## 4 Simulation Results and Complexity Analysis
### Simulation Results and Discussion
Figure 10: BER performance comparison for different detection methods with \(N_{R}\times N_{T}=128\times 16\) and number of iterations = 2 for **(a)** OFDM and **(b)** UFMC. According to the discussion in Section 3, convergence means that the signal vector gradually tends to a stable state after iteration; in other words, the BER reaches a stable value when the state no longer changes. To facilitate subsequent discussion, we define a convergence parameter that we call the BER altered rate \(\delta^{(k)}\) for each iteration, as follows:  
\[\delta^{(k)}=\frac{\psi^{(k)}-\psi^{(k+1)}}{\psi^{(k)}} \tag{41}\]  
Figure 12: BER performance comparison for different detection methods with \(N_{R}\times N_{T}=256\times 16\) and number of iterations = 1 for **(a)** OFDM and **(b)** UFMC. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{2}{c}{**OFDM**} & \multicolumn{2}{c}{**UFMC**} \\ \hline  & \begin{tabular}{c} **CHSOR’s** \\ **BER** \\ \end{tabular} & \begin{tabular}{c} **Chebyshev-** \\ **MAOR’s** **BER** \\ \end{tabular} & \begin{tabular}{c} **Improvement** \\ **Rate (\%)** \\ \end{tabular} & \begin{tabular}{c} **CHSOR’s** \\ **BER** \\ \end{tabular} & \
\begin{tabular}{c} **Chebyshev-** \\ **MAOR’s** **BER** \\ \end{tabular} &
\begin{tabular}{c} **Improvement** \\ **Rate (\%)** \\ \end{tabular} \\ \hline \(N_{R}\)=\(128\) & \(3.737\times 10^{-3}\) & \(6.892\times 10^{-2}\) & 94.578 & \(1.951\times 10^{-4}\) & \(6.471\times 10^{-2}\) & 99.699 \\ \hline \(N_{R}\)=\(192\) & \(9.476\times 10^{-4}\) & \(2.592\times 10^{-2}\) & 96.344 & \(4.764\times 10^{-5}\) & \(2.184\times 10^{-2}\) & 99.782 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: BER numerical results improvement rate of proposed CHSOR and Chebyshev-MAOR at SNR = 30 dB for \(N_{R}\times N_{T}=128\times 16\) and \(N_{R}\times N_{T}=192\times 16\) under OFDM and UFMC. where \(\psi^{(k)}\) represents the BER obtained at the \(kth\) iteration and \(\delta^{(k)}\) represents the altered rate in BER between the current iteration and the next iteration. In this paper, without loss of generality, we select a small enough \(\delta^{(k)}\) variation of 1%; therefore, the iteration schemes can be convergent as long as they meet the following conditions:  
\[\delta^{(k)}<1\ \%. \tag{42}\]  
To verify and compare the iteration convergence among all detectors in this study, we enumerate the BER altered rate for each iteration in Tables 4-7 and plot the convergence curve in Figures 13-16 with antenna configurations of \(N_{R}\times N_{\mathrm{T}}=64\times 16\), \(N_{R}\times N_{\mathrm{T}}=128\times 16\), \(N_{R}\times N_{\mathrm{T}}=192\times 16\), and \(N_{R}\times N_{\mathrm{T}}=256\times 16\), respectively. For Figure 13, the antenna configuration \(N_{R}\times N_{T}\) is \(64\times 16\) and the SNR is 37 dB, in which Figure 13a,b show the experiments conducted with the OFDM and UFMC systems, respectively. From the overall numerical analysis and observation of Table 4 and Figure 13a, it can be seen that the CHSOR method always performs the best and converges within five iterations. Compared to the second-best performing CSOR method, the BER improves by 90.329%. The results for the UFMC system are shown in Figure 13b. When the number of iterations is 5, the BER performance of the CHSOR method is improved by 99.849% compared with the CSOR method. When the \(N_{R}\) is increased to 128 and the SNR is 34 dB, the proposed method converges in three iterations, while the other methods fail to converge even after five iterations. The simulation results for \(N_{R}\) increased to 192 and SNR of 32 dB are shown in Figures 15a,b for the OFDM and UFMC systems, respectively. Similar to the results for \(N_{R}\) of 128, Table 6b shows that the proposed method converges in three iterations for UFMC; furthermore, Table 6a shows that CHSOR converges after just two iterations for OFDM. When the number of iterations is 2, the proposed scheme has an improvement rate of 96.789% and 99.989% over Chebyshev-MAOR in the OFDM and UFMC systems, respectively.