## 3 Charrelation Matrix Based Blind Detector for DS-CDMA System
### Blind User Separation and Blind Spreading/Chip Sequence Estimation  
Taking into account the problem of blind user separation in a DS-CDMA system, the whitening process will be carried out first in order to simplify the blind separation problem and suppress noise. Using the model Equation (7), the whitening processing is executed as follows:  
The autocorrelation of the observations is described as:  
\[\begin{array}{l}{\bf{R}}={\it{E}}\left[{\bf{X}}{\bf{X}}^{{}^{\rm{{}^{\rm{T}}}} }\right]={\bf{G}}{\it{E}}\left[{\bf{B}}{\bf{B}}^{{}^{\rm{{}^{\rm{T}}}}}\right]{ \bf{G}}^{{}^{\rm{{}^{\rm{T}}}}}+{\sigma}^{2}{\bf{I}}\\ ={\bf{U}}{\bf{\Sigma}}{\bf{U}}^{{}^{\rm{{}^{\rm{T}}}  
The model Equation (24) is a whitened BSS model, \(\mathbf{Q}=\boldsymbol{\lambda}_{\text{B}}^{-\
icefrac{{1}}{{2}}}\mathbf{U}_{\text{B}}^ {T}\) is whitening matrix same as in Equation (9), and \(\tilde{\mathbf{G}}\) is an orthogonal matrix. Therefore, the problem of Equation (16) is a converted into a orthogonal joint diagonalization to seek a mixing matrix. Assume that the separation matrix is \(\mathbf{W}\) obtained after the optimization of Equation (16). Then the goal of blind user separation is realized by:  
\[\tilde{\mathbf{B}}=\mathit{sign}\big{(}\mathbf{W}\tilde{\mathbf{X}}\big{)} \tag{25}\]  
Ideally, there exists the relationship of \(\mathbf{W}\tilde{\mathbf{G}}=\mathbf{I}\). In fact, there exists an inherent ambiguity problem in blind source separation. Therefore, \(\mathbf{V}=\mathbf{W}\tilde{\mathbf{G}}\) is not an identity matrix but rather a generalized permutation matrix. The ambiguity problem includes amplitude and order ambiguity. The amplitude ambiguity will be eliminated if we assume the covariance of \(\mathbf{B}\) satisfies \(E\big{[}\mathbf{B}\mathbf{B}^{T}\big{]}=\mathbf{I}\). After the whitening operation, this condition can be obtained easily. The order ambiguity can be solved based on the following principle:  
The recovered source signals can be given by:  
\[\hat{\mathbf{B}}=\mathbf{W}\tilde{\mathbf{G}}\mathbf{B}=\mathbf{V}\mathbf{B} \tag{26}\]  
where \(\mathbf{V}\) is a generalized permutation matrix, or global matrix, where each column (or row) contains only one non-zero element whose absolute value is 1. It is worthwhile to note that:  
\[E\big{[}\hat{\mathbf{B}}\mathbf{B}^{T}\big{]}=\mathbf{V}E\big{[}\mathbf{B} \mathbf{B}^{T}\big{]} \tag{27}\]  
The global matrix \(\mathbf{V}\) can be estimated by finding the cross-correlation matrix between the vectors of separated signal \(\hat{\mathbf{B}}\) and source signal \(\mathbf{B}\). If an estimate of \(\tilde{\mathbf{V}}\) is acquired, the source signal with proper order can be obtained as:  
\[\mathbf{B}=\tilde{\mathbf{V}}^{T}\hat{\mathbf{B}} \tag{28}\]  
In summary, we use the following steps which are shown in Table 1 to overcome the order ambiguity, where we assume each user transmits the short length \(M_{{}_{p}}\) of the pilot symbols. Next, we will discuss the blind spreading sequence estimate using BSS. According to Equation (7), when the noise \(\mathbf{N}\) is not considered except simulation experiments, we can arrive at:  
\[\mathbf{X}=\mathbf{G}\mathbf{B} \tag{29}\]  
\begin{table}
\begin{tabular}{l l} \hline (a) & Normalize the pilot symbols so that: \\  & \(\Big{(}\
icefrac{{1}}{{\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\  
The autocorrelation of \(\mathbf{X}\) is denoted as \(\mathbf{R}_{\mathbf{X}}\), then \(\mathbf{R}_{\mathbf{X}}\) is executed singular value decomposition (SVD), _i.e._,:  
\[\mathbf{R}_{\mathbf{X}}=\mathbf{UDU}^{T} \tag{30}\]  
we assume that the \(\mathbf{U}_{\text{B}}\in\mathbb{R}^{C\times K}\) denotes the matrix composed of \(K\) main eigenvectors, then the expanded space of column vector of \(\mathbf{U}_{\text{B}}^{T}\in\mathbb{R}^{K\times C}\) belongs to the same space as the expanded space of the column vector of \(\mathbf{G}^{T}\).