## 2 Blind Source Separation Techniques
### Complexity Pursuit Algorithm (CP)
#### 2.3.3 Extracting a Single Signal  
Equation (7) can be incorporated into Equation (10) as follows:  
\[F(\mathbf{y}_{i})=F(\mathbf{w}_{i},\mathbf{x})=\log\frac{V(\mathbf{w}_{i},\mathbf{x})}{U(\mathbf{w}_{ i},\mathbf{x})}=\log\frac{\mathbf{w}_{i}\overline{\mathbf{R}}\mathbf{w}_{i}^{\mathrm{T}}}{ \mathbf{w}_{i}\mathbf{R}\mathbf{w}_{i}^{\mathrm{T}}}, \tag{13}\]  
where \(\overline{\mathbf{R}}\) and \(\hat{\mathbf{R}}\) represent the long-term and short-term covariance between the mixtures, respectively. These matrices are square matrices with dimensions \(n\times n\). Note that maximizing only \(\mathbf{V}_{i}\) would result in a high variance signal with no constraints on its temporal structure. In contrast, minimizing only \(\mathbf{U}\) would result in a DC signal. In both cases, trivial solutions would be obtained for \(\mathbf{w}_{i}\) because \(\mathbf{V}_{i}\) can be maximized by setting the norm of \(\mathbf{w}_{i}\) to be large, and \(\mathbf{U}\) can be minimized by setting \(\mathbf{w}_{i}=0\). In contrast, the ratio \(\mathbf{V}_{i}/\mathbf{U}_{i}\) can be maximized only if two constraints are both satisfied: (1) \(\mathbf{y}\) has a nonzero range (i.e., high variance); (2) the values in \(\mathbf{y}\) change slowly over time. Note also that the value of \(\mathbf{F}\) is independent of the norm of \(\mathbf{w}_{i}\), such that only changes in the direction of \(\mathbf{w}_{i}\) affect the value of \(\mathbf{F}^{4}\). The elements of these matrices are defined as  
\[ \begin{array}{l}\overline{r}_{ij}=\sum\limits_{t=1}^{N}\big{(}x_{i}(t)- \overline{x}_{i}(t)\big{)}\big{(}x_{j}(t)-\overline{x}_{j}(t)\big{)}\\ \hat{r}_{ij}=\sum\limits_{t=1}^{N}\big{(}x_{i}(t)-\hat{x}_{i}(t)\big{)}\big{(}x _{j}(t)-\hat{x}_{j}(t)\big{)}.\end{array}  \tag{14}\]  
Calculation of the covariance matrices can be achieved through fast convolution, which is a computational technique that facilitates efficient processing of large datasets. Utilizing this technique, the CP learning rule can be applied to a group of mixtures \(\mathbf{x}(t)\), enabling the search for the de-mixing vector \(\mathbf{w}_{i}\) that maximizes the objective function \(F(\cdot)\). This process can use the classic gradient ascent technique. On the basis of Equation (13), the derivative of \(F\) with respect to \(\mathbf{w}_{i}\) can be expressed as  
\[\
abla_{\mathbf{w}_{i}}F=\frac{2\mathbf{w}_{i}}{\mathbf{V}_{i}}\overline{\mathbf{R}}-\frac{2 \mathbf{w}_{i}}{\mathbf{U}_{i}}\hat{\mathbf{R}}. \tag{15}\]  
The objective function \(F\) in Equation (15) can be expressed as a ratio of quadratic forms, which has a unique global maximum or minimum. To extract all latent sources simultaneously, [PERSON] [26] proposed an effective algorithm, which is described below. The gradient of \(F\) reaches zero in the solution, where  
\[\
abla_{\mathbf{w}_{i}}F=\frac{2\mathbf{w}_{i}}{\mathbf{V}_{i}}\overline{\mathbf{R}}-\frac{2 \mathbf{w}_{i}}{\mathbf{U}_{i}}\hat{\mathbf{R}}=0. \tag{16}\]  
This can be rewritten as  
\[\overline{\mathbf{R}}\mathbf{w}_{i}=\frac{\mathbf{V}_{i}}{\mathbf{U}_{i}}\hat{\mathbf{R}}\mathbf{w}_{ i}. \tag{17}\]  
The matrix \(\mathbf{W}=(\mathbf{w}_{1},\,\mathbf{w}_{2},\,...,\,\mathbf{w}_{M})^{\text{T}}\) can be obtained using a generalized eigenvalue routine, such as using the eigenvalue function \(\mathbf{W}^{\text{T}}=\text{eig}\,(\overline{\mathbf{R}},\hat{\mathbf{R}})\). As mentioned above, the eigenvector defines the maximum in \(F\). Therefore, by leveraging temporal predictability, CP can accurately extract the time-domain modal responses \(\mathbf{q}(t)\) that represent the component sources that are concealed within the system motion mixture \(\mathbf{x}(t)\):  
\[\mathbf{q}(t)=\mathbf{W}\mathbf{x}(t). \tag{18}\]  
On the basis of Equation (5), it is possible to obtain the mode shape \(\Phi\):  
\[\Phi=\mathbf{W}^{-1}. \tag{19}\]  
Figure 2 depicts the general flow of the mode extraction process using CP.