# Performance of Dense Wireless Networks in 5G and beyond Using Stochastic Geometry
## 3 CF of the Aggregated Interference in LOS
### Specific Cases  
UCA: It is a special case of UcyIA for \(N_{v}=1\) (i.e., \(G_{v}(\theta)=1\)). The statistical distribution of aggregated interference \(I\) for UCA can be adapted by considering \(R_{max}\to\infty\), although it can be extended to \(R_{max}<\infty\). In UCA, there is no radial mitigation of the interference, but it is only along the azimuth. After simplifying the relation (15) and resolving the singularity (Appendix A), it yields:  
\[\Psi_{I}(\omega)=\exp\biggl{(}-\frac{\pi\lambda}{C_{\alpha}}|\omega|^{\alpha }(1-j\text{sign}(\omega)\tan\frac{\pi\alpha}{2})P_{G_{c}}(\omega,\alpha,h)+ \pi\lambda h^{2}\biggr{)}, \tag{19}\]  
where  
\[P_{G_{c}}(\omega,\alpha,h)=\frac{\bar{\beta}^{2\alpha}}{K}\sum_{k=1}^{K}G_{c} ^{2\alpha}(\Phi_{k})P\biggl{(}-\alpha,-j\omega\frac{G_{c}^{2}(\Phi_{k})\bar{ \beta}^{2}}{h^{2b}}\biggr{)}. \tag{20}\]  
The limit for \(K\to\infty\) angular sectors is  
\[P_{G_{c}}(\omega,\alpha,h)=\frac{\bar{\beta}^{2\alpha}}{2\pi}\int\limits_{0}^ {2\pi}G_{c}^{2\alpha}(\phi)P\biggl{(}-\alpha,-j\omega\frac{G_{c}^{2}(\phi)\bar {\beta}^{2}}{h^{2b}}\biggr{)}d\phi, \tag{21}\]and this integral can be evaluated numerically. Considering \(h\to 0\) as a special case, \(P_{G_{c}}(\omega,\alpha,h)\to 1\), so the frequency dependence of \(P_{G_{c}}(\omega,\alpha,h)\) vanishes, and the aggregated interference in the \(k\)th angular sector is skewed alpha-stable  
\[G_{c}^{2}(\Phi_{k})\sum_{i\in\Phi_{k}}\frac{|\beta_{i}|^{2}}{R_{i}^{2b}}\sim \mathcal{S}\bigg{(}\alpha=\frac{1}{b},\gamma_{k}\bigg{)}, \tag{22}\]  
where \(\gamma_{k}=\frac{\Delta\phi_{k}}{2}\lambda G_{c}^{2\alpha}(\Phi_{k})\frac{ \tilde{\beta}^{2\alpha}}{C_{\alpha}}\) and \(C_{\alpha}^{-1}=\Gamma(1-\alpha)\cos(\pi\alpha/2)\). \(\mathcal{S}(\alpha,\gamma)\) denotes the skewed stable distribution with characteristic exponent \(\alpha\in(0,2]\), unitary skewness, and scale parameter (or dispersion) \(\gamma\geq 0\) with a characteristic function  
\[\mathbb{E}[e^{i\omega x}]= \begin{cases}\exp[-\gamma|\omega|^{\alpha}(1-j{\rm sign }(\omega)\tan\frac{\pi\alpha}{2})]&\alpha\
eq 1\\ \exp[-\gamma|\omega|^{\alpha}(1+j\frac{2}{\pi}{\rm sign}(\omega)\ln|\omega|)]& \alpha=1\end{cases} .\]  
The overall interference reduces to the sum of skewed stable random terms (straightforwardly from ([49], eq. (1.8)))  
\[I=\sum_{k}l_{k}\sim\mathcal{S}\bigg{(}\alpha=\frac{1}{b},\gamma_{c}\bigg{)}, \tag{23}\]  
where the total dispersion for UCA becomes  
\[\gamma_{c}=\pi\lambda\frac{\tilde{\beta}^{2\alpha}}{C_{\alpha}}\cdot\frac{ \int_{0}^{2\pi}G_{c}^{2\alpha}(\phi)d\phi}{2\pi}, \tag{24}\]  
assuming sectors \(\Delta\phi_{k}\to 0\). Therefore, the aggregated interference for UCA and \(h=0\) is skewed alpha-stable. However, when increasing the height \(h\), the distribution deviates from alpha-stable, as detailed later. Comparing this result with [11], one notices an additional term that depends on the UCA array gain \(G_{c}^{2}(\phi)\), which mitigates the mean level of interference in skewed stable distribution. As before, the fading powers of the interferers are embodied in the derivations, but they will average out in summation of the interference power over all of the UEs (the fading amplitude of each signal from each UE depends on the chosen fading model), while the fading power from the user of interest remains effective in the calculation of the service probability. Point antenna: a single point antenna can be considered as a special case of a UCA, where \(N_{c}=1\) leads to isotropic gain. The radiation gain \(G_{c}(\Phi_{k})=1\) for any azimuth, and the choice of \(K=1\) can be chosen, as it is not necessary to decompose the area into different cones. Placing the antenna at height \(h>0\), the CF of the aggregated interference power follows by simplifying (19) as  
\[\Psi_{I}(\omega)=\exp\biggl{(}-\frac{\pi\lambda}{C_{\alpha}}|\omega|^{\alpha} \tilde{\beta}^{2\alpha}P(-\alpha,-j\omega\frac{\tilde{\beta}^{2}}{h^{2b}})(1- j{\rm sign}(\omega)\tan\frac{\pi\alpha}{2})+\pi\lambda h^{2}\biggr{)}. \tag{25}\]  
This CF (25) generalizes the CF for \(h=0\) in [11].