## 4 Numerical Results and Discussions
### Comparison between the Proposed D2D-CRS-SM and VBLAST MIMO with ZF Detection  
Comparative results between the D2D-CRS-SM and spatial multiplexing MIMO (e.g., VBLAST) with ZF Detection, in terms of BER and SE are presented here. Recall that modulation order (i.e., symbol information) is the desired information of FU and NU in S\({}_{1}\) and S\({}_{2}\), respectively. On the contrary, antenna indices of BS is the desired information of NU and FU in S\({}_{1}\) and S\({}_{2}\), respectively. Therefore, for a fair comparison between the proposed D2D-CRS-SM and VBLAST-ZF, only the incorrect estimation of symbol information of FU in S\({}_{1}\) and NU in S\({}_{2}\) is considered in Figures 21 and 22, respectively. From both figures, it is apparent that for the same set of modulation order and antenna configuration, the proposed SM-based model (i.e., D2D-CRS-SM) outperforms the VBLAST-ZF for any value of SNR, although higher number of bits can be transmitted by VBLAST-ZF than D2D-CRS-SM. Please note that the SE loss is not limitation of our system. Rather, it is the limitation of SM-based system [12; 55]. Furthermore, it should be mentioned that SE obtained by SM-based system is sub-optimal because only one antenna of transmitter is activated at a particular time instant to transmit information. As a result, SM-based system can achieve lower SE than spatial multiplexing MIMO technique (e.g., VBLAST). Although SM-based systems provide sub-optimal SE, it has many advantages including simpler transmitter design, reduced receiver complexity, lower transmit power supply and so on. However, there are two alternative ways for SM-based system to achieve the SE as capacity achieving VBLAST: (i) having the same modulation order in both systems, SM-based systems should use a larger number of transmit antennas, or (ii) having the same antenna configuration in both systems, SM-based systems should use a higher modulation order [55]. In Table 4, the maximum achievable theoretical sum SE comparison is summarized between the proposed D2D-CRS-SM and VBLAST-ZF. From the table, it is apparent that the proposed system can achieve a competitive sum SE as in VBLAST-ZF by increasing modulation order for the same antenna configuration. Moreover, as shown in the simulation figures of the proposed system, by keeping the same modulation order, exploiting a larger number of transmit antennas can also achieve a better SE. The maximum achievable theoretical sum SE of D2D-CRS-SM in S\({}_{1}\) and S\({}_{2}\) is given respectively by  
\[R_{\text{SE,sum},S_{1}}^{\text{D2D-CRS-SM}}= \frac{1}{2}\left\{\frac{\log_{2}A_{\text{BS}}}{\text{NU}}+\frac{ \min(\log_{2}M_{1},\log_{2}M_{2})}{\text{FU}}+\frac{\log_{2}A_{\text{D}_{1}}} {\text{D}_{2}}\right\}, \tag{14}\] \[R_{\text{SE,sum},S_{2}}^{\text{D2D-CRS-SM}}= \frac{1}{2}\left\{\frac{\log_{2}M_{1}}{\text{NU}}+\frac{\min(\log _{2}A_{\text{BS}},\log_{2}M_{2})}{\text{FU}}+\frac{\log_{2}A_{\text{D}_{1}}} {\text{D}_{2}}\right\}. \tag{15}\]  
Figure 20: Sum SE comparison between S\({}_{1}\) and S\({}_{2}\). On the contrary, maximum achievable theoretical sum SE of VBLAST-ZF is given by  
\[R_{\text{SE,sum}}^{\text{VBLAST}}= \frac{1}{4}\bigg{\{}\frac{(A_{\text{BS}}\times N_{1})}{\text{NU}}+ \frac{\text{min}((A_{\text{BS}}\times N_{1}),(A_{\text{D}_{1}}\times N_{1}))}{ \text{FU}}+\frac{(A_{\text{D}_{1}}\times N_{1})}{\text{D}_{2}}\bigg{\}}. \tag{16}\]  
It should be mentioned that the duplexing loss is taken into account due to two-hop communication model. That is why, overall SE is divided by 1/2 in D2D-CRS-SM and 1/4 in VBLAST-ZF. This means that four time-slots in total are required to receive data by all the three users (NU, FU, and D\({}_{2}\)) if the considered network scenarios exploit VBLAST-ZF system, considering time-division multiple access. However, only two time slots are required in the proposed D2D-CRS-SM. Please note that for VBLAST-ZF, S\({}_{1}\) and S\({}_{2}\) are same. Figure 21: A comparative study between the proposed D2D-CRS-SM and VBLAST-ZF in terms of BER of FU in S\({}_{1}\).