## 3 Proposed Algorithm  
The channel-hardening phenomenon can be leveraged in large MIMO systems due to the significant number of antennas, effectively mitigating the impact of fast fading [31; 32; 33]. This phenomenon becomes more pronounced when the number of antennas at the BS significantly exceeds the number of users. Furthermore, the matrix to be inverted in the MMSE detector, i.e., the MMSE filtering matrix, is positive definite and Hermitian in massive MIMO systems. These properties justify the application of iterative approaches in large wireless systems, enabling the achievement of approximate solutions with substantially lower complexity. Consequently, several approximate iterative methods have been proposed or enhanced to improve performance while reducing computational demands. The AOR iterative technique is a promising scheme for solving linear equations. Hence, the AOR iterative technique has been applied to approximate the linear MMSE detection, thereby lowering computational complexity by circumventing extensive matrix inversions [22]. However, the conventional AOR-based detection algorithm exhibits slow convergence and requires more iterations to achieve the desired detection performance, which, in turn, increases the overall complexity. In this section, we design two efficient iterative detection algorithms, namely SAOR and CGA-SAOR, which address these challenges by enhancing convergence speed and maintaining low computational costs. By rewriting the MMSE estimator (2) as  
\[\mathbf{A\hat{s}}=\mathbf{y}^{\mathrm{MF}}, \tag{4}\]  
Equation (2) is equivalent to solving a system of linear equations. By applying iterative methods, we can solve these systems without computing the matrix inverse, thus streamlining the detection process. We first apply the SAOR scheme to solve (2) iteratively. The SAOR method operates through the following steps:  
**Step 1:** Since \(\mathbf{A}\) is a Hermitian positive definite and symmetric matrix in a massive MIMO system [12], it can be decomposed as  
\[\mathbf{A}=\mathbf{U}+\mathbf{D}+\mathbf{L} \tag{5}\]  
where \(\mathbf{U}\) is the strictly upper triangular matrix, \(\mathbf{D}\) is the diagonal matrix, and \(\mathbf{L}\) is the strictly lower triangular matrix of \(\mathbf{A}\). **Step 2:** Select the initial solution \(\mathbf{s}^{(0)}\). A good initial estimate significantly impacts the convergence rate and can notably reduce the overall number of iterations required to achieve the desired detection accuracy. Given that the computational complexity of each iteration is high in large MIMO systems due to the massive number of antennas, a well-chosen initial solution is essential. We utilize diagonal initialization instead of the classical zero-vector-based approach, as the diagonal dominance of the MMSE filtering matrix in massive MIMO enables this method to achieve the desired performance with fewer iterations. The diagonal initialization can be expressed as  
\[\mathbf{s}^{(0)}=\mathbf{D}^{-1}\mathbf{y}^{\mathrm{MF}} \tag{6}\]  
**Step 3:** Compute the first half-iteration as follows:  
\[\mathbf{s}^{(i+1/2)}=\mathbf{L}(r,w)\mathbf{s}^{(i)}+w(\mathbf{D}-r\mathbf{L} )^{-1}\mathbf{y}^{\mathrm{MF}} \tag{7}\]  
where \(i\) is the iteration number, \(\mathbf{s}^{(0)}\) is the initial solution, and \(r\) and \(w\) are the acceleration and over-relaxation parameters, respectively. The selection of these parameters is critical, as they greatly influence both convergence and convergence rates.