# A Novel Complex-Valued Blind Source Separation and Its Applications in Integrated Reception
## 4 Objective Function
Unitary matrices can be expressed in the form of the continuous multiplication of Givens matrices and extended to the complex field to express the following complex Givens matrices. \[ \begin{array}{c}\mathbf{W}=T_{K-1}T_{K-2}\cdots T_{1},\\ \mathbf{T}_{1}=T_{1,K}T_{1,K-1}\cdots T_{1,2},\\ \mathbf{T}_{2}=T_{2,K}T_{2,K-1}\cdots T_{2,3},\cdots\\ \mathbf{T}_{K-1}=T_{K-1,K2}\end{array}  \tag{32}\]  
where  
\[\mathbf{T}_{\rho,\mu}= \begin{pmatrix}1&\cdots&0&\cdots&0&\cdots&0\\ \vdots&\ddots&\vdots&&\vdots&&\vdots\\ 0&\cdots&\mathrm{c}\mathrm{e}^{\mathrm{i}\theta_{1}}&\cdots&-d\mathrm{e}^{ \mathrm{i}\theta_{2}}&\cdots&0\\ \vdots&&\vdots&\ddots&\vdots&&\vdots\\ 0&\cdots&d\mathrm{e}^{\mathrm{i}\theta_{3}}&\cdots&c\mathrm{e}^{\mathrm{i} \theta_{4}}&\cdots&0\\ \vdots&&\vdots&&\vdots&\ddots&\vdots\\ 0&\cdots&0&\cdots&0&\cdots&1\end{pmatrix} _{N\times N} \tag{33}\]  
where \(K\) is the matrix order, i.e., the number of sources. \(c=\cos\theta\),\(d=\sin\theta\),\(\theta\in(0,2\pi)\) is the rotation angle and satisfies \(\theta_{1}+\theta_{4}=\theta_{2}+\theta_{3}\). Because of the constraint \(\theta_{1}+\theta_{4}=\theta_{2}+\theta_{3}\), one unknown parameter can be reduced. Therefore, the number of unknown parameters of each complex Givens matrix is 4. \(\mathbf{T}_{1}\) contains \(K-1\) complex Givens matrices, \(\mathbf{T}_{2}\) contains \(2K\) complex Givens matrices, and so on; \(\mathbf{T}_{N-1}\) contains \(1\) complex Givens matrix. Each complex Givens matrix contains 4 unknown parameters, so the number of unknown parameters of \(\mathbf{W}\) is \[ \begin{split}(1+2+\cdots+(K-2)+(K-1))\cdot 4\\ =\frac{(1+(K-1))(K-1)}{2}\cdot 4\\ =\frac{K(K-1)}{2}\cdot 4=4C_{K}^{2}\end{split}  \tag{34}\]  
where C is a combination. If \(W\) is not represented by the concatenation of complex Givens matrices, since \(W\) is of order \(K\) and each element is in complex form with 2 unknown parameters, the number of unknown parameters included is \(2K^{2}\). Therefore, the number of unknown parameters that can be reduced \(2K^{2}-4C_{K}^{2}=2K\). For 4 source signals, \(W=T_{3}T_{2}T_{1}=T_{12}T_{13}T_{14}T_{23}T_{24}T_{34}\) can be expressed as follows:  
\[ \begin{split} W=&\left[\begin{array}{cccc}\cos\varphi_{ 1}e^{i\varphi_{2}}&-\sin\varphi_{1}e^{i\varphi_{3}}&0&0\\ \sin\varphi_{1}e^{i\varphi_{4}}&\cos\varphi_{1}e^{i(\varphi_{3}+\varphi_{4}- \varphi_{2})}&0&0\\ 0&0&1&0\\ 0&0&0&1\end{array}\right]\cdot\left[\begin{array}{cccc}\cos\varphi_{5}e^{i \varphi_{6}}&0&-\sin\varphi_{5}e^{i\varphi_{7}}&0\\ 0&1&0&0\\ \sin\varphi_{5}e^{i\varphi_{8}}&0&\cos\varphi_{5}e^{i(\varphi_{7}+\varphi_{8}- \varphi_{6})}&0\\ 0&0&0&1\end{array}\right]\cdot\\ &\left[\begin{array}{cccc}\cos\varphi_{9}e^{i\varphi_{10}}&0&0&-\sin \varphi_{9}e^{i\varphi_{11}}\\ 0&1&0&0\\ 0&0&1&0\\ \sin\varphi_{9}e^{i\varphi_{12}}&0&0&\cos\varphi_{9}e^{i(\varphi_{12}+\varphi_ {11}-\varphi_{10})}\\ 0&0&0&1\end{array}\right]\cdot\left[\begin{array}{cccc}1&0&0&0\\ 0&\cos\varphi_{13}e^{i\varphi_{14}}&-\sin\varphi_{13}e^{i\varphi_{15}}&0\\ 0&\sin\varphi_{13}e^{i\varphi_{16}}&\cos\varphi_{13}e^{i(\varphi_{16}+\varphi_ {15}-\varphi_{14})}&0\\ 0&0&0&1\end{array}\right]\cdot\\ &\left[\begin{array}{cccc}1&0&0&0\\ 0&\cos\varphi_{17}e^{i\varphi_{18}}&0&-\sin\varphi_{17}e^{i\varphi_{19}}\\ 0&0&1&0\\ 0&0&1&0\\ 0&\sin\varphi_{17}e^{i\varphi_{20}}&0&\cos\varphi_{17}e^{i(\varphi_{20}+ \varphi_{19}-\varphi_{18})}\end{array}\right]\cdot\left[\begin{array}{cccc}1&0 &0&0\\ 0&1&0&0\\ 0&0&\cos\varphi_{21}e^{i\varphi_{22}}&-\sin\varphi_{21}e^{i\varphi_{23}}\\ 0&0&\sin\varphi_{21}e^{i\varphi_{24}}&\cos\varphi_{21}e^{i(\varphi_{24}+ \varphi_{23}-\varphi_{22})}\end{array}\right]\end{split} \right] \tag{35}\]  
Based on the analysis above, the proposed algorithm converts the solution into the rotation angle in the complex Givens matrix. This algorithm not only guarantees the orthogonality of the separation matrix in the complex domain and enhances the accuracy of the solution, but also reduces the number of unknown parameters.