## 4 Numerical Results  
In this section, to validate the derived mathematical expressions in the previous sections, we provide analytical results together with the Monte-Carlo simulation results for comparison. For ease of presentation, both S and R use two transmitting antennas, that is \(N_{\mathrm{f}}=2\), while the number of receiving antennas \(N_{\mathrm{FR}}\) and \(N_{\mathrm{D}}\) are set to be equal and varies from 2 to 4 for performance evaluations. The SIC capability used for evaluation is \(\bar{\Omega}=\{-10,-5,0\}\) dB. For an LTE relay, the typical transmission power ranges from 23 dBm to 30 dBm. Taking 30 dBm for consideration, the RSI levels are \(\sigma_{\mathrm{RSI}}^{2}=\{20,25,30\}\) dBm, respectively. In all figures, we define the average SNR for the case without optimization as follows: \(\mathrm{SNR}=\frac{P_{\mathrm{S}}}{\sigma^{2}}=\frac{P_{\mathrm{R}}}{\sigma^{2}}\). In the case with optimization, the average SNR is defined by SNR at R, that is, \(\mathrm{SNR}=\frac{P_{\mathrm{S}}}{\sigma^{2}}\). The analytical curves are plotted using Equation (17) while the markers refer to Monte-Carlo simulation results. The simulation results were obtained using \(10^{6}\) channel realizations. Figure 2 plots the SEP of the considered SM-MIMO-FD relay system versus the SNR in dB for two modulation schemes, that is, BPSK (\(a=1\), \(b=2\)) and 4-QAM (\(a=2\), \(b=1\)). We should remind that \(a\) and \(b\) are constants, which depend on the types modulation scheme. These values for each type of modulation are given in Table 6.1 of Reference [26]. As shown in Figure 2, similar patterns of SEPs can be observed in both BPSK and 4-QAM modulation schemes. However, the system with 4-QAM modulation has higher SEP than the system with BPSK modulation. Moreover, the benefit of optimization is also reduced as the modulation order increases. Therefore, although our analysis method can be applied for all modulation types, we use the BPSK modulation in the following figures to clearly show the advantage of our proposed optimization algorithm in reducing the SEP of the considered SM-MIMO-FD relay system. Figure 3 compares the SEPs of the SM-MIMO-FD relay system in two cases, that is \(\alpha=1\) (without optimal power allocation, \(P_{\mathrm{R}}=P_{\mathrm{S}}\)) and \(\alpha=\alpha^{*}\) (with optimal power allocation, \(P_{\mathrm{R}}=P_{\mathrm{R}}^{*}\)) in (30) for different numbers of receiving antennas of R and D. Although we have used \(N_{\mathrm{FR}}=N_{\mathrm{D}}=2,3,4\) to obtain this figure, it is worth noting that we can use different numbers of receiving antennas at R and D for numerical calculation using the closed-form expression of SEP. The modulation used for evaluation is BPSK with parameters \(a=1,b=2\). The typical SIC capability of \(\bar{\Omega}=-10\) dB is used for calculation. It is easy to see from the figure that the analytical results perfectly match the simulation ones. Although the SEPs of both the cases with and without optimal power allocation suffer the same error floors in high SNR regime, the SEP with optimal power allocation is significantly lower than that  
without optimal power allocation in low SNR regime. For example, when \(\text{SNR}=8\) dB, the SEP in the case with \(\alpha^{*}\) is approximately ten times less than the case without \(\alpha^{*}\). Figure 4 shows the impact of the RSI on the SEP of the system with and without optimal power allocation for different SIC capabilities, that is, \(\tilde{\Omega}=-10,-5,0\) dB. When the RSI is small (\(\tilde{\Omega}=-10\) dB), the difference in SEPs in the cases with and without using \(\alpha^{*}\) is large. However, when the RSI becomes larger (\(\tilde{\Omega}=-5\) dB), the benefit of using \(\alpha^{*}\) decreases. Therefore, it is necessary to combine SIC techniques with optimal power allocation to achieve the best performance for this system.