# Transmission Power Adaption for Full-Duplex Relay-Aided Device-to-Device Communication
## 3 The Optimal Power Adaption Strategies  
Let us then define the objective function as:  
\[(p_{B}^{*},p_{R}^{*},\alpha^{*})=\arg\max_{(p_{B},p_{R},\alpha)}R_{D}^{FD} \tag{21}\]  
\[s.\ t.\qquad R_{\text{C}}^{FD}=R^{re} \tag{21a}\] \[0\leq p_{B}\leq 1\] (21b) \[0\leq p_{R}\leq 1\] (21c) \[0\leq\alpha\leq 1 \tag{21d}\]  
Where \((p_{B}^{*},p_{R}^{*},\alpha^{*})\) represents the optimal power allocation for BS and the D2D relay and the power split factor for the FD relaying D2D. Next we propose transmission power adaptation as a technique to mitigate the effect of self-interference on D2D relay. Without optimal transmit power adaptation at the D2D relay, it implies that the D2D transmitter simply uses the maximum allowed power, i.e., \(p_{R}=1\). **Remark 1**: _For extension of cellular coverage, the BS should always use the maximum transmission power, i.e., \(p_{B}=1\). In this way, it will enlarge the SINR for the first relaying hop, namely \(\gamma_{R}=p_{B}\gamma_{BK}/(p_{R}\gamma_{L}+1)\). Moreover, because typically we have \(\gamma_{L}\gg\gamma_{BC}\), the worsening effect on \(\gamma_{C}=\varpi p_{R}\gamma_{RC}/(p_{B}\gamma_{BC}+(1-\alpha)p_{R}\gamma_{RC }+1)\) will be insignificant with the maximum transmission power usage for BS. Thus, the best performance under constraint \(0\leq p_{B}\leq 1\) is achieved when \(p_{B}=1\) and we may concentrate on power adaptation for D2D relay individually. Transmission power adaptation in FD relays is motivated by the observation that the end-to-end performance is limited by the weakest hop. Thus, if the limiting factor is the first hop due to excessive loop interference, the end-to-end performance can be, in fact, improved by decreasing the relay transmit power._  
To solve the optimal problem, we first obtain \(\gamma_{FD}=\gamma\) from \(R_{C}^{FD}=R^{re}\). **Remark 2**: _from (16), we know that \(\gamma_{FD}\) is decided by \(\gamma_{R}\) and \(\gamma_{C}\) jointly. By getting the first class partial differential equation of \(\gamma_{FD}\) in term of \(\gamma_{R}\), namely, \(\partial\gamma_{FD}/\partial\gamma_{R}\), we can get the expression as:_  
\[\frac{\partial\gamma_{FD}}{\partial\gamma_{R}}=\frac{\gamma_{C}^{2}+1}{(\gamma _{C}+\gamma_{R}+1)^{2}} \tag{22}\]  
The equation above is obviously constant positive. Therefore, \(\gamma_{FD}\) is monotonous increasing in terms of \(\gamma_{R}\). We can derive that this is the case for \(\gamma_{C}\) also. It is easy to understand from this result that the end-to-end SINR of two hops FD relay will be improved by either of any one link with better channel quality. So, for the sake of fulfilling the rate requirement of cellular, we should enhance one or both of \(\gamma_{R}\) and \(\gamma_{C}\). Let us then analyze the corresponding received SINR in the D2D relay and in the destination of the cellular user which are expressed as shown in (12) and (15) respectively. In the first hop between BS and D2D relay, the SINR will be improved by decreasing the transmitting power of the relay due to the excessive loopback interference. In contrast, the SINR at the destination of the cellular user is a monotonous increasing function in terms of the transmitting power of relay. Then we know that there will be an optimal transmitting power of relay. In terms of power splitting factor of \(\alpha\), the SINR at the destination of the cellular user will also be improved following \(\alpha\) increasing. On the contrary, the SINR for D2D receiver will decrease as the spilt factor increases. Now we provide a new insight for the objective function which considers fulfilling the rate requirement of the cellular user in the primary and improves the D2D transmitting rate in the secondary. We have observed that an optimal power at the D2D relay exists such that a higher or lower value will result in the deterioration of the end-to-end SINR of cellular user. Therefore, by getting the optimal transmitting power of relay, the energy consumption for D2D users who usually handle equipment such as a mobile phone will decrease.