## 3 Proposed Scheme
### Overview of Iteration Method  
#### 3.1.1 Original SOR Method  
Assume a linear system model described by the linear equation  
\[\mathbf{A}\mathbf{x}=\mathbf{b}, \tag{9}\]  
where \(\mathbf{A}\) is a symmetric positive definite matrix, \(\mathbf{x}\) is an unknown complex vector, and \(\mathbf{b}\) is the matched filter output of the received signal \(\mathbf{y}\). Then, decompose \(\mathbf{A}\) into  
\[\mathbf{A}=\mathbf{D}-\mathbf{L}-\mathbf{U}, \tag{10}\]  
where \(\mathbf{D}\), \(-\mathbf{L}\), and \(-\mathbf{U}\) are respectively the diagonal matrix, strict lower triangular matrix, and strict upper triangular matrix of \(\mathbf{A}\). In [17,54], [PERSON] et al. illustrated that the recursive relation of the iteration equation of the original SOR can be written as follows:  
\[\mathbf{x}^{(k)}=(\mathbf{D}-\omega\mathbf{L})^{-1}\{[(1-\omega)\mathbf{D}+ \omega\mathbf{U}]\mathbf{x}^{(k-1)}+\omega\mathbf{b}\} \tag{11}\]  
where \(\omega\) is a relaxation parameter and \(k\) is the iteration index. To make this more approachable, we replace \((\mathbf{D}-\omega\mathbf{L})\) and \([(1-\omega)\mathbf{D}+\omega\mathbf{U}]\) with \(\mathbf{M}_{SOR}\) and \(\mathbf{N}_{SOR}\), respectively. Therefore, we can define the iteration matrix of SOR (denoted \(\mathbf{G}_{SOR}\)) as follows:  
\[\mathbf{G}_{SOR}=\mathbf{M}_{SOR}^{-1}\mathbf{N}_{SOR}. \tag{12}\]  
In addition, we can say that \(\mathbf{d}_{SOR}\) is the compensation vector of SOR, defined as  
\[\mathbf{d}_{SOR}=\mathbf{M}_{SOR}^{-1}\cdot\omega\mathbf{b}. \tag{13}\]  
It is worth mentioning that \(\mathbf{d}_{SOR}\) in the above equation can be adjusted according to the solution direction corresponding to \(\mathbf{b}\), allowing us to compensate for the offset error generated during the iteration process. Thus, Equation (11) can be streamlined as follows:  
\[\mathbf{x}^{(k)}=\mathbf{G}_{SOR}\mathbf{x}^{(k-1)}+\mathbf{d}_{SOR}. \tag{14}\]  
In terms of convergence, the spectral radius is usually used to determine whether the linear iteration method is convergent [55]. The spectral radius \(\rho(\mathbf{G})\) for any iteration matrix \(\mathbf{G}\) is defined as follows [52,55]:  
\[\rho(\mathbf{G})\triangleq\max_{\lambda\in\sigma(\mathbf{G})}|\lambda| \tag{15}\]where \(\sigma(\mathbf{G})\) is called the spectrum of \(\mathbf{G}\) (which is the set of eigenvalues of \(\mathbf{G}\)) and \(\lambda\) is \(\mathbf{G}\)'s eigenvalue. In order for Equation (14) to converge, \(\rho(\mathbf{G}_{SOR})\) needs to satisfy the following condition:  
\[\rho(\mathbf{G}_{SOR})<1. \tag{16}\]  
In other words, a spectral radius less than 1 ensures the convergence of the iterative method. To better explain the condition expressed in Equation (16), we can observe the error vector \(\varepsilon^{(k)}\) and describe it succinctly as explained below. First, consider the error vector \(\varepsilon^{(k)}=\mathbf{x}^{(k)}-\mathbf{x}^{*}\) at the \(kth\) iteration, where \(\mathbf{x}^{(k)}\) is the result at iteration \(k\) and \(\mathbf{x}^{*}\) is the target solution. In addition, the error vector \(\varepsilon^{(k)}\) has the following relationship with the iteration matrix \(\mathbf{G}_{SOR}\):  
\[\varepsilon^{(k)}=\mathbf{G}_{SOR}\varepsilon^{(k-1)}=\mathbf{G}_{SOR}^{2} \varepsilon^{(k-2)}=\cdots=\mathbf{G}_{SOR}^{k}\varepsilon^{(0)}.\]  
From the above equation, we can take the norm to obtain the following Cauchy inequality:  
\[\left\|\varepsilon^{(k)}\right\|_{2}=\left\|\mathbf{G}_{SOR}^{k}\varepsilon^{ (0)}\right\|_{2}\leq\left\|\mathbf{G}_{SOR}^{k}\right\|_{2}\left\|\varepsilon^ {(0)}\right\|_{2}.\]  
We can now find that \(\left\|\mathbf{G}_{SOR}^{k}\right\|_{2}\left\|\varepsilon^{(0)}\right\|_{2}\) is the upper bound of \(\left\|\varepsilon^{(k)}\right\|_{2}\). When \(k\rightarrow\infty\), we only need to confirm \(\left\|\mathbf{G}_{SOR}^{k}\right\|_{2}<1\) in order to ensure that \(\left\|\varepsilon^{(k)}\right\|_{2}\) equals zero (which indicates convergence) and that \(\left\|\mathbf{G}_{SOR}^{k}\right\|_{2}=\rho(\mathbf{G}_{SOR}^{k})=(\rho( \mathbf{G}_{SOR}))^{k}\).