## 3 Optimal Power Allocation for FD Mode
\begin{split}\text{SEP}=&\frac{a\sqrt{b}}{2\sqrt{2 \pi}}\left[\int\limits_{0}^{\infty}\frac{e^{-bx/2}}{\sqrt{x}}dx-\int\limits_{0 }^{\infty}\frac{e^{-bx/2}}{\sqrt{x}}e^{-\frac{x}{\gamma_{\rm R}}-\frac{x}{ \gamma_{\rm D}}}\sum_{i=0}^{N_{\rm R}-1}\sum_{j=0}^{N_{\rm D}-1}\frac{1}{i!j! }\frac{x^{i+j}}{\gamma_{\rm R}^{i}\gamma_{\rm D}^{j}}dx\right]\\ =&\frac{a}{2}-\frac{a\sqrt{b}}{2\sqrt{2\pi}}\sum_{i= 0}^{N_{\rm R}-1}\sum_{j=0}^{N_{\rm D}-1}\frac{\Gamma(i+j+\frac{1}{2})}{i!j! \gamma_{\rm R}^{i}\gamma_{\rm D}^{j}(\frac{1}{\gamma_{\rm R}}+\frac{1}{\gamma _{\rm D}}+\frac{b}{2})^{i+j+\frac{1}{2}}}.\end{split}  \tag{17}\]  
It is worth noting that we have used equations ([29] (3.361.2)) and ([29] (3.381.4))to solve the first and second integrals in (17), respectively. For the purpose of improving system performance and reducing the impact of the RSI in FD mode, we can calculate the optimal transmission power of R to minimize the system SEP. The optimal transmission power of R for minimizing the system SEP, denoted by \(P_{\rm R}^{*}\), is defined as  
\[P_{\rm R}^{*}=\arg\min_{\rm R}\text{SEP}. \tag{18}\]  
To explicitly determine the minSEP in (18), we begin with two terms in (17), that is, SEP\({}_{1}\) and SEP\({}_{2}\), as follows  
\[\text{SEP}=\underbrace{\frac{a}{2}}_{\text{SEP}_{1}}-\underbrace{\frac{a\sqrt {b}}{2\sqrt{2\pi}}\sum_{i=0}^{N_{\rm R}-1}\sum_{j=0}^{N_{\rm D}-1}\frac{\Gamma (i+j+\frac{1}{2})}{i!j!\gamma_{\rm R}^{i}\gamma_{\rm D}^{j}(\frac{1}{\gamma_{ \rm R}}+\frac{1}{\gamma_{\rm D}}+\frac{b}{2})^{i+j+\frac{1}{2}}}}_{\text{SEP}_{ 2}}, \tag{19}\]  
where  
\[\text{SEP}_{1}=\frac{a}{2}, \tag{20}\]  
and  
\[\text{SEP}_{2}=\frac{a\sqrt{b}}{2\sqrt{2\pi}}\sum_{i=0}^{N_{\rm R}-1}\sum_{j =0}^{N_{\rm D}-1}\frac{\Gamma(i+j+\frac{1}{2})}{i!j!\gamma_{\rm R}^{i}\gamma_ {\rm D}^{j}(\frac{1}{\gamma_{\rm R}}+\frac{1}{\gamma_{\rm D}}+\frac{b}{2})^{ i+j+\frac{1}{2}}}. \tag{21}\]  
As mentioned in Section 3, after (7), since \(a\) and \(b\) are constants, SEP\({}_{1}=\frac{a}{2}\) is also constant. Therefore, we have  
\[\min\text{SEP}=\min(\text{SEP}_{1}-\text{SEP}_{2})=\text{SEP}_{1}-\max\text{ SEP}_{2}=\max\text{SEP}_{2}. \tag{22}\]  
Next, (22) can be rewritten as  
\[\min\text{SEP}=\max\text{SEP}_{2}=\max\left(\frac{a\sqrt{b}}{2\sqrt{2\pi}} \sum_{i=0}^{N_{\rm R}-1}\sum_{j=0}^{N_{\rm D}-1}\frac{\Gamma(i+j+\frac{1}{2})}{ i!j!\gamma_{\rm R}^{i}\gamma_{\rm D}^{j}(\frac{1}{\gamma_{\rm R}}+\frac{1}{ \gamma_{\rm D}}+\frac{b}{2})^{i+j+\frac{1}{2}}}\right), \tag{23}\]and (23) can be presented as  
\[\min\text{SEP}=\max\Bigg{(}\frac{a\sqrt{b}}{2\sqrt{2\pi}}\sum_{i=0}^{N_{\text{R}}- 1}\sum_{j=0}^{N_{\text{D}}-1}\frac{\Gamma(i+j+\frac{1}{2})}{i!j!}\times\frac{1} {\hat{\gamma}_{\text{R}}^{i}\hat{\gamma}_{\text{D}}^{j}(\frac{1}{\mathcal{T}_{ \text{R}}}+\frac{1}{\mathcal{T}_{\text{D}}}+\frac{b}{2})^{i+j+\frac{1}{2}}} \Bigg{)}. \tag{24}\]  
It is noted that, in (24), \(a\), \(b\), \(N_{\text{FR}}\) and \(N_{\text{D}}\) are constants; \(i\) and \(j\) are antenna indices which do not depend on the transmission power of S and R. Moreover, \(\Gamma(i+j+\frac{1}{2})\) is also a constant for the certain values of \(i\) and \(j\). Therefore, (24) is maximized when  
\[\frac{1}{\hat{\gamma}_{\text{R}}^{i}\hat{\gamma}_{\text{D}}^{j}( \frac{1}{\mathcal{T}_{\text{R}}}+\frac{1}{\mathcal{T}_{\text{D}}}+\frac{b}{2 })^{i+j+\frac{1}{2}}} \tag{25}\]  
is maximized or  
\[\hat{\gamma}_{\text{R}}^{i}\hat{\gamma}_{\text{D}}^{j}(\frac{1}{ \mathcal{T}_{\text{R}}}+\frac{1}{\mathcal{T}_{\text{D}}}+\frac{b}{2})^{i+j+ \frac{1}{2}} \tag{26}\]  
is minimized.