# Transmission Power Adaption for Full-Duplex Relay-Aided Device-to-Device Communication
## 3 The Optimal Power Adaption Strategies
Moreover, our proposed scheme also makes sense of the demand of power split factor \(\alpha\) at the D2D relay. Then the D2D relay can spare more power to transmit its own signals. It should be made clear that the optimal transmitting power at the relay is not the global optimization as the achievable rate of D2D link is monotonously increased as follow \(p_{R}\). We can demonstrate the first class partial differential of \(\gamma_{D}\) in term of \(p_{R}\):  
\[ \begin{array}{rl}\frac{\partial\gamma_{D}}{\partial P_{R}}&=\frac{(1-\alpha) \gamma_{RD}(\gamma_{RD}+\varpi p_{R}\gamma_{RD}+1)-\alpha\gamma_{RD}(1-\alpha)p _{R}\gamma_{RD}}{(\gamma_{RD}+\varpi p_{R}\gamma_{RD}+1)^{2}}\\ &=\frac{(1-\alpha)\gamma_{RD}(\gamma_{RD}+1)^{2}}{(\gamma_{RD}+\varpi p_{R} \gamma_{RD}+1)^{2}}\\ \end{array}  \tag{23}\]In the situation that \(0\leq\alpha\leq 1\), \(\gamma_{RD}\) and \(\gamma_{BD}\) is a constant greater than zero, so \(\partial\gamma_{D}/\partial P_{R}\geq 0\) is permanently satisfied. It is obvious that the SINR will be greater as the transmitter power is larger when the channel to noise ratio is a constant. Therefore, in order to improve the energy efficiency of D2D relay, we adapt the optimal power for guaranteeing cellular rate requirement which is equivalent to the minimum transmission power usage for D2D relay. The optimal FD relay transmission power is:  
\[p_{R}^{*}=\min\left\{1,\sqrt{\frac{(\gamma_{BR}+1)(\gamma_{BC}+1)}{\gamma_{L} \gamma_{RC}}}\right\} \tag{24}\]  
**Proof:** By substituting Equation (12), (15) into (16), the end-to-end SINR for cellular can be given by (we can convert into the following form):  
\[\gamma_{FD}=\frac{\gamma_{BR}\gamma_{RD}p_{R}}{\gamma_{RD}\gamma_{L}p_{R}^{2}+ \gamma_{RD}p_{R}+\gamma_{BD}\gamma_{L}p_{R}+\gamma_{L}p_{R}+\gamma_{BR}\gamma_{ BD}+\gamma_{BD}+1} \tag{25}\]  
by solving the one class partial differential equation of \(\gamma_{FD}\) in terms of \(p_{R}\), we can get the numerator expression of \(\partial\gamma_{FD}/\partial p_{R}\) as:  
\[\gamma_{BR}^{2}\gamma_{RC}\gamma_{BC}+\gamma_{BR}^{2}\gamma_{RC}+\gamma_{BR} \gamma_{RC}\gamma_{BC}+\gamma_{BR}\gamma_{RC}-\gamma_{BR}\gamma_{L}\gamma_{RC} ^{2}p_{R}^{2}=0 \tag{26}\]  
which is equivalent to the one class partial differential equation \(\partial\gamma_{FD}/\partial p_{R}\). The optimal transmission power can be obtained by solving the equation above. Consequently, Equation (24) reveals that the maximum transmission power will be chosen, namely \(p_{R}^{*}=1\) when:  
\[\gamma_{L}\leq\frac{(\gamma_{BR}+1)(\gamma_{BC}+1)}{\gamma_{RC}} \tag{27}\]  
the inequalities above show that the maximum transmit power is optimal only with weak loopback interference. Otherwise, the performance can be improved by backing off from the maximum power as reasoned before. In practice, we apply constrained power adaptation for which the SINRs coincide with those of the unconstrained case given the above whenever the conditions in (27) are not satisfied. By further substituting \(p_{R}^{*}\) into \(\gamma_{FD}=\gamma\) where the equation guarantees the rate requirement of cellular we can obtain the optimal power split factor (also the minimum power the cellular share from the D2D relay) as:  
\[\alpha^{*}=\min\left\{1,\frac{\gamma(\gamma_{BC}+1+\gamma_{RC}p_{R}^{*})( \gamma_{BR}+1+\gamma_{L}p_{R}^{*})}{2\gamma_{BR}\gamma_{RC}p_{R}^{*}}\right\} \tag{28}\]  
Up to now, we have solved the optimal transmit powers allocated at the BS and at the D2D transmitter as well as the optimal power splitting factor at the D2D transmitter. The optimal power allocation could be found at the BS in a centralized manner, but the global channel state information is required. Via a dedicated feedback channel, e.g., the cognitive pilot channel proposed by the E2R2/E3 consortium in [12], the channel state information and the optimal power allocation can be conveyed between the BS and the distributed UE reliably.