## 3 Proposed Scheme
### Iteration Method Review
#### 3.1.2 AOR Method  
The accelerated over-relaxation (AOR) iterative algorithm [37] can be regarded as an extension of the successive over-relaxation (SOR) iterative algorithm, and its iterative equation is as follows:  
\[\mathbf{x}^{(k+1)} =(D-\gamma L)^{-1}[(1-\omega)D+\omega U+(\omega-\gamma)L]\mathbf{ x}^{(k)}\] \[+\omega(D-\gamma L)^{-1}\mathbf{b}, \tag{13}\]  
where \(\omega\) and \(\gamma\) are the relaxation parameter and the acceleration parameter, respectively. Furthermore, for specific values of \(\omega\) and \(\gamma\), when \(\omega=\gamma\) or \(\omega=\gamma=1\), the AOR iterative algorithm can reduce to the SOR or the Gauss-Seidel iterative algorithm, respectively. According to _Lemma_1 in [27], in uplink large-scale MIMO, the conventional MMSE filtering matrix \(W\) is a symmetric positive definite matrix, and it can be decomposed as  
\[W=D+L+L^{H}. \tag{14}\]where \(D\), \(L\), and \(L^{H}\) are the diagonal, strict lower part, and strict upper part of \(W\). Therefore, we can use the AOR iterative algorithm to rewrite the conventional MMSE algorithm as follows:  
\[\mathbf{x}^{(k+1)} =(D+\gamma L)^{-1}[(1-\omega)D-\omega L^{H}-(\omega-\gamma)L] \mathbf{x}^{(k)}\] \[+\omega(D+\gamma L)^{-1}\mathbf{y}^{MF}, \tag{15}\]  
and the relaxation parameter \(\omega\) and the acceleration parameter \(\gamma\) of the AOR iterative method are given by [37] as follows:  
\[\omega = \frac{1}{\sqrt{1-\mu^{2}}}, \tag{16}\] \[\gamma = \frac{2}{1+\sqrt{1-\mu^{2}}}, \tag{17}\]  
where \(\mu=\left.\rho(G_{\gamma,\omega})\right|_{\gamma=0,\omega=1}\); we will also define \(\rho(G_{\gamma,\omega})\) in later sections. AOR Convergence Analysis  
We know that in any iterative algorithm in the form of Equation (10), when \(k\) reaches infinity, \(\mathbf{x}^{(k+1)}\) will be equal to \(\mathbf{x}^{(k)}\). Noting Equation (15), we can use Equation (10) to define \(G_{\gamma,\omega}=(D+\gamma L)^{-1}[(1-\omega)D-\omega L^{H}-(\omega-\gamma)L]\) and \(\mathbf{d}=\omega(D+\gamma L)^{-1}\mathbf{y}^{MF}\); afterward, Equation (15) can be rewritten as  
\[\mathbf{x}^{(k+1)}=G_{\gamma,\omega}\mathbf{x}^{(k)}+\mathbf{d}, \tag{18}\]  
and the spectral radius of \(G_{\gamma,\omega}\) is defined as the non-negative number as follows:  
\[\rho(G_{\gamma,\omega})\triangleq\max_{\lambda\in\rho(G_{\gamma,\omega})}| \lambda|, \tag{19}\]  
in which \(\lambda\) is the eigenvalue of \(G_{\gamma,\omega}\). Moreover, if the spectral radius \(\rho(G_{\gamma,\omega})\) satisfies  
\[\rho(G_{\gamma,\omega})=\max_{1<n<N_{T}}|\lambda_{n}|<1, \tag{20}\]  
Therefore, Equation (15) will converge [31]. According to the theorem of eigenvalues, we can obtain  
\[+\omega(D-\gamma L)^{-1}\mathbf{b}, \tag{13}\]0  
where \(\mathbf{x}\) is an arbitrary \(N_{T}\times 1\) non-zero real-valued vector and Equation (21) also can be rewritten as  
\[+\omega(D-\gamma L)^{-1}\mathbf{b}, \tag{13}\]1  
Multiplying both sides of Equation (22) by the conjugate transpose of \(\mathbf{x}\), we can obtain  
\[+\omega(D-\gamma L)^{-1}\mathbf{b}, \tag{13}\]2  
Next, we take the conjugate transpose on both sides of Equation (23), in which \(D\) is a diagonal matrix, so \(D=D^{H}\), and then we can obtain an updated equation as  
\[+\omega(D-\gamma L)^{-1}\mathbf{b}, \tag{13}\]3  
Then, adding Equations (23) and (24) will obtain  
\[+\omega(D-\gamma L)^{-1}\mathbf{b}, \tag{13}\]4 \[+\omega(D-\gamma L)^{-1}\mathbf{b}, \tag{13}\]5Substituting Equation (14) into Equation (25), we can obtain  
\[+\omega(D-\gamma L)^{-1}\mathbf{b}, \tag{13}\]6  
Since the filter matrix \(W\) of conventional MMSE is positive definite, as demonstrated in [27], \(D\) is also positive definite, i.e., the diagonal matrix of \(W\). Thus, we have \(\mathbf{x}^{H}D\mathbf{x}>0\) and \(\mathbf{x}^{H}W\mathbf{x}>0\). Moreover, if both \((\gamma-2)\) and \((\lambda_{n}-1)\) are less than zero, i.e., \(0<\gamma<2\) and \(\lambda_{n}<1\), then  
\[+\omega(D-\gamma L)^{-1}\mathbf{b}, \tag{13}\]7  
and \(\lambda_{n}\gamma+(2\omega-\gamma)>0\).