## 2 Blind Source Separation Techniques
### Generalized Eigen Decomposition (GED)
Using the matrix pencil (\(R_{x1}\), \(R_{x2}\)), the generalized eigenvalue (GED) statement is  
\[R_{x2}E=R_{x1}E\mathbf{D}, \tag{24}\]  
where \(\mathbf{D}\) is a diagonal \(m\times m\) matrix with the eigenvalues of the matrix pencil (\(R_{x1}\), \(R_{x2}\)) and \(\mathbf{E}\) is a \(m\times m\) eigenvector matrix. Considering that \(\mathbf{X}\) is an instantaneous mixture of the source signals, i.e., \(\mathbf{X}=\mathbf{A}\mathbf{s}\), the equation can be rewritten as  
\[\mathbf{A}R_{s2}\mathbf{A}^{\rm T}\mathbf{E}=\mathbf{A}\mathbf{R}_{s1}\mathbf{A}^{\rm T}\mathbf{E}\mathbf{D}. \tag{25}\]  
When the instantaneous mixing matrix \(\mathbf{A}\) is invertible, we can multiply both sides of the GED statement by \(\mathbf{A}^{-1}\), set as follows:  
\[\mathbf{E}_{s}=\mathbf{A}^{\rm T}\mathbf{E}, \tag{26}\]  
which leads to the eigen decomposition of the source pencil:  
\[\mathbf{R}_{s2}\mathbf{E}_{s}=\mathbf{R}_{s1}\mathbf{E}_{s}\mathbf{D}. \tag{27}\]  
Using above relationship, we can relate the columns of the eigenvector matrix \(\mathbf{E}_{s}\) of the source matrix pencil (\(\mathbf{R}_{s1}\), \(\mathbf{R}_{s2}\)) to the columns of the eigenvector matrix \(\mathbf{E}\) of the observed mixtures. When the instantaneous mixing matrix \(\mathbf{A}\) is an \(m\times n\) matrix with \(m>n\), Equation (25) can be written using block matrix notation. Specifically, we can divide \(\mathbf{A}\) into two blocks, \(\mathbf{A}_{H}\) and \(A_{L}\), where \(\mathbf{A}_{H}\) is an \(n\times n\) matrix and \(A_{L}\) is an \((m-n)\times n\) matrix. Similarly, we can divide \(\mathbf{E}\) into two blocks, \(\mathbf{E}_{H}\) and \(\mathbf{E}_{L}\), where \(\mathbf{E}_{H}\) is an \(n\times m\) matrix and \(\mathbf{E}_{L}\) is an \((m-n)\times m\) matrix. Then, using matrix block operations, Equation (25) can be written as  
\[ \begin{array}{l}\mathbf{A}_{H}\mathbf{R}_{s2}\mathbf{\Omega}=\mathbf{A}_{H}\mathbf{R}_{s1}\mathbf{ \Omega}\mathbf{D},\\ \mathbf{A}_{L}\mathbf{R}_{s2}\mathbf{\Omega}=\mathbf{A}_{L}\mathbf{R}_{s1}\mathbf{\Omega}\mathbf{D},\end{array}  \tag{28}\]  
where  
\[\mathbf{\Omega}=\mathbf{A}_{H}^{\rm T}\mathbf{E}_{H}+\mathbf{A}_{L}^{\rm T}\mathbf{E}_{L}=\mathbf{A}^ {\rm T}\mathbf{E}. \tag{29}\]  
If \(\mathbf{A}_{H}\) is an invertible matrix, then the first equation in Equation (28) can be used to obtain the eigen decomposition of the source matrix pencil (\(\mathbf{R}_{s1}\), \(\mathbf{R}_{s2}\)). When \(\mathbf{A}_{H}\) is not invertible, then \(\mathbf{\Omega}\) has rank less than \(m\), and we cannot obtain the complete eigen decomposition of the source matrix pencil using the GED method. In this case, \(\mathbf{\Omega}\) has \((m-n)\) zero columns, corresponding to the null space of \(\mathbf{A}_{H}^{\rm T}\), and the eigenvalues in \(\mathbf{D}\) that do not belong to the eigenvalue decomposition of (\(\mathbf{R}_{s1}\), \(\mathbf{R}_{s2}\)) are not paired with eigenvectors of the pencil. For BSS problems, the separation matrix \(\mathbf{W}\) is obtained from the transpose of the eigenvector matrix \(E^{\rm T}\):  
\[\hat{\mathbf{s}}=\mathbf{W}\mathbf{x}=E^{\rm T}\mathbf{x}. \tag{30}\]  
If the source eigenvector matrix \(\mathbf{E}_{s}\) is normalized to have unit length, then \(\hat{\mathbf{s}}\) will be the estimated value of the source signal, but the magnitude and ordering are uncertain.