## 3 The Full-Duplex Dual Antenna Energy Harvesting
### The Optimal Power Allocation Scheme  
Above, we have proved that when \(0<\alpha<\alpha_{1}\), the optimal time-division factor for maximizing energy efficiency is \(\alpha^{*}\). When \(\alpha_{1}<\alpha<1\), the optimal segmentation factor for maximizing energy is \(\alpha_{1}\). Therefore, when \(\alpha\in(0,1)\), for any \(P_{s}\), the optimal segmentation factor for maximizing energy efficiency is \(\alpha^{*}\in 0,\alpha_{1}\). At this time, \(\text{SINR}=\frac{\eta P_{s}\left(|h_{1}|^{2}+|h_{2}|^{2}\right  
Since the denominator of Equation (27) is greater than zero, the molecule is not judged positive or negative, so we make the molecule of Equation (27)  
\[G(P_{s})=\eta\big{(}|h_{1}|^{2}+|h_{2}|^{2}\big{)}|g|^{2}\alpha(\beta P_{s}+P_{c })-\beta\big{(}(1-\alpha)N_{0}+\eta P_{s}\big{(}|h_{1}|^{2}+|h_{2}|^{2}\big{)}|g| ^{2}\alpha\big{)}|\omega g_{2}\bigg{(}1+\frac{\eta P_{s}\big{(}|h_{1}|^{2}+|h_{ 2}|^{2}\big{)}|g|^{2}\alpha}{N_{0}(1-\alpha)}\bigg{)}|\eta n2. \tag{28}\]  
For the Equation (28) derivation function, the simplification is  
\[\frac{\partial G(P_{s})}{\partial P_{s}}=-\beta\eta\Big{(}|h_{1}|^{2}+|h_{2}| ^{2}\Big{)}|g|^{2}\alpha log_{2}\Bigg{(}1+\frac{\eta P_{s}\big{(}|h_{1}|^{2}+|h _{2}|^{2}\big{)}\big{|}g|^{2}\alpha}{N_{0}(1-\alpha)}\bigg{)}|n2. \tag{29}\]  
Since \(\frac{\partial G(P_{s})}{\partial P_{s}}\) is less than zero for any \(P_{s}\in(0,P_{max})\), \(G(P_{s})\) monotonically decreases in the range of \(P_{s}\in(0,P_{max})\), due to \(G(0)=\eta\Big{(}|h_{1}|^{2}+|h_{2}|^{2}\Big{)}|g|^{2}\alpha P_{c}>0\). If \(G(P_{S1})=0\), \(\xi(P_{S})\) is a monotonically increasing function of \(P_{S}\) in the region \((0,P_{S1}]\) and \(\xi(P_{S})\) is a monotonically decrease function of \(P_{S}\) in the region \((P_{S1},P_{max})\), where \(P_{S1}\) is optimal \(P_{S}{}^{*}\). Otherwise, \(P_{max}\) will be optimal \(P_{S}{}^{*}\).  
If Equation (29) is equal to zero, we get  
\[\begin{split}&\eta\big{(}|h_{1}|^{2}+|h_{2}|^{2}\big{)}|g|^{2} \alpha\beta P_{s}+\eta\Big{(}|h_{1}|^{2}+|h_{2}|^{2}\Big{)}|g|^{2}\alpha P_{c} -\beta\Big{(}(1-\alpha)N_{0}+\eta P_{s}\big{(}|h_{1}|^{2}+|h_{2}|^{2}\big{)}|g |^{2}\alpha\big{)}\big{)},\\ &\hskip-28.452756 ptln\Big{(}N_{0}(1-\alpha)+\eta P_{s}\big{(}|h _{1}|^{2}+|h_{2}|^{2}\Big{)}|g|^{2}\alpha\big{)}+\beta\Big{(}(1-\alpha)N_{0}+ \eta P_{s}\big{(}|h_{1}|^{2}+|h_{2}|^{2}\big{)}|g|^{2}\alpha\big{)}|\eta(N_{0} (1-\alpha))=0.\end{split} \tag{30}\]  
Equation (34) can be converted into  
\[\frac{\beta(\alpha-1)N_{0}+\eta\Big{(}|h_{1}|^{2}+|h_{2}|^{2}\Big{)}|g|^{2} \alpha P_{c}}{\beta e(1-\alpha)N_{0}}=exp\bigg{(}ln\frac{y_{2}}{e(1-\alpha)N_{ 0}}\bigg{)}|\eta\frac{y_{2}}{e(1-\alpha)N_{0}}, \tag{31}\]  
where \(y_{2}=N_{0}(1-\alpha)+\eta P_{s}\big{(}|h_{1}|^{2}+|h_{2}|^{2}\big{)}|g|^{2}\alpha\).  
Using the Lambert W function, Equation (35) can be written as  
\[ln\frac{y_{2}}{e(1-\alpha)N_{0}}=W\Bigg{(}\frac{\beta(\alpha-1)N_{0}+\eta \big{(}|h_{1}|^{2}+|h_{2}|^{2}\big{)}|g|^{2}\alpha P_{c}}{\beta e(1-\alpha)N_{ 0}}\Bigg{)}. \tag{32}\]  
Therefore, we the optimal power allocation \(P_{s}{}^{*}\) can be written as  
\[P_{s}{}^{*}=\frac{(1-\alpha)N_{0}\bigg{(}exp\bigg{(}W\bigg{(}\frac{\beta( \alpha-1)N_{0}+\eta\big{(}|h_{1}|^{2}+|h_{2}|^{2}\big{)}|g|^{2}\alpha P_{c}}{ \beta e(1-\alpha)N_{0}}\bigg{)}+1\bigg{)}+\beta\bigg{)}}{\eta\Big{(}|h_{1}|^{2} +|h_{2}|^{2}\Big{)}|g|^{2}\alpha}. \tag{33}\]