## 4 Simulation Results and Complexity Analysis
### Simulation Results and Discussion
While CHSOR already demonstrates near-MMSE performance with only two iterations at \(N_{R}\) of 192, its advantage becomes even better when the number of antennas is further increased to 256. As shown in the BER vs. iteration curves in Figure 16, the line for the CHSOR scheme is almost flat for both OFDM and UFMC systems, indicating that the proposed method converges almost within one iteration. With a large enough number of antennas, such as 192 or 256, it can significantly aid the HSOR stage to reduce the burden on the Chebyshev stage. In contrast, although Chebyshev-MAOR remains the fastest among the other methods, it still requires 3-4 iterations to reach a similar BER level, as evidenced in Table 7.  
In summary, among the related detectors discussed in this paper, our proposed CHSOR algorithm has fast convergence speed and presents the best BER performance. Moreover, the experimental results in Figures 13-16 and Tables 4-7 along with Appendix A theoretically prove the convergence of the proposed scheme and verify convergence from the obtained experimental data. To analyze the BER performance of different detectors varies with different ratios of \(N_{R}\) to the total number of user antennas \(N_{T}\), we denote this as \(\beta\), or the antenna ratio [50; 59]. Figures 17 and 18 show the results of the experiments with one and two iterations, respectively, when the SNR is 30 dB. In Figure 17, it can be seen tht when the antenna ratio is low, no method can achieve the BER performance of MMSE with one iteration. As \(\beta\) increases, the BER of the proposed CHSOR scheme decreases significantly and gradually approaches the MMSE detector. Even though the BER of other methods also improves, the declining trend is quite slow compared with our CHSOR scheme. When \(\beta\) is equal to 16, the CHSOR scheme is remarkably close to the MMSE detector. Next, the results with two iterations are shown in Figure 18. Compared with Figure 17, the BER performance of each method improves, but all are still inferior to our proposed method. Moreover, the curve of the CHSOR method almost coincides with the MMSE curve at any antenna ratio \(\beta\). To show the performance differences more intuitively, we enumerate the BER differences between each method and the MMSE detector under the same conditions in Tables 8 and 9 based on Figures 17 and 18. Table 8a for the OFDM system shows that when \(\beta\) is 12 and 16, the distances are \(2.463\times 10^{-4}\) and \(1.453\times 10^{-5}\), respectively. Table 8b for the UFMC system shows that when \(\beta\) is 12 and 16, the gap can be reduced to \(3.604\times 10^{-5}\) and \(4.883\times 10^{-7}\), respectively. Moreover, when the number of iterations is 2 and with a sufficiently high \(\beta\), the gap between our method and MMSE can be reduced to approximately order \(10^{-9}\) for both the OFDM or UFMC systems, as enumerated in Table 9.  
Figure 13: BER performance vs. number of iterations with \(N_{R}\times N_{T}=64\times 16\) and SNR = 37 dB for (**a**) OFDM and (**b**) UFMC. Thus far, the discussion on the impact of the number of receive antennas and antenna ratios on the uplink receiver of the BER performance has been well established. Due to the advantage actuated by the spatial diversity gain of massive antennas, as long as the antenna ratio \(\beta\) becomes higher, the BER of any method will decrease without exception. It is worth noting that the effect is particularly significant for our proposed CHSOR scheme. Moreover, this performance echoes what was inferred earlier; when \(N_{R}\) increases, BER performance can be improved and convergence acceleration can be achieved with reduced complexity. This means that the experimental results are reasonable and confirms that the proposed method can adequately and efficiently utilize the spatial diversity gain, resulting in performance quite close to that of the MMSE baseline. Moreover, the sensitivity of the proposed method is not very significant.