## 3 Charrelation Matrix Based Blind Detector for DS-CDMA System
### Charrelation Matrix Based Blind Source Separation
As a consequence, deriving the charrelation matrix \(\Psi_{{}_{\mathbf{i}}}\left(\mathbf{u}\right)\) which is obtained by calculating the second derivative of \(\varphi_{{}_{\mathbf{i}}}\left(\mathbf{u}\right)\) with respect to \(\mathbf{u}\), we can obtain the following charrelation matrix \(\Psi_{{}_{\mathbf{i}}}\left(\mathbf{u}\right)\) (the details are illustrated in Appendix A):  
\[\Psi_{{}_{\mathbf{i}}}\left(\mathbf{u}\right)=\mathbf{G}\Psi_{{}_{\mathbf{b}} }\left(\mathbf{G}^{{}_{\mathbf{f}}}\mathbf{u}\right)\mathbf{G}^{{}_{\mathbf{f}}} \tag{14}\]  
with:  
\[\Psi_{{}_{\mathbf{i}}}\left(\mathbf{u}\right)=\
abla_{{}_{\mathbf{u}^{\mathbf{ \mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{ \mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{     }}}}}}}}}}}}}}} \left[\
abla_{{}_{\mathbf{u}}}\varphi_{{}_{\mathbf{i}}}\left(\mathbf{u} \right)\right]=\frac{\partial}{\partial\mathbf{u}^{{}_{\mathbf{f}}}}\Bigg{[} \frac{\partial\varphi_{{}_{\mathbf{i}}}\left(\mathbf{u}\right)}{\partial \mathbf{u}}\Bigg{]} \tag{15}\]  
It is worthwhile to note that \(\Psi_{{}_{\mathbf{b}}}\left(\mathbf{G}^{{}_{\mathbf{f}}}\mathbf{u}\right)\) is a diagonal matrix (the details are illustrated in Appendix B). In a determined blind separation model, source signals are usually separated by multiplying the observations with the pseudo-inverse/inverse of mixing matrix estimate. The estimation of the mixing matrix can be carried out by approximate joint diagonalization (AJD) of a series of charrelation matrices. By choosing a set of processing points \(\left\{\mathbf{u}_{{}_{\mathbf{i}}},\mathbf{u}_{{}_{\mathbf{2}}},\ldots, \mathbf{u}_{{}_{L}}\right\}\), we can construct a series of charrelation matrices that obey the transformation Equation (14). Then the estimation problem of the mixing matrix can be described as the following joint diagonalization [26, 27]:  
\[\min_{\mathbf{G}}\sum_{i=1}^{L}\left|\left|\Psi_{{}_{\mathbf{i}}}\left( \mathbf{u}_{{}_{\mathbf{i}}}\right)-\mathbf{G}\Psi_{{}_{\mathbf{b}}}\left( \mathbf{G}^{{}_{\mathbf{f}}}\mathbf{u}_{{}_{\mathbf{i}}}\right)\mathbf{G}^{{} _{\mathbf{f}}}\right|\right|_{{}_{F}}^{2} \tag{16}\]  
where \(\left|\left|\right|\right|_{{}_{F}}^{2}\) denotes the squared Frobenius norm. In BSS, the diagonal matrices \(\Psi_{{}_{\mathbf{b}}}\left(\mathbf{G}^{{}_{\mathbf{f}}}\mathbf{u}\right)\) contain some statistical or structural properites of sources. The \"target matrices\" \(\Psi_{{}_{\mathbf{f}}}\left(\mathbf{u}\right)\) usually denotes estimates of similar matrices pertaining to the observed mixtures. The diagonalization of the matrices \(\Psi_{{}_{\mathbf{b}}}\left(\mathbf{G}^{{}_{\mathbf{f}}}\mathbf{u}\right)\), which is often attributed to the statistical independence of the sources, serves as the key to identifiability of mixing matrix \(\mathbf{G}\) from the matrices \(\Psi_{{}_{\mathbf{f}}}\left(\mathbf{u}\right)\). The optimization of Equation (16) is a well-known joint diagonalization problem. A number of joint diagonalization methods have been reported, such as in the literatures [26, 27]. Among these, a state-of-the-art algorithm called weight exhaustive diagonalization using Gauss iteration (WEDGE) is used to minimize the criterion in Equation (16). The WEDGE approach will not be illustrated in this paper, as details are provided in [26].