## 4 Numerical Results
### Error Rate Performance
Linear MMSE and zero-forcing detectors are included as benchmarks. It should be noted that the ESOR method requires a higher number of iterations to achieve near-MMSE and zero-forcing performance compared to the IGS and proposed CGA-SAOR methods. While the IGS method achieves performance close to that of the CGA-SAOR method for \(i=3\) iterations, the proposed method outperforms both the ESOR and IGS methods in terms of convergence rate and performance. Moreover, although the IGS-based detection method achieves good results, it presents significant challenges when it comes to implementation in parallel systems. This limitation arises from the inherently sequential nature of its iteration structure, which makes it less suitable for highly parallelized environments, such as those required in real-time massive MIMO systems. In contrast, the proposed CGA-SAOR method not only offers better performance but also  
Figure 4: Performance comparison for an \(N\times K=256\times 16\) massive MIMO uplink system utilizing the 64-QAM modulation technique. Figure 3: Comparison of SER performance between the proposed detector and existing iterative detectors for a MIMO system employing \(N=256\) antennas at the BS and 64 users with 16-QAM. allows for more efficient parallelization, making it more suitable for practical large-scale MIMO deployments. Next, we present the results under imperfect CSI to demonstrate the robustness of the proposed CGA-SAOR method. Figure 6 shows the results for the considered massive MIMO system with \(N\times K=256\times 32\) antennas employing 64-QAM modulation. The SER performance is compared to that of the linear MMSE detection method under both perfect and imperfect CSI. As expected, both the proposed algorithm and the MMSE detection algorithm experience performance degradation under imperfect CSI compared to the perfect CSI case. This is due to the inherent inaccuracies in channel estimation, which led to degraded detection performance for both the proposed algorithm and the MMSE algorithm. However, it is noteworthy that the proposed algorithm consistently approaches near-MMSE performance with significantly fewer iterations, regardless of whether the CSI was perfect or imperfect. This demonstrates the robustness of the proposed algorithm under both scenarios. To further evaluate the performance of the algorithms, we provide the mean evaluation time (in seconds) for various signal detection algorithms across two different antenna settings, i.e., \(N\times K=256\times 16\) and \(N\times K=256\times 64\), in Table 2. The evaluation was conducted on a desktop equipped with an Intel Core i5-6500 CPU running at 3.20 GHz and 12 GB of RAM, using a 64-bit operating system (Windows 10 Pro, Version 22H2). A total of 10,000 trials were performed, and four iterations were considered for each configuration. The results show that the mean evaluation time differed across the algorithms, with the MMSE algorithm requiring the most time for both configurations. The AOR and SOR algorithms also demonstrated varying evaluation times, while the CGA-SAOR algorithm provided a balanced evaluation time across different antenna settings. It is important to note that these evaluation times can vary based on several factors: (1) implementing parallel loops allows multiple iterations or computations to be processed simultaneously, particularly when leveraging multi-core processors or GPUs; (2) using vectorized operations instead of for-loops can exploit highly optimized libraries for matrix operations, resulting in faster computations; (3) leveraging specialized numerical libraries, optimized for performance, can speed up core matrix operations in iterative methods; and (4) efficient memory usage and avoiding unnecessary data copies can minimize overhead and enhance execution speed. These approaches, among others, can contribute to reducing the mean evaluation time in practical implementations.