## 3 Proposed Scheme
### Proposed CHSOR Method  
The review in the previous subsections helps in comprehending the convergence conditions and parameter range for the iteration equations of the SOR and MSOR schemes along with the convergence properties of the recurrence relation for Chebyshev polynomials. This prompts us to develop the highly efficient two-stage detection procedure shown in Figure 4. In the first stage, we combine the advantages of the original SOR and MSOR with the improved convergence speed provided by the relaxation parameter \(\omega\) to find the optimal convergence direction and starting point for the second stage. In the second stage, we use the recursive Chebyshev acceleration in an approach we call hybrid SOR-based Chebyshev acceleration (CHSOR). The proposed CHSOR scheme speeds up convergence, helping to reduce complexity and improve BER performance. To explain the algorithms used in the first and second stages in a more streamlined and clear way, we list their process steps in Algorithm 1. As mentioned in Section 3.1.2, MSOR is a variant of SOR. Below, we describe the procedure in detail and explain the reason for combining these approaches in our proposed CHSOR scheme. The combined equations are shown afterwards. First, we can utilize the expressions \(\mathbf{N}_{SOR}=[(1-\omega)\mathbf{D}+\omega\mathbf{U}]\) and \(\mathbf{N}_{MSOR}=\mathbf{U}+(1-\omega)\mathbf{L}\) to observe the main differences between SOR and MSOR. In these expressions, \(\mathbf{N}_{SOR}\) can be regarded as a generalized weighting of the upper triangular matrix and \(\mathbf{N}_{MSOR}\), similar to removing diagonal elements and then adding a weighting lower triangular matrix of \(\mathbf{N}_{SOR}\); in other words, \(\mathbf{N}_{MSOR}\) is a matrix that has no diagonal elements. Thus, we can expect that mixing the two iterated matrices can bring the output towards a better convergence state by integrating their complementarity. In addition, from their definition we know that \(\mathbf{M}_{SOR}\) and \(\mathbf{M}_{MSOR}\) are both the same as \((\mathbf{D}-\omega\mathbf{L})\). This means that \((\mathbf{D}-\omega\mathbf{L})\) does not need to repeat the operation when combining the SOR and MSOR algorithms, which reduces the complexity of the proposed method. Therefore, it is worthwhile to anticipate integrating the MSOR and SOR into one matrix in order to produce a confident estimate parameter and initial state, allowing us to obtain a better detection signal at almost no cost in terms of complexity. We call this first stage the hybrid SOR (HSOR) stage. Figure 4: Block diagram of the CHSOR method. As shown in Figure 5, we denote the MSOR input as \(\mathbf{\tilde{x}}^{(0)}\), then rewrite the output of MSOR as \(\mathbf{x}^{\prime}\) according to Equation (20) and substitute it into Equation (14) to produce the output \(\mathbf{\tilde{x}}^{(1)}\) of the HSOR stage. The expression is as follows:  
\[\mathbf{\tilde{x}}^{(1)} =\mathbf{G}_{\mathit{SOR}}\mathbf{x}^{\prime}+\mathbf{d}_{\mathit{ SOR}} \tag{37}\] \[=\mathbf{G}_{\mathit{SOR}}(\mathbf{G}_{\mathit{MSOR}}\mathbf{ \tilde{x}}^{(0)}+\mathbf{d}_{\mathit{MSOR}})+\mathbf{d}_{\mathit{SOR}}\] \[=\mathbf{G}_{\mathit{SOR}}\mathbf{G}_{\mathit{MSOR}}\mathbf{ \tilde{x}}^{(0)}+\mathbf{G}_{\mathit{SOR}}\mathbf{d}_{\mathit{MSOR}}+\mathbf{ d}_{\mathit{SOR}}.\]  
Next, we define the iteration matrix \(\mathbf{G}_{\mathit{HSOR}}\) and compensation vector \(\mathbf{d}_{\mathit{HSOR}}\) for the HSOR stage as \(\mathbf{G}_{\mathit{SOR}}\mathbf{G}_{\mathit{MSOR}}\) and \((\mathbf{G}_{\mathit{SOR}}\mathbf{d}_{\mathit{MSOR}}+\mathbf{d}_{\mathit{SOR}})\), respectively.