## 2 Theories of BSS Based on BCA
### BSS Based on BCA  
Compared to the ICA algorithm, the biggest advantage of the BCA algorithm is that it can separate non-independent signals, which brings more universality to the blind source separation algorithm. Its basic principle is similar to the ICA algorithm, and it also requires some necessary assumptions [21; 22]:  
1. Assume that the mixing matrix A has rank n, i.e., column full rank, i.e., the number of sensors should be greater than or equal to the number of source signals. 2. Assume that the distribution of source signals is bounded. 3. Assume that the branch set S, consisting of the joint distribution of all source signals, can be expressed by the Cartesian product of the branch set Si of each source signal: \[S=S_{1}\otimes S_{2}\otimes\cdots\otimes S_{n}\] (6) where \(n\) is the number of source signals and \(\otimes\) denotes the Cartesian product. Assumptions (2) and (3) are weaker and more loosely constrained compared to the assumptions of ICA. Assumption (3) is automatically satisfied when the signals are independent of each other, so the blind source separation algorithm based on BCA has stronger generalizability. With the help of the analysis above, it is clear that the BCA algorithm's steps are to find the separation matrix \(M\) when the mixed signal \(X\) is known, and then multiply \(M\) by \(X\) to determine \(Y\), which is the estimated value of the source signal \(S\). Then, the process of the BCA algorithm can be simply summarized as solving the separation matrix to approximate the independent source signal, and the specific approximation method is actually an optimization search of the established objective function. Thus, the algorithm can be simply described as follows:  
\[\text{BCA algorithm}=\text{optimization criterion}+\text{optimization-seeking algorithm}\]  
In fact, this criterion is to construct some objective function and then use this function for separation. The selection of the objective function determines the statistical performance of the algorithm, and the optimization algorithm determines the convergence performance, storage requirements, and stability of the algorithm. In this paper, the normalized boundary objective function is selected as follows:  
\[J(y)=\frac{\overset{\cdot}{R(y)}}{\sqrt{\text{E}\{y^{2}\}}} \tag{7}\]  
\(y\) denotes the separation signal, \(\text{E}\{y^{2}\}\) denotes the variance of \(y\), and \(\overset{\cdot}{R}(y)\) is the boundary operator. \[\overset{\cdot}{R}(y)=max(y)-min(y) \tag{8}\]  
where \(max(y)\) and \(min(y)\) denote the maximum values of \(y\) and the minimum values of \(y\)  
Figure 1: Model of linear mixed signal separation. The objective function \(J(y)\) is minimized if and only if the global vector \(W\) is a perfect separator, that is, when \(W\) has only one non-zero component. The proof is given below. Combining with the previous Equation (5) and assumption (3), Equation (8) can be written as  
\[\hat{R}(y)=(W_{+}-W_{-})(max(s)-min(s)) \tag{9}\]  
where \(s\) denotes the source signals, \(max(s);min(s)\) denote the maximum values of \(s\) and the minimum values of \(s\); \(W_{+}\) denotes extracting positive components of \(W\) while setting others to zero; and \(W_{-}\) denotes extracting the negative components of \(W\) while setting others to zero. We define \(\forall=diag(max(s)-min(s))\), and then combining the matrix theory knowledge, we can obtain [6] the following:  
\[\hat{R}(y)=\left\|W\forall\right\|_{1}=\left\|\ell\right\|_{1} \tag{10}\]  
We define \(\ell=W\forall\). The vector \(\ell\) can be written as  
\[\ell=\underset{i=1}{\overset{N}{\sum}}\ell_{i}\varepsilon_{i} \tag{11}\]  
where \(\varepsilon_{i}\) denotes the basic unit vector.