# Transmission Power Adaption for Full-Duplex Relay-Aided Device-to-Device Communication
## 2 System Model  
In Figure 1 we consider a two-antenna deployment scenario, where a macro BS is intended to communicate with a cellular user who is in the edge of traditional cell and suffering severe deep fading or shadowing. This means the direct link between macro BS and cellular is weak and unable to guarantee the rate requirement of cellular user. Moreover, there is a pair of D2D users located nearby the cellular user and there is a good channel quality from BS and cellular respectively. We assume that the transmitter of the D2D pair is equipped with isolated receiving and transmitting antennas and caches the date that intends to send to the D2D receiver. We can automatically see that the system performance will be beneficial from deploying D2D transmitter as a relay for BS and the cellular user. We assume that the D2D communication is allowed to share the downlink spectral resource allocated to the cellular user while remaining under the control of the BS. We set the D2D relay to operate in FD mode, suffering from the backhaul interference is inevitable due to the simultaneously receiving and transmitting signals which is an inherent nature of FD mode. It also should be made clear that this scenario corresponds to the situation that radio resource management has taken place and spectrum resources have been allocated for a particular BS to serve a particular cellular user.  
Figure 1: Relay-aided device-to-device (D2D) communication model in full-duplex (FD) mode.  
We assume that the system operates in the downlink in frequency division duplex (FDD) mode with a bandwidth of \(W\) Hz. We denote the channel impulse response for the link from BS to cellular user, BS to the relay, namely, the D2D transmitter, from BS to the D2D receiver, from the relay to cellular, from the relay to the D2D receiver and the loopback link from the transmitting antenna to the receiving antenna as \(h_{BC}\), \(h_{BR}\), \(h_{BD}\), \(h_{RC}\), \(h_{RD}\) and \(h_{L}\) respectively. These are all complex variables. The system is parametrized with channel gain-to-noise ratios, which are defined as:  
\[\begin{split}\gamma_{BR}=|h_{BR}|^{2}/\sigma^{2},\;\gamma_{BC}=|h _{BC}|^{2}/\sigma^{2},\;\gamma_{BD}=|h_{BD}|^{2}/\sigma^{2}\\ \gamma_{RC}=|h_{RC}|^{2}/\sigma^{2},\;\gamma_{RD}=|h_{RD}|^{2}/ \sigma^{2},\;\gamma_{L}=|h_{L}|^{2}/\sigma^{2}\end{split} \tag{1}\]  
Where \(\sigma^{2}\) denotes the variance of the zero-mean additive white Gaussian noise (AWGN). As we can see, each of these variables actually corresponds to the signal-to-noise ratio (SNR) of one-hop transmission over the particular channel with maximum transmission power. In other words, we set \(p_{B}=1\) and \(p_{R}=1\). All of the channels we used are modeled to be narrow band quasi-static state and frequency flat fading. Note that we assume that the nodes in our scenario are generally stationary or moving at moderate speed to allow channel state information acquisition.  
The transmitting power of BS and relay node are denoted by \(p_{B}\) and \(p_{R}\) respectively. These have been normalized and are subject to independent constraints by \(0\leq p_{B}\leq 1\) and \(0\leq p_{R}\leq 1\) instead of aggregate transmitting power constraint \(p_{B}+p_{R}=2\) as does in [27]. We assume that the D2D relay adapts amplify-and-forward protocol to aid the downlink transmission of a cellular user and the relay uses a fraction of its transmitting power to forward the cellular user's signals.