## 3 Charrelation Matrix Based Blind Detector for DS-CDMA System
### Performance Analysis  
In this subsection, we evaluate the blind separation performance of the proposed algorithm compared with the conventional scheme (see details in the simulation analysis) and the HOS-based JADE algorithm. Due to the fact the charrelation matrix incorporates HOS characteristics, which can suppress the Gaussian noise [25]. Therefore, the proposed blind scheme can improve the system performance compared to the conventional scheme in the noise environment. Moreover, the proposed algorithm exploits a \"hybrid statistics\" (SOS and HOS) manner to extract statistical information, which can acquire more perfect estimated information compared to the JADE with HOS-based manner when the length of signal samples is not enough. In addition, we know that the principle of JADE is joint diagonalization of estimated fourth-order cumulant eigenmatrices [1]. The principle of the proposed method is joint \begin{table}
\begin{tabular}{l l} \hline (a) & The whitening preprocessing of the observation using Equations (21) and (24); \\ (b) & Initialize processing point \(\mathbf{u}_{{}_{i}},i=1...L\)  from the range of [-1, 1]; \\ (c) & Estimate charrelation matrices \(\mathbf{\Psi}_{\text{r}}\left(\mathbf{u}\right)\) of the whitened signal using Equation (20); \\ (d) & Carry out WEDGE joint diagonalization for optimizing problem Equation (16); \\ (e) & Seek separation matrix and estimate users’ original signal of DS-CDMA using \\  & Equation (25), order ambiguity eliminated by Equations (27) and (28); \\ (f) & Based on the model (7), to estimate spreading sequence of DS-CDMA using \\  & Equations (30)–(32). \\ \hline \end{tabular}
\end{table} 
Table 2: Blind Adaptive Detector for DS-CDMA system. diagonalization of estimated charrelation matrices. From the perspective of computational complexity, the cumulant matrix needs four derivatives of the second characteristic function compared to the two derivatives in charrelation matrix. Finally, the computational complexity of the joint diagonalization method is also considered for both these blind separation algorithms. The computational complexity of the joint diagonalization method of the JADE and the proposed method is \(O\left(KM^{{}^{4}}\right)\)[1] and \(O\left(KM^{{}^{3}}\right)\)[28] respectively. Thus we can know the new method is computed simply and extracts perfect information from samples of observation signals. From the previous analysis, we can evaluate the improved blind separation performance that can be obtained. In order to carry out more performance assessments, the performance in terms of ISR is displayed in mathematical analysis. The ISR is computed from the estimated mixing matrix by post-multiplying its inverse by the true mixing matrix and averaging the minimum to maximum power ratio in each row of the results. More specifically, if we defined \(\mathbf{V}=\mathbf{W}\tilde{\mathbf{G}}\) as the resulting overall \"contamination matrix\", \(ISR_{{}_{ij}}\triangleq E\left[\mathbf{V}_{{}_{ij}}^{2}\right]\) is the residual mean contaminating power of source \(j\) in the reconstruction of source \(i\). Thus, in the vicinity of a non-mixing condition \(\tilde{\mathbf{G}}=\mathbf{I}_{{}_{K}}\), it is easy to observe that \(ISR_{{}_{ij}}\approx E\left[\mathbf{W}_{{}_{ij}}^{2}\right]\). Under the small-errors assumption and sub-Gaussian source signal considered (most communication signals are sub-Gaussian signals), the covariance in the estimation of the elements of \(\tilde{\mathbf{G}}\), and hence the ISR can be predicted.