## 4 Numerical Results and Discussion
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**SNRs (dB)**} & \multicolumn{3}{c}{**FPP-SCA Scheme**} & \multicolumn{3}{c}{**Low-Complexity Scheme**} \\ \cline{2-7}  & \(\beta=0.3\) & \(\beta=0.5\) & \(\beta=0.7\) & \(\beta=0.3\) & \(\beta=0.5\) & \(\beta=0.7\) \\ \hline
**0** & 0.35 & 0.25 & 0.2 & 0.80 & 0.77 & 0.74 \\
**5** & 0.12 & 0.06 & 0.03 & 0.66 & 0.62 & 0.59 \\
**10** & 0.01 & 0.003 & \(9.3\times 10^{-4}\) & 0.52 & 0.47 & 0.45 \\
**15** & \(1.8\times 10^{-4}\) & \(1.0\times 10^{-5}\) & 0.0 & 0.38 & 0.33 & 0.31 \\
**20** & 0.0 & 0.0 & 0.0 & 0.25 & 0.22 & 0.19 \\
**25** & 0.0 & 0.0 & 0.0 & 0.14 & 0.11 & 0.09 \\
**30** & 0.0 & 0.0 & 0.0 & 0.06 & 0.04 & 0.03 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The effects of \(\beta\) variation. Figure 3: The total mean square error (total-MSE) versus the iterations for \(N=M=2\). Accordingly, in comparison to the results in Figure 4 and the two tables, we can see that for the proposed FPP-SCA algorithm, the performance is always higher than the SDR-based one for different antennas and SNRs, and we can make the conclusion that our proposed FPP-SCA-based scheme performs better than the traditional SDR-based scheme. More intriguingly, when \(N=M\), the low-complexity scheme achieves a comparable performance to that of the FPP-SCA one and yields a better performance than that of \(N>M\) (e.g., 0.0 vs. 0.04 for \(SNR=30\) dB shown in Tables 2 and 3). Combined with the low complexity of the low-complexity scheme, it is more applicable than the FPP-SCA one in the \(N=M\) case. However, when \(N>M\), in comparison to the FPP-SCA scheme, the performance of the low-complexity scheme is a bit worse owing to the influence of the enhancive diversity gain. In summary, when the number of antennas at the relay node and source nodes are different, it is more beneficial to choose the FPP-SCA scheme. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**SNRs (dB)** & **Unaided Scheme** & **FPP-SCA Scheme** & **Low-Complexity Scheme** & **SDR Scheme** \\ \hline
**0** & 0.44 & 0.39 & 0.44 & 0.42 \\
**5** & 0.26 & 0.19 & 0.21 & 0.20 \\
**10** & 0.12 & 0.04 & 0.047 & 0.05 \\
**15** & 0.03 & 0.003 & 0.0043 & 0.004 \\
**20** & 0.0027 & 5.0 \(\times\) 10\({}^{-6}\) & 4.5 \(\times\) 10\({}^{-5}\) & 9.0 \(\times\) 10\({}^{-6}\) \\
**25** & 3.5 \(\times\) 10\({}^{-5}\) & 0.0 & 0.0 & 0.0 \\
**30** & 0.0 & 0.0 & 0.0 & 0.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The BER performance for different schemes when \(M=N=2\). Figure 4: BER versus SNR for 50 iterations. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**SNRs (dB)** & **Unaided Scheme** & **FPP-SCA Scheme** & **Low-Complexity Scheme** & **SDR Scheme** \\ \hline
**0** & 0.80 & 0.25 & 0.77 & 0.28 \\
**5** & 0.67 & 0.06 & 0.62 & 0.07 \\
**10** & 0.52 & 0.003 & 0.47 & 0.005 \\
**15** & 0.38 & 2.5 \(\times\) 10\({}^{-5}\) & 0.33 & 4.0 \(\times\) 10\({}^{-5}\) \\
**20** & 0.27 & 0.0 & 0.22 & 0.0 \\
**25** & 0.18 & 0.0 & 0.11 & 0.0 \\
**30** & 0.12 & 0.0 & 0.04 & 0.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The BER performance for different schemes when \(M=2\), \(N=4\). From Figure 5 and Table 4, we can see that the performance increases with the number of antennas for both schemes under \(M=N\) or \(N>M\). When the number of antennas increases, more antennas can be used to suppress multipath fading with antenna diversity, to increase the system capacity, and to improve the performance. Considering the cost of computing of the low-complexity scheme and comparing the existing work proposed in Reference [35], we choose to use \(M=N=2\) and \(M=2,N=4\) for both schemes. In detail, in Table 4, \(A^{M,N}\), \(B^{M,N}\), \(C^{M,N}\), and \(D^{M,N}\) correspond to \(M=N=2\); \(M=2,N=4\); \(M=N=4\); and \(M=4,N=8\) for the FPP-SCA scheme and \(E^{M,N}\), \(F^{M,N}\), \(G^{M,N}\), and \(H^{M,N}\) correspond to \(M=N=2\); \(M=2,N=4\); \(M=N=4\); and \(M=4,N=8\) for the low-complexity scheme.