## 3 Proposed Scheme
### Proposed CHSOR Method
Further, we can rewrite the Equation (37) as follows:  
\[\mathbf{\tilde{x}}^{(1)}=\mathbf{G}_{\mathit{HSOR}}\mathbf{ \tilde{x}}^{(0)}+\mathbf{d}_{\mathit{HSOR}}. \tag{38}\]  
Similar, to the SOR and MSOR methods, the spectral radius of the HSOR iteration matrix \(\rho(\mathbf{G}_{\mathit{HSOR}})\) must be less than 1 to ensure that Equation (38) converges, as described below:  
\[\rho(\mathbf{G}_{\mathit{HSOR}})\triangleq\max_{\lambda\in\mathcal{O}( \mathbf{G}_{\mathit{HSOR}})}|\lambda|<1. \tag{39}\]  
Therefore, we can find the mathematical equations for the theoretical value of the relaxation parameter \(\omega_{\mathit{theory}}\) as follows [23; 56]:  
\[\omega_{\mathit{theory}}=\frac{1}{\sqrt{1-\mu^{2}}} \tag{40}\]  
where \(\mu\) is \(\rho(\mathbf{G}_{\mathit{HSOR}})|_{\omega=1}\). To guarantee the convergent condition, we need to define the interval of the relaxation parameter \(\omega\). The derivation shown in Appendix A demonstrates that the HSOR method converges when \(\omega\) is in the interval [1; 2]. To verify the above inference and choose the best relaxation parameter \(\omega\), Figure 6 shows the BER performance of the HSOR scheme under different values of \(\omega\) with the antenna configuration \(N_{R}\times N_{T}=64\times 16\) and the SNR at 35 dB. This experiment obtains an optimal \(\omega\) value of 1.1 and verifies that our inferred convergence interval of \(\omega\) is between 1 and 2. More importantly, the theoretical value of the relaxation parameter \(\omega_{\mathit{theory}}\) obtained using Equation (40) is consistent with the \(\omega_{\mathit{estimate}}\) experimental results, i.e., \(\omega_{\mathit{theory}}\) is equal to \(\omega_{\mathit{estimate}}\). In addition, the optimal value of 1.1 used in this paper applies to both the OFDM and UFMC waveforms. Specifically, the spectral radius \(\rho(\mathbf{G}_{\mathit{HSOR}})\) corresponding to the selected relaxation parameter \(\omega_{\mathit{estimate}}\) is 0.648, while the experiment further confirms the stability and convergence of our proposed method. In the second stage, we adopt the recurrent Chebyshev acceleration algorithm to refine the BER performance. To help explain the processing steps, the streamlined Chebyshev acceleration process is shown in the second stage of Algorithm 1. As shown in Steps 2 and 3 of the second stage of Algorithm 1, this method uses the recurrence characteristics  
Figure 5: Block diagram of the HSOR stage. of Chebyshev polynomials to update and adjust the coefficients of each iteration, which gradually suppresses errors and significantly improves iteration efficiency. In brief, the first stage makes use of HSOR to obtain a preliminary estimate through MSOR with lower complexity, then makes use of SOR to produce a more precise result with better BER performance. This result is then passed to the second Chebyshev acceleration  
Figure 6: BER vs. \(\omega\) for OFDM and UFMC using the HSOR method when \(N_{R}\times N_{T}=64\times 16\) and SNR = 35 dB. stage, providing an improved initial status for iteration. Moreover, the first stage initializes parameters such as \(\mu\), \(k\), \(\mu^{(0)}\), \(\mu^{(1)}\), \(\tilde{\mathbf{x}}^{(0)}\), and \(\tilde{\mathbf{x}}^{(1)}\). These parameters are delivered to the Chebyshev acceleration stage along with the iteration matrix \(\mathbf{G}_{HSOR}\) and compensation vector \(\mathbf{d}_{HSOR}\). In the second stage, the Chebyshev acceleration method uses the parameters and products from the HSOR stage to efficiently refine the BER performance. It is worth mentioning that because we utilize two SOR-based methods in the first stage, the same parts do not need to be calculated repeatedly, which can further reduce the complexity burden and cost.