# A Novel Complex-Valued Blind Source Separation and Its Applications in Integrated Reception
## 4 Objective Function  
At present, the objective function commonly used in CBSS is kurtosis maximization in a complex domain. However, kurtosis needs to be averaged many times, which requires a large amount of computation when there are many snapshots. Therefore, this paper chooses the F-norm in joint diagonalization as the objective function. It only needs to multiply the matrix, which can greatly reduce the running time. For a given set of diagonalizable target matrices \(M=\left\{\mathbf{M}^{i},l=1,\ldots,L\right\}\), where \(\mathbf{M}^{i}\) is a square matrix of \(K\times K\), the goal of the F-norm objective function is to solve \(\mathbf{W}\) by minimizing the following equation  
\[J(\mathbf{W})=\sum_{k=1}^{K}\text{off}\left(\mathbf{W}\mathbf{M}^{k}\mathbf{W}^{\text{H}}\right) \tag{26}\]  
where \(\text{off}(\cdot)\) is the sum of the F-norms of all non-diagonal elements, expressed as  
\[\text{off}(\mathbf{F})\triangleq\parallel\mathbf{F}-diag(\mathbf{F})\parallel_{\text{F}} ^{2}=\sum_{i\
eq i}\!F_{ij}^{2} \tag{27}\]  
where \(\mathbf{F}\) is an arbitrary matrix. Obviously, the F-norm cost function is to measure the degree to which the matrix group \(\mathbf{W}\mathbf{M}^{k}\mathbf{W}^{H}\) deviates from the diagonal matrix to solve the optimal solution. \(\mathbf{W}=\mathbf{0}\) is a global minimal solution satisfying this cost function if there is no additional constraint. However, \(\mathbf{W}=\mathbf{0}\) is called a trivial solution. Constrained diagonalization matrix \(\mathbf{W}\) is a unitary matrix, i.e., \(\mathbf{W}\mathbf{W}^{\text{H}}=\mathbf{I}\). Obviously, the diagonalized matrix can avoid a trivial solution. At present, there are four-order cumulants of statistically independent sources and second-order statistics of statistically uncorrelated sources to obtain joint diagonalized matrices. Because the fourth-order cumulant of Gaussian white noise is zero, it has a strong anti-noise ability. Moreover, by whitening and selecting multiple cumulant matrices for joint diagonalization, a very robust solution can be obtained. Therefore, the fourth-order cumulant of the whitening signal is used to obtain the joint diagonalization matrix group. The (\(p\),\(\rho\)) element in the fourth-order cumulant matrix \(\mathbf{Q}_{z}(\mathbf{M})\) of the whitening signal \(\mathbf{z}(t)\) is  
\[\left[\mathbf{Q}_{z}(\mathbf{M})\right]_{p,q}=\sum_{i=1}^{K}\sum_{o=1}^{K}\mathbf{Q}_{z}(p,q, i,o)m_{i,o} \tag{28}\]  
where \(\mathbf{M}\) is any matrix of \(K\times K\) and \(m_{i,o}\) is the (\(i\),\(\rho\)) element of matrix \(\mathbf{M}\). The fourth-order cumulant of whitening signal \(\mathbf{z}(t)\) is  
\[ \begin{array}{l}\mathbf{Q}_{z}(p,q,i,o)=cum\big{(}z_{p}(t),z_{i}^{*}(t),z_{q}(t),z_{o}^{*}(t)\big{)},\\ p,q=1,2,\cdots,K\end{array}  \tag{29}\]  
where \(z^{*}\) represents the conjugate of \(z\), and \(z_{q}(t)\) is the \(i\)-th line of whitening signal \(\mathbf{z}(t)\). \(cum(\cdot)\) is the fourth-order cumulant operation  
\[ \begin{array}{c}cum\big{(}z_{p}(t),z_{i}^{*}(t),z_{q}(t),z_{o}^{*}(t)\big{)} \\ =E\big{(}z_{p}(t)z_{q}(t)z_{i}^{*}(t)\big{)}-E\big{(}z_{p}(t)z_{i}^{*}(t)\big{)} E\big{(}z_{q}(t)z_{o}^{*}(t)\big{)}\\ -E\big{(}z_{p}(t)z_{o}^{*}(t)\big{)}E\big{(}z_{q}(t)z_{i}^{*}(t)\big{)}-E\big{(} z_{p}(t)z_{q}(t)\big{)}E\big{(}z_{i}^{*}(t)\big{)}\end{array}  \tag{30}\]  
Obviously, the number of elements of \(\mathbf{Q}_{z}(p,q,i,o)\) is \(K\times K\times K\times K\). \(\mathbf{Q}_{z}\) is constructed as an \(K^{2}\times K^{2}\) dimensional matrix \(\mathbf{H}_{z}\), where  
\[\mathbf{H}_{z}(n_{1},n_{2})=\mathbf{Q}_{z}(p,q,k,l) \tag{31}\]  
where \(n_{1}=p+(q-1)K\), \(n_{2}=o+(i-1)K\). The \(K^{2}\times 1\) eigenvectors corresponding to \(K\) maximum eigenvalues \(\lambda_{k}\) of matrix \(\mathbf{H}_{z}\) are \(\mathbf{u}_{k}(k=1,2,\cdots K)\), and \(\mathbf{u}_{k}\) is the performed matrix operation to obtain \(vec^{-1}(\mathbf{u}_{k})\). By the joint diagonalization of \(\mathbf{M}^{k}=\lambda_{k}vec^{-1}(\mathbf{u}_{k})\), the unmixing matrix can be obtained.