## 4 Simulation Results and Complexity Analysis
### Simulation Results and Discussion
\begin{table}
\begin{tabular}{c l} \hline \hline
**Scheme** & **Brief Description** \\ \hline SOR [27] & SOR is a method of solving a linear system of equations derived by extrapolating the Gauss–Seidel method. \\ modified SOR [28] & Modified SOR is a variant of SOR, which changes certain parameters in the SOR algorithm and is nearly unaffected by the relaxation factor \(\omega\) in lower modulation order. \\ SSOR [29] & SSOR combines two SOR sweeps in such a way that the resulting iteration matrix is similar to a symmetric matrix. \\ AOR [30] & The AOR iterative algorithm can be regarded as an extension of the SOR iterative algorithm, which is a two-parameter generalization of the SOR method. \\ CAOR & The CAOR method combines the AOR iterative algorithm and the recursive characteristics of the Chebyshev polynomials.  
\end{table} 
Table 2: A brief comparison of our proposed scheme and previously published work. Figures 4-6 depict the BER performance comparison with five different detection schemes, and \(N_{R}\) is 64, 128, and 256, respectively. In Figure 4, we can roughly observe that the CAOR method is closest to the performance of conventional MMSE when the iteration number is equal to five. Moreover, when SNR = 45 dB, the BER performance of CAOR and AOR is approximately \(0.6\times 10^{-3}\) and \(0.8\times 10^{-3}\), respectively. In other words, compared to AOR, CAOR can improve BER performance by 25%. When increasing \(N_{R}\) to 128 and the number of iterations \(i\) is 3, shown in Figure 5, we continued to observe that the BER performance of the proposed method almost overlaps with the conventional MMSE scheme, only requiring three iterations. Moreover, Figure 6 shows that when \(N_{R}\) increases to 256 and is iterated twice only, we can note that the proposed method has the best BER performance in comparison with other detectors. Figure 3: BER performance of CAOR method against \(\omega\) with \(\text{SNR}=40\) dB, \(N_{R}\times N_{T}=64\times 16\), and number of iterations \(i=4\). Figure 2: BER performance of CAOR method against \(\gamma\) with \(\text{SNR}=40\) dB, \(N_{R}\times N_{T}=64\times 16\), and number of iterations \(i=4\). Figure 4: BER performance comparison for different detection schemes, \(N_{R}\times N_{\rm T}=64\times 16\). Figure 5: BER performance comparison for different detection schemes, \(N_{R}\times N_{\rm T}=128\times 16\). Figure 6: BER performance comparison for different detection schemes, \(N_{R}\times N_{\rm T}=256\times 16\). We turn now to the ratio (\(\beta\)) analysis, which is the ratio between the number of base station antennas (\(N_{R}\)) and the total number of user antennas (\(N_{T}\)) [24]. As shown in Figure 7, the BER performance changes in terms of the \(\beta\) of each detection scheme. Alongside this, when \(\beta\) equals 16 and SNR is 40 dB, the BER performance of SSOR and CAOR is equal to \(1.75\times 10^{-4}\) and \(9.27\times 10^{-5}\), respectively. Taking it for granted, relative to the higher \(\beta\) value, the BER performance of all detectors is improved due to the spatial diversity gain of a large number of antennas without exception. Moreover, we know that, regardless of the \(\beta\) value, for the BER performance, SSOR is satisfactory, as is AOR, but our proposed scheme CAOR surpasses them all. It is worth mentioning that when the number of antennas continues to increase, the number of iterations required is relatively reduced; simultaneously, the BER performance of our proposed method improves more definitely. On the whole, this is a good result for a 5G system with massive antennas. Moreover, in Table 3, we compare the degree of the BER performance approaching conventional MMSE among all detectors in terms of dB value (i.e., to obtain a more demarcated numerical comparison, we have taken \(10 log(\cdot)\) for the BER distance between the individual detector and conventional MMSE) under different \(\beta\), where the CAOR method has the smallest distance with conventional MMSE.