## 4 Simulation Results and Complexity Analysis
### Computational Complexity Analysis
#### 4.2.2 Complexity Quantitative Analysis  
For more intuitive comparison and quantitative analysis, the exact parameter values are substituted in Table 10, with numerical values enumerated in Table 11 and presented using a bar chart in Figure 19.  
Next, computational complexity and BER performance are discussed to provide a comprehensive comparison. It can be seen from Figure 9a,b that with four iterations the proposed method improves the BER by 91.624% and 99.638% compared with CSOR in the OFDM and UFMC systems, respectively. For complexity, Table 11 shows that it only increases by 22.381% regardless of whether the OFDM or UFMC system is used.  
Moreover, Figure 13 along with Tables 4 and 11 allow us to observe and compare each method's iterative procedure and required complexity when it reaches convergence under the antenna configuration with \(N_{R}\times N_{T}=64\times 16\) and SNR at 37 dB. Evidently, the CHSOR method converges at the fourth iteration and the complexity load is 5040 CMAs. Among the other methods, the CSOR scheme has the best performance; however, it still has not converged after six iterations, and its complexity burden of 5544 CMAs has exceeded that of the CHSOR method after four iterations. Although at first glance our proposed CHSOR scheme seems slightly higher than the CSOR method with the same number of iterations, the BER performance of the CHSOR scheme can be much better than that of the CSOR method with only a few iterations.  
As previously mentioned, Figures 13-16 show the relationship between the number of iterations \(i\) and BER performance among all schemes for different antenna configurations. From these results, we can conclude that faster the convergence is achieved when there are more antennas at the receiving end. Taking the proposed method as an example, when \(N_{R}\) increases from 64 to 256, the number of iterations required for convergence can be reduced from 4 to 2 or even 1; in addition, the BER performance is significantly better than other methods. Therefore, the proposed CHSOR detector achieves a strong balance between computational complexity and BER performance. When \(N_{R}\) increases, the proposed CHSOR detector takes full advantage of the spatial diversity gain of M-MIMO and the optimal initial conditions of the HSOR stage to provide excellent convergence speed without sacrificing any computational complexity, thereby considering both computational complexity and BER performance.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Detection Methods** & \begin{tabular}{c} **CMAs** \\ \(i=1\) \\ \end{tabular} & \begin{tabular}{c} **CMAs** \\ \(i=2\) \\ \end{tabular} & \begin{tabular}{c} **CMAs** \\ \(i=3\) \\ \end{tabular} & \begin{tabular}{c} **CMAs** \\ \(i=4\) \\ \end{tabular} & \begin{tabular}{c} **CMAs** \\ \(i=5\) \\ \end{tabular} & \begin{tabular}{c} **CMAs** \\ \(i=6\) \\ \end{tabular} \\ \hline \begin{tabular}{c} SOR \\ MSOR \\ AOR \\ SOR \\ CSOR \\ Cebbyshev-MAOR \\ Chebyshev-RI \\ CHSOR \\ \end{tabular} & \begin{tabular}{c} 1176 \\ 1144 \\ 1556 \\ 3360 \\ \end{tabular} & \begin{tabular}{c} 1704 \\ 1672 \\ 2304 \\ 3408 \\ 3920 \\ 3920 \\ 3920 \\ 3920 \\ 3920 \\ 3920 \\ 3920 \\ 3920 \\ 3920 \\ \end{tabular} & \begin{tabular}{c} 2232 \\ 22760 \\ 2200 \\ 2200 \\ 3920 \\ 3921 \\ 384 \\ 3840 \\ 5040 \\ 5040 \\ 5040 \\ 5040 \\ 5040 \\ 5040 \\ 5040 \\ \end{tabular} & \
\begin{tabular}{c} 3288 \\ 3816 \\ 3784 \\ 3576 \\ 3782 \\ 3554 \\ 5632 \\ 5600 \\ 5600 \\ \end{tabular} &
\begin{tabular}{c} 3816 \\ 3784 \\ 3784 \\ 37632 \\ 10752 \\ 5544 \\ 5632 \\ 5632 \\ 6160 \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table}
Table 11: Numerical complexity comparison for different detectors with \(N_{T}=16\).