\mathbf{R}_{x2}=\frac{1}{N-d}\mathbf{X}\mathbf{H}^{\mathrm{T}}\mathbf{X}^{\mathrm{T}}, \tag{22}\]  
where \(\mathbf{R}_{x2}\) is obtained by applying a lower diagonal matrix \(\mathbf{H}\) to the observed mixtures \(\mathbf{X}\). Specifically, for a delay of \(d\) samples, the matrix \(\mathbf{H}\) has only one nonzero diagonal, which is the \(d\)-th diagonal below the main diagonal. If \(d=0\), the matrix \(\mathbf{H}\) becomes the identity matrix and Equation (21) is obtained, which represents the correlation matrix of the observed mixtures without any time delay. The sensor pencil is correlated with the pencil calculated from the source signal through the mixing matrix \(\mathbf{A}\) as follows:  
\[\mathbf{R}_{x1}=\mathbf{A}\mathbf{R}_{x1}\mathbf{A}^{\mathrm{T}},\mathbf{R}_{x2}=\mathbf{A}\mathbf{R}_{x2 }\mathbf{A}^{\mathrm{T}}, \tag{23}\]  
Figure 2: CP in mode extraction. where (\(R_{x1}\), \(R_{x2}\)) is the sensor pencil, (\(R_{s1}\), \(R_{s2}\)) is the source matrix pencil, and \(A\) denotes the instantaneous mixing matrix.