## 3 Proposed Algorithm
\begin{cases}\rho_{1}=1,\\ \rho_{i+1}=\left[1-\frac{v_{i+1}}{v_{i}}\left(\frac{\mathbf{\Sigma}^{(i)} \mathbf{\Sigma}^{(i)}}{\left(\mathbf{\Sigma}^{(1)}\cdot\mathbf{\Sigma}^{(1)} \cdot\mathbf{\Sigma}^{(1)}\right)}\right)\frac{1}{\rho_{i}}\right]^{-1}.\end{cases}  \tag{17}\]  
Importantly, the proposed CGA-SAOR detection method maintains a history of two vectors instead of just one, which enhances stability and performance. This recursive nature of the CGA-SAOR detector leads to significantly improved results compared to conventional methods. From (15) and (17), we have  
\[\mathbf{\Sigma}^{(i+1)}=\rho_{i+1}\Big{[}v_{i+1}\mathbf{X}(r,w)\mathbf{\Sigma }^{(i)}+(1-v_{i+1})\mathbf{\Sigma}^{(i)}\Big{]}+(1-\rho_{i+1})\mathbf{\Sigma} ^{(i-1)} \tag{18}\]  
To compute the iteration and parameters of the proposed algorithm, there are specific initial settings required, such as \(\rho_{1}\), \(\mathbf{\Sigma}^{(0)}\), \(\mathbf{\Sigma}^{(1)}\), and \(v_{1}\). It is shown in the results section that the proposed CGA-SAOR detector is more robust to higher-order modulation techniques and can be effectively applied for various antenna settings compared to conventional iterative methods. The proposed CGA-SAOR detection method is summarized in Algorithm 2. ```
1 Input: iter, \(\mathbf{H}\), \(\mathbf{y}^{MF}\), \(\sigma^{2}\), \(E_{s}\) Output: Estimated signal, \(\mathbf{\hat{s}}\);
3 Initializations: \(\mathbf{\hat{y}}=\mathbf{H}^{H}\mathbf{y}^{MF}\) \(\mathbf{A}=\mathbf{H}^{H}\mathbf{H}+\frac{\sigma^{2}}{E_{s}}\mathbf{I}_{ \mathbf{K}}\) \(\mathbf{A}=\mathbf{U}+\mathbf{D}+\mathbf{L}\) \(\mathbf{s}^{(0)}=\mathbf{D}^{-1}\mathbf{\hat{y}}\) Select acceleration parameter \(r\) Select over-relaxation parameter \(w\) \(\rho_{1}=1\) \(\mathbf{L}(r,w)=(\mathbf{D}-r\mathbf{L})^{-1}[(1-w)\mathbf{D}+(w-r)\mathbf{L}+ w\mathbf{U}]\) \(\mathbf{U}(r,w)=(\mathbf{D}-r\mathbf{U})^{-1}[(1-w)\mathbf{D}+(w-r)\mathbf{U}+ w\mathbf{L}]\) \(\mathbf{X}(r,w)=\mathbf{U}(r,w)\mathbf{L}(r,w)\) \(\mathbf{t}=w\mathbf{U}(r,w)(\mathbf{D}-r\mathbf{L})^{-1}\mathbf{y}^{MF}+w( \mathbf{D}-r\mathbf{U})^{-1}\mathbf{y}^{MF}\) \(\mathbf{\Sigma}^{(\mathbf{0})}=\mathbf{X}(r,w)\mathbf{s}^{(0)}+\mathbf{t}- \mathbf{s}^{(0)}\) \(\mathbf{s}^{(1)}=\mathbf{X}(r,w)\mathbf{s}^{(0)}+\mathbf{t}\) \(\mathbf{\Sigma}^{(\mathbf{1})}=\mathbf{X}(r,w)\mathbf{s}^{(1)}+\mathbf{t}- \mathbf{s}^{(1)}\) \(v_{1}=\left[1-\frac{(\mathbf{x}^{(0)},\mathbf{X}(r,w)\mathbf{\Sigma}^{(0)})}{( \mathbf{\Sigma}^{(0)},\mathbf{\Sigma}^{(0)})}\right]^{-1}\) Iteration: for\(i=1,\ldots,iter\)do \(v_{i+1}=\left[1-\frac{(\mathbf{x}^{(i)}\mathbf{X}(r,w)\mathbf{\Sigma}^{(i)})}{( \mathbf{\Sigma}^{(i)}\mathbf{\Sigma}^{(i)})}\right]^{-1}\) \(\rho_{i+1}=\left[1-\frac{v_{i+1}}{v_{1}}\frac{(\mathbf{\Sigma}^{(1)}, \mathbf{\Sigma}^{(1)})}{\left(\mathbf{\Sigma}^{(0)},\mathbf{\Sigma}^{(0)} \frac{1}{\rho_{1}}\right)}\right]^{-1}\) \(\mathbf{s}^{(i+1)}=\rho_{i+1}\Big{(}v_{i+1}\mathbf{\Sigma}^{(1)}+\mathbf{s}^ {(1)}\Big{)}+(1-\rho_{i+1})-\mathbf{s}^{(0)}\) \(\mathbf{\Sigma}^{(\mathbf{i+1})}=\rho_{i+1}\Big{[}v_{i+1}\mathbf{X}(r,w) \mathbf{\Sigma}^{(\mathbf{1})}+(1-v_{i+1})\mathbf{\Sigma}^{(1)}\Big{]}+(1- \rho_{i+1})\mathbf{\Sigma}^{(\mathbf{0})}\) \(\mathbf{\Sigma}^{(\mathbf{0})}=\mathbf{\Sigma}^{(\mathbf{1})}\) \(\mathbf{\Sigma}^{(\mathbf{1})}=\mathbf{\Sigma}^{(\mathbf{i+1})}\) \(\rho_{1}=\rho_{i+1}\) \(v_{1}=v_{i+1}\) End for
```  
**Algorithm 2**Proposed CGA-SAOR algorithm