# Optimized Gateway Placement for Interference Cancellation in Transmit-Only LPWA Networks
## 3 Single GW Placement for Two SNs
### IC and the Decoding Crescent
Define  
\[F(\alpha,\gamma)=\frac{D_{r_{1},r_{2}}^{2}-D_{c_{1},c_{2}}^{2}}{\|s_{1}-s_{2} \|^{2}}. \tag{21}\]  
\[\Longleftrightarrow\tau^{\frac{1}{\alpha}}\leq\frac{d_{2}}{d_{1}}\leq(z* \tau)^{-\frac{1}{\alpha}}, \tag{14}\]0  
For \(\tau\geq 1\), it is obvious that \(\alpha=\tau^{2/n}\geq 1\). We know that \(z\leq\frac{1}{\tau^{2}}\); then, \((\tau z)\leq 1/\tau\leq 1\), and \(\alpha\gamma=(z\tau)^{2/n}\leq 1\). Therefore, \(F(\alpha,\gamma)\geq 0\). Similarly, set \(\alpha=\tau^{2/n},\gamma=z^{2/n}\) and define  
\[\Longleftrightarrow\tau^{\frac{1}{\alpha}}\leq\frac{d_{2}}{d_{1}}\leq(z* \tau)^{-\frac{1}{\alpha}}, \tag{14}\]1  
Then,  
\[\Longleftrightarrow\tau^{\frac{1}{\alpha}}\leq\frac{d_{2}}{d_{1}}\leq(z* \tau)^{-\frac{1}{\alpha}}, \tag{14}\]2  
\[\Longleftrightarrow\tau^{\frac{1}{\alpha}}\leq\frac{d_{2}}{d_{1}}\leq(z* \tau)^{-\frac{1}{\alpha}}, \tag{14}\]3  
where \(\gamma=z^{2/n}\) and \(0\leq z\leq 1\); then \((1-\gamma)\geq 0\). Because \(z\leq\frac{1}{\tau^{2}}\) and \((1-\alpha^{2}\gamma)\geq 0\), then, \(G(\alpha,\gamma)\geq 0\). Therefore, it is proven that the ICD circle will be inside of the capture circle. Furthermore, \(z\leq\frac{1}{\tau^{2}}\leq\frac{1}{\tau}\), for \(\tau\geq 1\), there exists a region which satisfies capture and IC requirements for GW according to Equations (8) and (12). Finally, Lemma 1 is proven as shown in Figure 3(a). There are two symmetric shaded regions. A GW located in the shadowed region on the left will decode \(s_{i}\) first and \(s_{j}\) second. A GW located in the shadowed region on the right will decode \(s_{j}\) first and \(s_{i}\) second. We refer to each shaded crescent-shaped region in Figure 3(a) as a capture and ICD crescent or just a decoding crescents. **Lemma 2**. : _For any pair of SNs \((S_{i},S_{j}),(i\
eq j)\), such that \(\frac{1}{\tau^{2}}<z\leq\frac{1}{\tau}\), the capture circle will be inside of the ICD circle and there are no locations that satisfy both capture and ICD requirements for GW, given \(\tau\geq 1\)._  
Proof. : According to Equation (22), for \(\tau\geq 1\), it is obvious that \(\alpha=\tau^{2/n}\geq 1\). Furthermore, because \(\frac{1}{\tau^{2}}<z\leq\frac{1}{\tau}\), then, \((\tau z)\leq 1\), and \(\alpha\gamma=(z\tau)^{2/n}\leq 1\). Therefore, \(F(\alpha,\gamma)\geq 0\) and this means one circle will be inside of another circle. Furthermore, from Equation (24), when \(\gamma=z^{2/n}\) and \(0\leq z\leq 1\), then \((1-\gamma)\geq 0\). Because \(\frac{1}{\tau^{2}}<z\leq\frac{1}{\tau}\), then \((1-\alpha^{2}\gamma)\leq 0\) and \(G(\alpha,\gamma)\leq 0\). Now, it is proven Finally, because \(\frac{1}{\tau^{2}}<z\leq\frac{1}{\tau}\), for \(\tau\geq 1\), there is no region that satisfies both capture and ICD requirements for GW according to Equations (8) and (12), and Lemma 2 is proven, as shown in Figure 4b. The symmetric shaded region shows that capture region is inside of the capture circle and the ICD region is outside of the ICD circle. There are no overlap regions because the capture circle is inside of ICD circle. **Lemma 3**: _For any pair of SNs \((S_{i},S_{j}),(i\
eq j)\), such that if \(z>\frac{1}{\tau}\), there will be no overlap of the capture circle and the ICD circle, given \(\tau\geq 1\)._  
Based on the same idea from Equation (22), for \(\tau\geq 1\), then \(\alpha=\tau^{2/n}\geq 1\). Because \(z>\frac{1}{\tau}\), then, \((\tau z)>1\), and \(\alpha\gamma=(z\tau)^{2/n}>1\). Therefore, \(F(\alpha,\gamma)<0\) and there is no intersection between the capture circle and the ICD circle, as shown in Figure 4c,d for different capture order by \(s_{i}\) and \(s_{j}\). Both packets transmitted by \(s_{i}\) and \(s_{j}\) in two-way collision will be decoded successfully by a GW inside of either one of the decoding crescents. Furthermore, it has to be noticed that, when \(z=\frac{1}{\tau^{2}}\), the ICD circle and capture circle for \((s_{i},s_{j})\) will be the same circle.