## 3 System Model
\begin{table}
\begin{tabular}{c l} \hline \hline
**Variable** & **Meaning** \\ \hline \(m_{t}\) & D2D transmitter \\ \(m_{r}\) & D2D receiver \\ \(r\) & D2D relay station \\ \(c\) & Cellular user \\ \(c_{f}\) & The cellular user whose resource blocks are reused in first hop D2D link \\ \(c_{s}\) & The cellular user whose resource blocks are reused in second hop D2D link \\ \(P_{m_{r}}\) & Transmit power of D2D transmitter \\ \(P_{r}\) & Transmit power of relay station \\ \(P_{c_{f}}\) & Transmit power of cellular user \(c_{f}\) \\ \(P_{c_{s}}\) & Transmit power of cellular user \(c_{s}\) \\ \(d_{m_{t}r}\) & Transmission distance between D2D transmitter and relay station \\ \(d_{c_{f}r}\) & Transmission distance between cellular user and relay station \\ \(h_{m_{t}r}\) & Channel response of the first hop D2D link \\ \(k_{c_{f}r}\) & Channel response of the interference link from cellular user \(c_{f}\) \\ \(\alpha\) & Path-loss exponent \\ \(h_{s_{t}m_{t}r}\) & Small-scale fading (Rayleigh) channel coefficients between D2D transmitter and relay station \\ \(h_{s_{c}r}\) & Small-scale fading (Rayleigh) channel coefficients between cellular user and relay station \\ \(N_{0}\) & One-sided power spectral density of the additive white Gaussian noise \\ \(\gamma_{r}\) & SINR received by relay station \(r\) \\ \(\gamma_{m_{r}}\) & SINR received by D2D receiver \(m_{r}\) \\ \(\gamma_{m_{r}m_{r}}\) & SINR of the two-hop D2D link \\ \(\gamma_{c_{f}}\) & SINR of the cellular user whose resource blocks are reused by the first hop D2D link \\ \(\gamma_{c_{s}}\) & SINR of the cellular user whose resource blocks are reused by the second hop D2D link \\ \(EE_{m_{r}r_{f}r_{s}}\) & Energy efficiency of the two-hop D2D link \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Nomenclature. Figure 1: System model of D2D cooperative relay communications underlying the cellular network. In other words, the proposed algorithm is adaptive to other more complicated path-loss models. The signal to interference plus noise ratio (SINR) received by RS \(r\) can be expressed as:  
\[\gamma_{r}=\frac{P_{m_{t}}h_{m_{t}r}^{2}}{P_{c_{f}}h_{c_{f}r}^{2}+N_{0}}=\frac{P_ {m_{t}}d_{m_{t}r}^{-\alpha}h_{s,m_{t}r}^{2}}{P_{c_{f}}d_{c_{f}r}^{-\alpha}h_{s, c_{f}r}^{2}+N_{0}}. \tag{1}\]  
Here, \(P_{m_{t}}\) and \(P_{c_{f}}\) are the transmit power of D2D TX \(m_{t}\) and CUE \(c_{f}\), respectively. \(h_{m_{t}r}\) and \(h_{c_{f}r}\) denote the channel responses of the first hop D2D link and the interference link from CUE \(c_{f}\), respectively. \(d_{m_{t}r}\) is the transmission distance between D2D TX \(m_{t}\) and RS \(r\), while \(d_{c_{f}r}\) is the distance between CUE \(c_{f}\) and RS \(r\). \(\alpha\) is the path-loss exponent corresponding to the large-scale fading of the transmission channel. \(h_{s,m_{t}r}\) and \(h_{s,c_{f}r}\) are the small-scale fading (Rayleigh) channel coefficients, which obey the complex Gaussian distribution \(\mathcal{CN}(0,1)\). \(N_{0}\) represents the one-sided power spectral density of the additive white Gaussian noise (AWGN). The SINR received by D2D RX \(m_{r}\) is given by:  
\[\gamma_{m_{r}}=\frac{P_{r}h_{m_{r}}^{2}}{P_{c_{s}}h_{c_{m_{t}}}^{2}+N_{0}}= \frac{P_{r}d_{rrrh}^{-\alpha}h_{s,m_{t}}^{2}}{P_{c_{s}}d_{c_{m}}^{-\alpha}h_{s,c_{m_{t}}}^{2}+N_{0}}, \tag{2}\]  
where \(P_{r}\) and \(P_{c_{s}}\) are the transmit power of D2D RS \(r\) and CUE \(c_{s}\), respectively. \(h_{rrh_{r}}\) and \(h_{c_{s}m_{r}}\) represent the channel responses of the second hop D2D link and the interference link from CUE \(c_{s}\), respectively. \(d_{rrh_{r}}\) is the transmission distance between RS \(r\) and D2D RX \(m_{r}\), and \(d_{c_{s}m_{r}}\) is the distance between CUE \(c_{s}\) and D2D RX \(m_{r}\). \(h_{s,rrh_{r}}\) and \(h_{s,c_{s}m_{r}}\) are the corresponding small-scale fading channel coefficients. **Lemma 1**.