## 4 Numerical Results
### Error Rate Performance  
We initially compare the SER performance of different detectors in a massive MIMO system comprising 50 users and 256 antennas at the BS, as depicted in Figure 1. In this simulation, a 16-QAM modulation scheme is employed. The plot clearly indicates that the performance of all detectors based on iterative techniques improve with an increasing number of iterations. However, the SOR exhibited inferior performance compared to the other detection schemes. It is noteworthy that the SAOR detection technique outperforms the conventional SOR and AOR iterative detection techniques in terms of the error rate within the considered antenna scenario. Moreover, the proposed CGA-SAOR technique outperforms the proposed SAOR technique and achieved nearly the same accuracy as the linear MMSE detector with \(i=2\), which shows that the performance of the CGA-SAOR method was better than that of the SAOR method. Therefore, we compare the proposed CGA-SAOR detector with conventional detectors in the subsequent results. We analyze the SER versus the SNR of the proposed CGA-SAOR detector and recently developed SOR- and AOR-based iterative detectors, as shown in Figure 2. In this scenario, we consider 16 users, and the BS is equipped with 128 antennas. The results using the 16-QAM modulation scheme demonstrate that the performance of the proposed detector is closely aligned with that of the linear MMSE detector with only two iterations. Note that the SOR and AOR exhibit inferior performance compared to the proposed technique. Thus, it is evident that the proposed algorithm converges more rapidly than existing iterative detectors. Next, both the number of users and the number of BS antennas are increased, as shown in Figure 3, to further validate the efficiency of the proposed detector. When comparing Figure 3 with Figures 1 and 2, it can be seen that all methods require more iterations to achieve the desired results. Compared to AOR and SOR, the proposed algorithm achieves much better performance for the same iteration count. However, it needs one extra iteration to achieve results close to those of the MMSE detector, as seen in Figures 1 and 2. The performance gap between the linear MMSE detector and the proposed algorithm is only 0.08 dB at \(10^{-4}\) for \(i=3\), which is negligible. Still, the complexity of the designed detector is one order of magnitude less than that of the MMSE detector with exact matrix inversion. \begin{table}
\begin{tabular}{c c} \hline
**Algorithm** & **Computations** \\ \hline AOR & \(i\left(\frac{3K^{2}+7K}{2K}\right)\) \\ SOR & \(i\left(2K^{2}+2K\right)\) \\ Proposed SAOR & \(i\left(3K^{2}+7K\right)\) \\ Proposed CGA-SAOR & \(i\left(5K^{2}+14K\right)\) \\ \hline \end{tabular}
\end{table} 
Table 1: Complexity in terms of number of multiplications. In Figure 4, under the 64-QAM modulation scheme, we compare the SER performance of the proposed method with that of recently proposed detection methods considering 256 antennas at the BS and 16 users. We can see that as the modulation order increases, the AOR- and SOR-based detectors experience some degree of performance degradation and require a higher number of iterations to obtain results close to those of the MMSE detector. However, the proposed method continues to exhibit better performance than the aforementioned detection algorithms, and only two iterations are required for the proposed detector to obtain results close to those of the MMSE detector. Figure 1: SER performance versus SNR of the proposed SAOR and CGA-SAOR algorithms, as well as conventional SOR- and AOR-based detection schemes, for a 256 \(\times\) 50 antenna system using the 16-QAM modulation technique. Figure 2: SER as a function of SNR for the antenna configuration of \(N\times K=128\times 16\) employing the 16-QAM modulation technique. In Figure 5, we compare the proposed detector with our recently proposed IGS and ESOR methods. An \(N\times K=128\times 16\) system employing 64-QAM modulation is considered in this case.