# Location-Based Relay Selection in Full-Duplex Random Relay Networks
## 4 Location-Based Relay Selection in FDRRNs
### Analysis of Relay Location in FDRRNs  
From (5), when the locations of the transmitter and receiver nodes are \(\mathbf{x}_{\text{tx}}=(x_{tx},y_{tx})\) and \(\mathbf{x}_{\text{rx}}=(x_{rx},y_{rx})\), respectively, the optimal relay location \(\mathbf{x}_{\text{o}}^{*}\) that minimizes the outage probability of an FDRRN is given by  
\[ \begin{split}(x_{o}^{*},y_{o}^{*})&=\arg\min_{(x,y )}p_{\text{o}}(\mathbf{x}_{\text{r}})\\ &=\arg\min_{(x,y)}g(x,y),\end{split}  \tag{6}\]where \(\mathbf{x}_{\mathrm{r}}=(x,y)\) is the location of the candidate relay node and \(g(x,y)\) can be given by  
\[ \begin{split} g(x,y)=&\frac{I_{S}D_{1}^{\mathrm{A}} \tau}{P_{\mathrm{r}}}+v_{\mathrm{a}}\tau^{2/\alpha}\Big{(}D_{1}^{2}\lambda_{1}+ D_{2}^{2}\lambda_{2}\Big{)}\\ &=\mu\Big{(}(x-x_{tx})^{2}+(y-y_{tx})^{2}\Big{)}^{\frac{\alpha}{2}} \\ &+\rho_{\mathrm{a}}\lambda_{1}\Big{(}(x-x_{tx})^{2}+(y-y_{tx})^{2 }\Big{)}\\ &+\rho_{\mathrm{a}}\lambda_{2}\Big{(}(x-x_{rx})^{2}+(y-y_{rx})^{2 }\Big{)},\end{split}  \tag{7}\]  
where \(\mu=\frac{I_{S}\mathrm{I}\tau}{P_{\mathrm{r}}}\) and \(\rho_{\mathrm{a}}=v_{\mathrm{a}}\tau^{2/\alpha}\). The optimal relay location refers to a specific position where, if the transmitter node selects a relay node located there, it can achieve optimal performance in terms of outage probability. Unfortunately, however, the generalized optimal point of (7) in terms of \(\alpha\) cannot be determined. Therefore, we propose a simplified algorithm to identify a potential relay location that sub-optimally reduces the outage probability of an FDRRN when \(\alpha\) is fixed. **Lemma 1**. : _When \(\alpha=4\), the potential relay location \(\mathbf{x}_{r}^{*}=(x^{*},y^{*})\) in the proposed algorithm can be expressed as follows:_  
\[\mathbf{x}_{r}^{*}=\left( \begin{array}{c}x_{tx}+\frac{\kappa}{6\sqrt[3]{2} \mu}-\frac{2\rho(\lambda_{1}+\lambda_{2})}{\sqrt[3]{4}\kappa},\\ y_{tx}+\frac{\kappa}{6\sqrt[3]{2}\mu}-\frac{2\rho(\lambda_{1}+\lambda_{2})}{ \sqrt[3]{4}\kappa}\end{array} \right), \tag{8}\]  
_where_  
\[\kappa=\sqrt[3]{\sqrt{\zeta_{l}^{2}+4(\rho\mu\lambda_{1}+\rho\mu\lambda_{2})^{ 3}+\zeta_{l}}}, \tag{9}\]  
\[\zeta_{l}=108\rho\lambda_{2}l_{rx}-108\rho\lambda_{2}l_{tx},\ l\in\{x,y\}. \tag{10}\]  
Proof. : To derive the potential relay location that provides a sub-optimal solution in terms of minimizing the outage probability, we calculate the coordinates \(x^{*}\) and \(y^{*}\) separately. The potential relay location can then be expressed as follows:  
\[(x_{o}^{*},y_{o}^{*})=\left(\arg\min_{x}g(x),\ \arg\min_{y}g(y)\right), \tag{11}\]  
where  
\[ \begin{split} g(x)&=\mu(x-x_{tx})^{4}+\rho_{4}\lambda _{1}(x-x_{tx})^{2}+\rho_{4}\lambda_{2}(x-x_{rx})^{2},\\ g(y)&=\mu(y-y_{tx})^{4}+\rho_{4}\lambda_{1}(y-y_{tx})^{2 }+\rho_{4}\lambda_{2}(y-y_{rx})^{2},\end{split}  \tag{12}\]  
respectively. Since \(g(x)\) and \(g(y)\) are convex functions of \(x\) and \(y\), respectively, the potential relay location can be obtained by solving \(\frac{\partial g(x)}{\partial x}=0\) and \(\frac{\partial g(y)}{\partial y}=0\), respectively. Note that for different values of \(\alpha\), we can simply extend the expressions by modifying \(g(x)\) and \(g(y)\) accordingly. Even though (8) provides the closed form of the potential relay location, it is difficult to gain insight into the relationship between the potential relay location and network parameters. In this regard, we further simplify (8) to gain further insight by assuming that \(\alpha=2\). **Lemma 2**. : _For the case when \(\alpha=2\), the potential relay location that minimizes the outage probability of an FDRRN is given by_  
\[\mathbf{x}_{r}^{*}=\bigg{(}\frac{(\mu+\rho_{2}\lambda_{1})x_{tx}+\rho_{2}\lambda_ {2}x_{rx}}{\mu+\rho_{2}\lambda_{1}+\rho_{2}\lambda_{2}},\frac{(\mu+\rho_{2} \lambda_{1})y_{tx}+\rho_{2}\lambda_{2}y_{rx}}{\mu+\rho_{2}\lambda_{1}+\rho_{2} \lambda_{2}}\bigg{)}. \tag{13}\]  
Proof.