## 4 Simulation Results and Complexity Analysis
### Computational Complexity Analysis  
#### 4.2.1 Complexity Formula Analysis  
In this subsection, we evaluate the computational complexity of the proposed method in terms of the number of required complex multiplications and additions (CMAs) and compare it with other detection methods mentioned in this paper [17; 18; 19; 20; 21; 22; 24; 25]. We use \(i\) and \(N_{T}\) to denote the number of iterations and total number of user antennas, respectively. We sequentially analyze the algebraic calculation of the proposed CHSOR detection method through its iterative equation as follows:  
\[\mathbf{\tilde{x}}^{(k)}=\frac{2\mu^{(k)}}{\mu^{(k-1)}\cdot\mu}(\mathbf{G}_{ HSOR}\mathbf{\tilde{x}}^{(k-1)}+\mathbf{d}_{HSOR})-\frac{\mu^{(k)}}{\mu^{(k-2)}} \mathbf{\tilde{x}}^{(k-2)} \tag{43}\]  
Figure 18: BER vs. \(\beta\) for different detection schemes when the number of iterations is 2 and the SNR is 30 dB for (**a**) OFDM and (**b**) UFMC. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Scheme** & \(\boldsymbol{\beta=4}\) & \(\boldsymbol{\beta=8}\) & \(\boldsymbol{\beta=12}\) & \(\boldsymbol{\beta=16}\) \\ \hline  & & (a) & & \\ \hline SOR & \(3.533\times 10^{-1}\) & \(3.242\times 10^{-1}\) & \(2.982\times 10^{-1}\) & \(2.778\times 10^{-1}\) \\ MSOR & \(3.583\times 10^{-1}\) & \(3.281\times 10^{-1}\) & \(3.012\times 10^{-1}\) & \(2.802\times 10^{-1}\) \\ AOR & \(3.506\times 10^{-1}\) & \(3.211\times 10^{-1}\) & \(2.956\times 10^{-1}\) & \(2.756\times 10^{-1}\) \\ SOR & \(3.047\times 10^{-1}\) & \(2.291\times 10^{-1}\) & \(1.753\times 10^{-1}\) & \(1.374\times 10^{-1}\) \\ SAOR & \(2.983\times 10^{-1}\) & \(2.284\times 10^{-1}\) & \(1.754\times 10^{-1}\) & \(1.377\times 10^{-1}\) \\ CSOR & \(2.473\times 10^{-1}\) & \(1.582\times 10^{-1}\) & \(1.032\times 10^{-1}\) & \(6.916\times 10^{-2}\) \\ Chebyshev-MAOR & \(3.621\times 10^{-1}\) & \(1.181\times 10^{-1}\) & \(2.506\times 10^{-2}\) & \(5.661\times 10^{-3}\) \\ Chebyshev-RI & \(2.713\times 10^{-1}\) & \(2.196\times 10^{-1}\) & \(1.740\times 10^{-1}\) & \(1.381\times 10^{-1}\) \\ CHSOR & \(7.368\times 10^{-2}\) & \(4.886\times 10^{-3}\) & \(2.463\times 10^{-4}\) & \(1.453\times 10^{-5}\) \\ \hline  & & (b) & & \\ \hline SOR & \(3.719\times 10^{-1}\) & \(3.276\times 10^{-1}\) & \(2.986\times 10^{-1}\) & \(2.776\times 10^{-1}\) \\ MSOR & \(3.769\times 10^{-1}\) & \(3.314\times 10^{-1}\) & \(3.016\times 10^{-1}\) & \(2.801\times 10^{-1}\) \\ AOR & \(3.693\times 10^{-1}\) & \(3.245\times 10^{-1}\) & \(2.961\times 10^{-1}\) & \(2.755\times 10^{-1}\) \\ SSOR & \(3.232\times 10^{-1}\) & \(2.319\times 10^{-1}\) & \(1.747\times 10^{-1}\) & \(1.360\times 10^{-1}\) \\ SAOR & \(3.167\times 10^{-1}\) & \(2.312\times 10^{-1}\) & \(1.749\times 10^{-1}\) & \(1.363\times 10^{-1}\) \\ CSOR & \(2.656\times 10^{-1}\) & \(1.608\times 10^{-1}\) & \(1.025\times 10^{-1}\) & \(6.779\times 10^{-2}\) \\ Chebyshev-MAOR & \(3.810\times 10^{-1}\) & \(1.195\times 10^{-1}\) & \(2.164\times 10^{-2}\) & \(3.637\times 10^{-3}\) \\ Chebyshev-RI & \(2.898\times 10^{-1}\) & \(2.230\times 10^{-1}\) & \(1.745\times 10^{-1}\) & \(1.381\times 10^{-1}\) \\ CHSOR & \(8.548\times 10^{-2}\) & \(2.900\times 10^{-3}\) & \(3.604\times 10^{-5}\) & \(4.883\times 10^{-7}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 8: BER difference rate between the BER of all detection methods and MMSE at different \(\beta\) when the number of iterations is 1 and the SNR is 30 dB for (**a**) OFDM and (**b**) UFMC. where the iteration matrix \(\mathbf{G}_{HOSR}\) is equal to \(\left[\mathbf{M}_{SOR}^{-1}\mathbf{N}_{SOR}\mathbf{M}_{MSOR}^{-1}\mathbf{N}_{MSOR }\right]\), which requires \(\left[6N_{T}^{2}-2N_{T}\right]\) CMAs, and the compensation vector \(\mathbf{d}_{HOSR}\) is \(\left[\mathbf{G}_{SOR}\mathbf{d}_{MSOR}+\mathbf{d}_{SOR}\right]\), which needs \(\left[3N_{T}^{2}+N_{T}\right]\) CMAs. Therefore, according to Equation (38), the operation complexity for the first stage can be obtained as a total of \(\left[11N_{T}^{2}-N_{T}\right]\) CMAs.