## 3 Charrelation Matrix Based Blind Detector for DS-CDMA System
### Charrelation Matrix Based Blind Source Separation  
Taking into account the blind source separation mixture model linking with DS-CDMA system from above discussion, the system model is rewritten as:  
\[\mathbf{r}\big{(}m\big{)}=\mathbf{Gb}\big{(}m\big{)}+\mathbf{n}\big{(}m\big{)},m =1,...,M \tag{8}\]  
from the perspective of blind separation framwork, the stochastic vector \(\mathbf{r}\big{(}m\big{)}\!\in\!\mathbb{R}^{{}^{C}}\) represents observation signals, and \(\mathbb{R}\) denotes real field. The components of the stochastic vector \(\mathbf{b}\big{(}m\big{)}\!\in\!\mathbb{R}^{{}^{K}}\) correspond to the unobserved source signals. \(\mathbf{n}\big{(}m\big{)}\!\in\!\mathbb{R}^{{}^{C}}\) denotes additive Gaussian white noise. The unknown mixing matrix characterizes the way the sources are combined in the observation. The goal of blind separation consists of estimating the mixing matrix \(\mathbf{G}\!\in\!\mathbb{R}^{{}^{C\!K}}\) from the observations and recovering the source signals, on the assumption that the source signal are non-Gaussian and statistically independent. From the single antenna/sensor DS-CDMA reception point of view, in order to have a standard blind separation model available in the receiver, the number of users \(K\) can be at most \(C\), _i.e._, \(K\!\leq\!C\). That is to say, the mixing matrix \(\mathbf{G}\) is full column rank. Equation (8) is an overdetermined or determined BSS model for \(K\!\leq\!C\) in mixing matrix \(\mathbf{G}\). In the process of blind separation, the whitening operation is always implemented to convert a model Equation (8) as:  
\[\mathbf{\tilde{r}}\big{(}m\big{)} =\mathbf{Q}\mathbf{r}\big{(}m\big{)} \tag{9}\] \[=\mathbf{\tilde{G}}\mathbf{b}\big{(}m\big{)}+\mathbf{\tilde{n}} \big{(}m\big{)}\]  
where \(\mathbf{Q}\) is a whitening matrix which is derived in following Section 3.3. After whitening operation, we can obtain that \(\mathbf{\tilde{r}}\big{(}m\big{)}\!\in\!\mathbb{R}^{{}^{K}}\), \(\mathbf{\tilde{G}}\!\in\!\mathbb{R}^{{}^{K\!K}}\) and \(\mathbf{\tilde{n}}\!\in\!\mathbb{R}^{{}^{K}}\). Next, the principle of blind separation based on charrelation matrices is illustrated. Let \(\mathbf{u}\!\in\!\mathbb{R}^{{}^{K}}\) denote an arbitrary (deterministic) vector, called \"processing point\". The generalized characteristic function and the generalized second characteristic function of the obesevation vector are defined respectively as:  
\[\phi_{\mathbf{\hat{r}}}\left(\mathbf{u}\right)\!\triangleq\!E\Big{[}\mathrm{ exp}\Big{(}\mathbf{u}^{\mathrm{ T}}\mathbf{\tilde{r}}\big{(}m\big{)}\Big{)}\!\Big{]} \tag{10}\]  
\[\varphi_{\mathbf{\hat{r}}}\left(\mathbf{u}\right)\!\triangleq\!\log\!\left( \phi_{\mathbf{\hat{r}}}\left(\mathbf{u}\right)\right) \tag{11}\]  
\(E\big{[}\cdot\big{]}\) denotes the expectation operation. Next we discuss the way to estimate the mixing matrix based on the charrelation matrix of the observations. Replacing \(\mathbf{\tilde{r}}\big{(}m\big{)}\) by its representation and negelecting the noise contribution gets:  
\[\phi_{\mathbf{\hat{r}}}\left(\mathbf{u}\right)=E\Big{[}\mathrm{exp}\Big{(} \mathbf{u}^{\mathrm{T}}\mathbf{Gb}\big{(}m\big{)}\Big{)}\!\Big{]}=\phi_{ \mathbf{b}}\left(\mathbf{G}^{\mathrm{T}}\mathbf{u}\right) \tag{12}\]  
Furthermore, using the independence property of source vector, the generalized second characteristic function can be rewritten as:  
\[\varphi_{\mathbf{\hat{r}}}\left(\mathbf{u}\right)=\varphi_{\mathbf{\hat{b}}} \left(\mathbf{G}^{\mathrm{T}}\mathbf{u}\right)=\sum_{i=1}^{K}\varphi_{\mathbf{ \hat{r}}_{\mathbf{\hat{b}}}}\left(\mathbf{g}_{i}^{\mathrm{T}}\mathbf{u}\right) \tag{13}\]where \(\mathbf{g}_{{}_{i}}\) is column vector of \(\mathbf{G}\).