## 3 Performance Analysis
### Bit-Error-Rate Computation  
Data transmission from BS to FU is a dual-hop transmission. As mentioned in the previous section, D\({}_{1}\) forwards the signal of FU during \(P_{2}\), only if the received signal from BS is correctly estimated during \(P_{1}\), otherwise D\({}_{1}\) remains silent during \(P_{2}\). Hence, to determine the BER at FU, we have to consider the estimated error of links BS\(\rightarrow\) D\({}_{1}\) and D\({}_{1}\rightarrow\)FU during \(P_{1}\) and \(P_{2}\), respectively. Let \(P_{c,\text{FU}}^{S,P_{1}}\) and \(P_{c,\text{FU}}^{A,P_{1}}\) denote the probability of incorrect estimation of transmitted data of FU at D\({}_{1}\) in S\({}_{1}\) and S\({}_{2}\), respectively during \(P_{1}\). Again, let \(P_{c,\text{FU}}^{S,P_{2}}\) denote the probability of incorrect estimation of transmitted \begin{table}
\begin{tabular}{c} **Transmit** \\ **Symbol** \\ \end{tabular} \\ \hline
10010 & 1 & 0010 & 10 (2) & 010 \\
10011 & 1 & 0011 & 10 (2) & 011 \\
10100 & 1 & 0100 & 10 (2) & 100 \\
10101 & 1 & 0101 & 10 (2) & 101 \\
10110 & 1 & 0110 & 10 (2) & 110 \\
10111 & 1 & 0111 & 10 (2) & 111 \\
11000 & 1 & 1000 & 11 (3) & 000 \\
11001 & 1 & 1001 & 11 (3) & 001 \\
11010 & 1 & 1010 & 11 (3) & 010 \\
11011 & 1 & 1011 & 11 (3) & 011 \\
11100 & 1 & 1100 & 11 (3) & 100 \\
11101 & 1 & 1100 & 11 (3) & 100 \\
11101 & 1 & 1101 & 11 (3) & 101 \\
11110 & 1 & 1101 & 11 (3) & 101 \\
11110 & 1 & 1110 & 11 (3) & 111 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: _Cont._ \begin{table}
\begin{tabular}{c} **Transmit** \\ **Symbol** \\ \end{tabular} \\ \hline
0000 & 000 (0) & 0 & 00 (0) & 00 \\
0001 & 000 (0) & 1 & 00 (0) & 01 \\
0010 & 001 (1) & 0 & 00 (0) & 10 \\
0011 & 001 (1) & 1 & 00 (0) & 11 \\
0100 & 010(2) & 0 & 01 (1) & 00 \\
0101 & 010(2) & 1 & 01 (1) & 01 \\
0110 & 011(3) & 0 & 01 (1) & 10 \\
0111 & 011 (3) & 1 & 01 (1) & 11 \\
1000 & 100(4) & 0 & 10 (2) & 00 \\
1001 & 100(4) & 1 & 10 (2) & 01 \\
1010 & 101(5) & 0 & 10 (2) & 10 \\
1011 & 101(5) & 1 & 10 (2) & 11 \\
1100 & 110(6) & 0 & 11 (3) & 00 \\
1101 & 110(6) & 1 & 11 (3) & 01 \\
1110 & 1111(2) & 0 & 11 (3) & 10 \\
1111 & 111(7) & 1 & 11 (3) & 11 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: An example of mapping table at D\({}_{1}\) for data transmission during \(P_{2}\). For \(N_{1}\)(\(=\) 1 and 2) bits of modulation symbol/antenna index from BS, D\({}_{1}\) uses 2\({}^{N_{1}}\)-QAM/PSK. data at FU during \(P_{2}\). Therefore, the end-to-end (e2e) probability of error at FU in S\({}_{1}\) and S\({}_{2}\) can be written respectively as  
\[ \begin{array}{ll}P_{\text{e2e,FU}}^{\text{S}_{1}}=&1-(1-P_{e,\text{FU}}^{ \text{S}_{1}})(1-P_{e,\text{FU}}^{\text{S}_{1}})\\ =&P_{e,\text{FU}}^{\text{S}_{1},\text{P}}+P_{e,\text{FU}}^{\text{S}_{2},\text{ P}}-P_{e,\text{FU}}^{\text{S}_{1},\text{P}}\times P_{e,\text{FU}}^{\text{S}_{1}, \text{P}_{2}},\\ P_{\text{e2e,FU}}^{\text{S}_{2}}=&1-(1-P_{e,\text{FU}}^{\text{A}_{1},\text{P}} )(1-P_{e,\text{FU}}^{\text{S}_{1},\text{P}_{2}})\\ =&P_{e,\text{FU}}^{\text{A}_{1},\text{P}}+P_{e,\text{FU}}^{\text{S}_{2},\text{ P}}-P_{e,\text{FU}}^{\text{A}_{1},\text{P}}\times P_{e,\text{FU}}^{\text{S}_{1}, \text{P}_{2}}.\\ \end{array}  \tag{5}\]  
Apart from the node FU, other communication nodes (e, NU and D\({}_{2}\)) in D2D-CRS-SM receive their data bits through the direct link. Hence, during \(P_{1}\), the probability of incorrect estimation of transmitted data at NU in S\({}_{1}\) and S\({}_{2}\) are denoted by \(P_{e,\text{NU}}^{\text{A}_{1},\text{P}_{1}}\) and \(P_{e,\text{NU}}^{\text{S}_{1},\text{P}_{1}}\), respectively. In addition, the probability of incorrect estimation of antenna index at D\({}_{2}\) is denoted by \(P_{e,\text{D}_{2}}^{\text{A}_{1},\text{P}_{2}}\) during \(P_{2}\). It is worth noting the following observations:  
* NU is only concerned about the correct estimate of antenna index and modulation symbol during \(P_{1}\) in S\({}_{1}\) and S\({}_{2}\), respectively.