# Location-Based Relay Selection in Full-Duplex Random Relay Networks
## 4 Location-Based Relay Selection in FDRRNs
### Analysis of Relay Location in FDRRNs
: From (6), the objective function \(g(x,y)\) for \(\alpha=2\) can be expressed as  
\[g(x,y) =(\mu+\rho_{2}\lambda_{1})\Big{(}(x-x_{tx})^{2}+(y-y_{tx})^{2} \Big{)} \tag{14}\] \[\qquad+\rho_{2}\lambda_{2}\Big{(}(x-x_{rx})^{2}+(y-y_{rx})^{2} \Big{)}. \tag{15}\]  
Since \(g(x,y)\) is a convex quadratic function with respect to \(x\) and \(y\), the potential relay location can be obtained as in (13) by solving \(\frac{\partial g(x,y)}{\partial x}=0\) and \(\frac{\partial g(x,y)}{\partial y}=0\), respectively. **Remark 1**. : _Lemma 2 reveals the influence of \(I_{S1}\) and \(\lambda_{i}\) on the potential relay location \(\mathbf{x}_{r}^{*}\). As \(\lambda_{1}\) increases, \(\mathbf{x}_{r}^{*}\) moves toward the transmitter node. Conversely, as \(\lambda_{2}\) increases, \(\mathbf{x}_{r}^{*}\) moves closer to the receiver node. This behavior can be attributed to the fact that the interference at the \(i\)th hop transmission increases as \(\lambda_{1}\) or \(\lambda_{2}\) increases. To compensate for the impact of interference, the potential relay node adjusts its position accordingly, moving closer to either the transmitter node or receiver node to maintain the desired transmission condition. Furthermore, it can be observed that as \(I_{S1}\) increases, \(\mathbf{x}_{r}^{*}\) shifts toward the transmitter node. This indicates that when the SIC capability at the relay node is inadequate, the relay node should be positioned closer to the transmitter node to ensure more reliable communication by minimizing the impact of the residual \(SI\)._  
Note that in this paper, we consider the cases where \(\alpha=2\) and \(\alpha=4\), representing free space and relatively lossy environments, respectively. Other values of the path-loss exponent can also be extended using the simplified algorithm. **Corollary 1**. : _For all values of \(\alpha\) where \(\alpha>2\), the proposed algorithm can determine the potential relay location \(\mathbf{x}_{r}^{*}=(x^{*},y^{*})\)._  
Proof. : For an arbitrary \(\alpha\), \(g(x)\) and \(g(y)\) are given, respectively, by  
\[ \begin{split} g(x)&=\mu(x-x_{tx})^{a}+\rho_{4} \lambda_{1}(x-x_{tx})^{2}+\rho_{a}\lambda_{2}(x-x_{rx})^{2},\\ g(y)&=\mu(y-y_{tx})^{a}+\rho_{a}\lambda_{1}(y-y_{ tx})^{2}+\rho_{a}\lambda_{2}(y-y_{rx})^{2}.\end{split}  \tag{16}\]  
We can then obtain the second derivative of \(g(x)\) with respect to \(x\) as follows:  
\[ \begin{split}\frac{\partial^{2}g(x)}{\partial x^{2}}& =\alpha(\alpha-1)(x-x_{tx})^{a-2}+\rho_{a}\lambda_{1}+\rho_{a} \lambda_{2},\\ \frac{\partial^{2}g(y)}{\partial y^{2}}&=\alpha( \alpha-1)(y-y_{tx})^{a-2}+\rho_{a}\lambda_{1}+\rho_{a}\lambda_{2}.\end{split}  \tag{17}\]  
Note that \(\alpha>2\), \(\rho_{a}\lambda_{1}>0\), and \(\rho_{a}\lambda_{2}>0\). Furthermore, the relay node should be positioned between the transmitter and receiver nodes, i.e., \(x_{tx}<x<x_{rx}\) and \(y_{tx}<y<y_{rx}\). As a result, we can obtain \(\frac{\partial^{2}g(x)}{\partial x^{2}}>0\) and \(\frac{\partial^{2}g(y)}{\partial y^{2}}>0\). This indicates that \(g(x)\) and \(g(y)\) each have a global minimum point, representing the potential relay location determined by the proposed algorithm. Consequently, we can conclude that the potential relay location identified by the proposed algorithm exists for all values of \(\alpha\) where \(\alpha>2\). Corollary 1 demonstrates the applicability of the proposed algorithm in practical scenarios, where the path-loss exponent varies with the environment, by determining the potential relay location based on the case when \(\alpha=4\).