## 2 System Model and Problem Formulation
### Full-Duplex Dual-Antenna Energy Harvesting  
In the energy harvesting phase, the relay node uses dual antennas for energy harvesting. The received signal of the relay R can be expressed as  
\[y_{r1}=\left(\begin{array}{c}h_{1}\\ h_{2}\end{array}\right)\sqrt{P_{S}}x_{1}+n_{r}, \tag{1}\]  
where \(P_{s}\) is the transmission power of the source and \(n_{r}\) is the Gaussian white noise of the relay node. The unit power is known to be transmitted to the relay node by the source node. The energy that the relay node obtains from the source node can be expressed as  
\[E=\eta\alpha TP_{s}\Big{(}|h_{1}|^{2}+|h_{2}|^{2}\Big{)}, \tag{2}\]  
where \(\eta(0<\eta<1)\) is the energy conversion efficiency. As in [27], this depends on the rectification process and the energy harvesting circuit. Here, we do not consider the noise power, which is much smaller than the power of the energy signal [28].  
As in [29; 30], we assume that the system works in the original architecture used by the collector. The energy collected during the energy harvesting phase is stored in a supercapacitor. Then, in the information transmission phase, the energy collected in the first phase is used when the relay node completely consumes the process of forwarding the source signal. Therefore, the transmission power of the relay node can be expressed as  
\[P_{r}=\frac{\eta\alpha TP_{s}\Big{(}|h_{1}|^{2}+|h_{2}|^{2}\Big{)}}{(1-\alpha)T }=\frac{\eta\alpha P_{s}\Big{(}|h_{1}|^{2}+|h_{2}|^{2}\Big{)}}{(1-\alpha)}. \tag{3}\]  
In the information transmission phase, the signal received by the relay node R can be represented as  
\[y_{r2}=h_{1}\sqrt{P_{s}}x_{2}+fx_{r}+n_{r}, \tag{4}\]  
Figure 2: (**a**) Two-phase SWIPT protocol. (**b**) Time-switching architecture at the FD relay.  
where \(x_{2}\) is the information signal transmitted by the source node to the relay node in the second phase, \(f\) is the self-interference coefficient that satisfies \(E\Big{\{}\big{|}f\big{|}^{2}\Big{\}}=1\), and \(n_{r}\) is the Gaussian white noise at the relay node, which obeys the distribution with a mean of zero and a variance of \(N_{0}\). Since the relay node can recognize its own signal, it eliminates the interference caused by part of itself [31]. Therefore, after interference cancellation, the signal received by the relay node can be expressed as  
\[\widehat{y_{r2}}=h_{1}\sqrt{P_{s}}x_{2}+\widehat{f\chi_{r}}+n_{r}. \tag{5}\]  
The residual self-interference coefficient caused by the imperfect self-interference cancellation is usually considered as the Rayleigh channel [32].  
In the second phase, we consider the DF protocol. \(x_{r}(n)\) is written as  
\[x_{r}(n)=\sqrt{\frac{P_{r}}{P_{s}}}x_{2}(n-\tau), \tag{6}\]  
where \(\tau\) is the delay. The signal received by the destination node can be expressed as  
\[y_{d}=gx_{r}(n)+n_{d}, \tag{7}\]  
where \(n_{d}\) is the Gaussian white noise at the destination node and \(g\) is the channel coefficient of the relay node to the destination node.  
Therefore, the signal to interference and noise ratio can be expressed as  
\[\text{SINR}=min\Bigg{\{}\frac{|h_{1}|^{2}(1-\alpha)}{\big{|}f\big{|}^{2}\big{(} |h_{1}|^{2}+|h_{2}|^{2}\big{)}\alpha},\frac{\eta P_{s}\Big{(}|h_{1}|^{2}+|h_{2 }|^{2}\Big{)}|g|^{2}\alpha}{N_{0}(1-\alpha)}\Bigg{\}}. \tag{8}\]  
Therefore, the spectrum efficiency of the system can be expressed as  
\[C=(1-\alpha)log_{2}(1+\text{SINR}). \tag{9}\]  
Therefore, the energy efficiency can be expressed as  
\[\xi=\frac{C}{\beta P_{s}+P_{c}}, \tag{10}\]  
where \(P_{c}\) is the constant circuit power consumption, which is independent of information transmission, and \(\beta\) is the power amplifier. The objective function is to maximize the system energy efficiency, i.e.,  
\[max\xi=\frac{(1-\alpha)log_{2}\bigg{(}1+min\bigg{\{}\frac{|h_{1}|^{2}(1-\alpha )}{\big{|}f\big{|}^{2}\big{(}|h_{1}|^{2}+|h_{2}|^{2}\big{)}\alpha},\frac{\eta P _{s}\big{(}|h_{1}|^{2}+|h_{2}|^{2}\big{)}|g|^{2}\alpha}{N_{0}(1-\alpha)} \bigg{\}}\bigg{)}}{\beta P_{s}+P_{c}}. \tag{11}\]