# Transmission Power Adaption for Full-Duplex Relay-Aided Device-to-Device Communication
## 2 System Model
### FD Relay-Aided D2D Communication Scheme  
Before making an explicit description of the proposed relaying based D2D scheme, we review the traditional cellular direct transmission (DT) mode. In a traditional cellular network, the cellular users have no choice but to communicate with the BS even the DT link suffers from severely deep fading or shadowing. Unfortunately, this situation does not happen in a small probability. When the cellular users are located in the cell edge or urban environment, it will be a frequent occurrence. Therefore, DT links from the BS to cellular users in a conventional cellular mode will lead to a poor performance of the system in this situation, and might even cause high outage probability or call drop for cellular users. Assume frequency-division multiple-access technology is used for the cellular downlink transmissions in DT mode. At any time instant, the BS transmits symbols to the cellular user over the allocated frequency band. Therefore, the achievable rate of direct link is given by:  
\[R_{BC}^{DT}=W\log_{2}(1+p_{B}\gamma_{BC}) \tag{2}\]  
where \(W\) corresponds to the width of the subchannel. Let \(R^{re}\) represent the rate requirement of the cellular user. When the channel impulse response \(h_{BC}\) suffers from deep fading or shadowing, it will lead to the situation that the achievable rate of the DT link will fall below the rate requirement even the BS use maximum transmitting power with \(p_{B}=1\), namely:  
\[R^{re}>W\log_{2}(1+\gamma_{BC}) \tag{3}\]  
Hence, for the sake of fulfilling the requirement rate of cellular user, we point out that the desired channel to noise ratio \(\gamma\) should satisfy:  
\[\gamma\geq\frac{\left(2^{R^{re}/W}-1\right)}{p_{B}} \tag{4}\]  
From the analysis above, we conclude that the conventional DT mode can do nothing to satisfy the cellular user's quality of service requirement in terms of transmitting rate when the DT link is weak. Therefore, we aim to find a novelty D2D relaying-aided scheme to guarantee the cellular user's rate requirement. In order to improve spectral efficiency, throughput and energy efficiency, BS allows UE who are in proximity to communicate directly without the help of BS. Moreover, for the sake of improving the poor performance of conventional cellular mode in which the cellular users suffer severely from deep fading or shadowing, we outline an FD relaying based D2D communication scheme to allow the downlink transmission from BS to cellular user via D2D relay and D2D direct communication from D2D relay to D2D receiver take place at the same instant. In our system model, it is assumed that the D2D relay is equipped with two isolate antennas. We now describe the relaying based D2D protocol. Let us denote \(x(i)\) and \(d(i)\) as the signals transmitted by BS and D2D relay, namely, D2D transmitter at time instant \(i\). Then we have:  
\[E\Big{\{}\left|x(i)\right|^{2}\Big{\}}=p_{B} \tag{5}\]  
\[E\Big{\{}\left|d(i)\right|^{2}\Big{\}}=(1-\alpha)p_{R} \tag{6}\]  
in which \(E\{\cdot\}\) denotes the expectation of variable and \(\alpha\) represents the power split factor at relay. We assume that the D2D relay is willing to share \(\alpha(0\leq\alpha\leq 1)\) fraction of its power to assist the cellular user's downlink transmission. Then the residual \((1-\alpha)\) fraction is used for its own signal transmission. In addition, the system will degrade to a conventional relay forward mode or a D2D underlaying mode when the power split factor \(\alpha\) is equal to zero or one respectively. Furthermore, considering the loopback interference from the relay transmitting antenna to the receiving antenna, the signal \(r(i)\) received at the D2D relay and the transmitting signal \(t(i)\) by D2D relay can be formulated as:  
\[r(i)=h_{BR}x(i)+h_{L}t(i)+n(i) \tag{7}\]  
\[t(i)=gr(i)+\sqrt{(1-\alpha)p_{R}}d(i) \tag{8}\]  
where \(n(i)\) is the AWGN at relay with mean value \(E\Big{\{}\left|n(i)\right|^{2}\Big{\}}=\sigma^{2}\).