## 3 Proposed Algorithm
The \(\mathbf{L}\) in (7) is the iteration matrix of the forward AOR scheme given by  
\[\mathbf{L}(r,w)=(\mathbf{D}-r\mathbf{L})^{-1}[(1-w)\mathbf{D}+(w-r)\mathbf{L}+w \mathbf{U}] \tag{8}\]  
**Step 4:** Compute the second half-iteration as follows:  
\[\mathbf{s}^{(i+1)}=\mathbf{U}(r,w)\mathbf{s}^{(i+1/2)}+w(\mathbf{D}-r\mathbf{ L})^{-1}\mathbf{y}^{MF} \tag{9}\]  
where \(\mathbf{U}\) is the iteration matrix of the backward AOR technique, represented as  
\[\mathbf{U}(r,w)=(\mathbf{D}-r\mathbf{U})^{-1}[(1-w)\mathbf{D}+(w-r)\mathbf{U} +w\mathbf{L}] \tag{10}\]  
Compared to the AOR detector, the SAOR detector converges faster since it is an improved version of AOR, as shown in (7) and (9). Moreover, it holds a significant advantage over AOR due to its symmetric iteration. This symmetry allows for the effective application of CG acceleration, which will be discussed later in this section. For convenience, (7) and (9) can be combined as follows:  
\[\mathbf{s}^{(i+1)}=\mathbf{X}(r,w)\mathbf{s}^{(i)}+\mathbf{t} \tag{11}\]  
where \(\mathbf{X}(r,w)\) is the SAOR iteration matrix given by  
\[\mathbf{X}(r,w)=\mathbf{U}(r,w)\mathbf{L}(r,w) \tag{12}\]  
and  
\[\mathbf{t}=w\mathbf{U}(r,w)(\mathbf{D}-r\mathbf{L})^{-1}\mathbf{y}^{MF}+w( \mathbf{D}-r\mathbf{U})^{-1}\mathbf{y}^{MF} \tag{13}\]  
We summarize the proposed SAOR detection method in Algorithm 1. ```
1 Input: iter, \(\mathbf{H}\), \(\mathbf{y}^{MF}\), \(\sigma^{2}\), \(E_{s}\)
2 Output: Estimated signal, \(\mathbf{\hat{s}}\);
3 Initializations:
4\(\mathbf{\hat{y}}=\mathbf{H}^{H}\mathbf{y}^{MF}\)
5\(\mathbf{A}=\mathbf{H}^{H}\mathbf{H}+\frac{\sigma^{2}}{E_{s}}\mathbf{I}_{ \mathbf{K}}\)
6\(\mathbf{A}=\mathbf{U}+\mathbf{D}+\mathbf{L}\)
7\(\mathbf{s}^{(0)}=\mathbf{D}^{-1}\mathbf{\hat{y}}\)
8 Select acceleration parameter \(r\)
9 Select over-relaxation parameter \(w\)
10\(\mathbf{L}(r,w)=(\mathbf{D}-r\mathbf{L})^{-1}[(1-w)\mathbf{D}+(w-r)\mathbf{L} +w\mathbf{U}]\)
11\(\mathbf{U}(r,w)=(\mathbf{D}-r\mathbf{U})^{-1}[(1-w)\mathbf{D}+(w-r)\mathbf{ U}+w\mathbf{L}]\)
12\(\mathbf{X}(r,w)=\mathbf{U}(r,w)\mathbf{L}(r,w)\)
13\(\mathbf{t}=w\mathbf{U}(r,w)(\mathbf{D}-r\mathbf{L})^{-1}\mathbf{y}^{MF}+w( \mathbf{D}-r\mathbf{U})^{-1}\mathbf{y}^{MF}\)
14 Iteration:
15 for\(i=1,\dots,iter\)do
16\(\mathbf{s}^{(i)}=\mathbf{X}(r,w)\mathbf{s}^{(i-1)}+\mathbf{t}\)
17 Endfor
```  
**Algorithm 1**SAOR detector Next, we propose an enhanced low-complexity detection algorithm that addresses the shortcomings of existing methods. We apply the CG acceleration procedure to the SAOR method to further improve convergence rates and computational efficiency. The \(i\)th iterated vector of the CGA-SAOR method can be expressed as  
\[\mathbf{A}=\mathbf{U}+\mathbf{D}+\mathbf{L} \tag{5}\]0  
It can be observed that the CGA-SAOR algorithm requires the computation of critical parameters, such as the pseudo-residual vector \(\mathbf{\Sigma}^{(i)}\), \(\rho_{i+1}\), and \(v_{i+1}\). Each of these parameters is essential in guiding the iterative process of the CGA-SAOR algorithm, which enhances its convergence properties. These parameters will be defined later. The pseudo-residual vector is defined by  
\[\mathbf{A}=\mathbf{U}+\mathbf{D}+\mathbf{L} \tag{5}\]1  
where \(\mathbf{t}\) is defined in (13). To compute \(\mathbf{s}^{(i+1)}\) and the pseudo-residual vector, we need an initial estimate \(\mathbf{s}^{(1)}\). This initial estimate can be obtained using the proposed SAOR algorithm, ensuring a robust starting point for iterative refinement. In subsequent iterations, the pseudo-residual vector is calculated to evaluate the accuracy of the current estimate and guide the update process toward the true solution. This iterative process is designed to continue until the estimates converge to a satisfactory level of accuracy, thus ensuring that the final result is both precise and computationally efficient. The CG parameters \(\rho_{i+1}\) and \(v_{i+1}\) in (14) are given as [34]  
\[\mathbf{A}=\mathbf{U}+\mathbf{D}+\mathbf{L} \tag{5}\]2  
and  
\[