## 4 Simulation Results and Complexity Analysis
### Computational Complexity Analysis  
In this subsection, we evaluate the proposed detectors in terms of the computational complexity based on the number of complex multiplications required compared with other representative counterparts [27; 28; 29; 30]. As in Table 6, we give the computational complexity \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Scheme** & \(\mathbf{i=2}\) & \(\mathbf{i=3}\) & \(\mathbf{i=4}\) \\ \hline SOR & 44.35 dB & 28.40 dB & 3.31 dB \\ modified SOR & 44.80 dB & 30.62 dB & 7.14 dB \\ SSOR & 34.63 dB & 10.05 dB & \(-\)7.90 dB \\ AOR & 43.98 dB & 25.91 dB & \(-\)0.21 dB \\ CAOR & 28.94 dB & 0.61 dB & \(-\)16.88 dB \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison of the degree to which the BER performance approaches that of conventional MMSE among all detectors in terms of dB value under different iteration numbers when \(N_{R}\times N_{T}=128\times 16\) and SNR is 45 dB. Figure 10: BER performance vs. number of iterations with \(N_{R}\times N_{T}=256\times 16\) and SNR \(=40\) dB. of different detection schemes, in which \(i\) and \(N_{T}\) denote the number of iterations and the total number of user antennas, respectively. In addition, the inverse matrix of each iteration method requires \(2N_{T}^{2}-N_{T}\) complex multiplications and additions (CMAs) [28]. We consider the CAOR iterative equation as follows:  
\[\tilde{\mathbf{x}}^{(k)}=M^{-1}\cdot\frac{2\mu_{k}}{\mu_{(k-1)}\rho}\cdot(N \tilde{\mathbf{x}}^{(k-1)}+\omega\mathbf{b})-\frac{\mu_{k}}{\mu_{(k-2)}}\tilde{ \mathbf{x}}^{(k-2)}, \tag{42}\]  
in which \(M^{-1}=\left(D+\gamma L\right)^{-1}\) and \(N=(1-\omega)D-(\omega-\gamma)L-\omega L^{H}\) require \(2N_{T}^{2}-N_{T}\) CMAs and \(N_{T}^{2}\) CMAs, respectively. Additionally, \(\omega\mathbf{b}\) requires \(N_{T}\) CMAs; as discussed above, we can obtain \(3N_{T}^{2}\) CMAs, which is part of the CAOR's computational complexity, as shown in Table 6. For the iteration part of CAOR, \(M^{-1}\cdot\frac{2\mu_{k}}{\mu_{(k-1)}\rho}\cdot(N\tilde{\mathbf{x}}^{(k-1)} +\omega\mathbf{b})\) and \(-\frac{\mu_{k}}{\mu_{(k-2)}}\tilde{\mathbf{x}}^{(k-2)}\) require \(i(3N_{T}^{2}+N_{T})\) CMAs and \(2 iN_{T}\) CMAs, respectively. Thus, the iteration part of the CAOR's computational complexity requires \(3i(N_{T}^{2}+N_{T})\) CMAs altogether. Further, the numerical comparison of the complexity is shown in Table 7.  
Regarding the complexity and performance factors, as shown in Figure 5, even though the BER performance of SSOR [29] is approximately \(2.5\times 10^{-3}\), which is also close to the conventional MMSE scheme, its complexity is approximately \(37.8\%\) higher than the CAOR method, which can be verified by the equation within Table 6. Meanwhile, we also know that the computational complexity of SSOR is the highest among all detectors, which is shown in Table 7.  
For a more detailed discussion, we first consider four iterations in Figure 8; the CAOR method has improvements of \(93\%\) in BER compared with the conventional AOR. Meanwhile, the complexity only increases by \(192\) CMAs (increases by \(5\%\)). On the other hand, the CAOR method with \(i=4\) only requires \(4032\) CMAs, which can achieve similar performance to the conventional AOR with \(i=5\) requiring \(4608\) CMAs. Alongside this, when \(N_{R}\times N_{T}=64\times 16\) and the iteration number is four, SSOR's complexity and BER performance both are worse than AOR, not to mention compared with the CAOR method. The more complete numerical computational complexity in terms of iteration number for different detectors is shown in Table 7.  
Secondly, to further observe the effect on BER and complexity when the number of antennas is increased, according to Figures 6 and 10, as \(N_{R}\) increases, the number of iterations required apparently gradually decreases. Furthermore, the BER performance and the computational complexity of CAOR decrease more than other detection methods.