# Performance of Dense Wireless Networks in 5G and beyond Using Stochastic Geometry
## 3 CF of the Aggregated Interference in LOS
### Uniform Cylindrical Arrays
One way to make the CF tractable is by uniformly decomposing the elevation angle \(\theta_{v}\in(0,\pi/2]\) into \(M\) angular sectors of \(\Delta_{v}=\pi/2M\) width; each sector is centered in \(\tilde{\theta}_{m}=\frac{\pi}{2}(\frac{2(M-m)+1}{2M})\); and the width \(\Delta_{v}\) is small enough to let the array gain in every angular sector (2) be constant \(G_{v}^{2}(\tilde{\theta}_{m})\). The array gain is constant on every annulus (ring)-shaped areas with unequal widths (non-uniform ring division for uniform elevations \(\tilde{\theta}_{m}\)). These rings are centered in the radial distance of the intersection of the \(m\)th bisector \(\tilde{\rho}_{m}=h/\tan(\theta_{m}-\frac{\pi}{4M})\), so that the array gain in the \(k\)th wedge and \(m\)th ring is \(\tilde{G}^{2}(\tilde{\rho}_{m},\Phi_{k})\). Within the \(k\)th wedge and \(m\)th ring, the array gain is constant and the interference originated \(I_{k,m}\) has the CF  
\[\Psi_{I_{k,m}}(\omega)=\exp\left(-\frac{\pi}{K}\alpha\lambda|\omega|^{\alpha} \int_{\tau_{m+1}}^{\tau_{m}}\left[\frac{1-\mathbb{E}_{|\beta|^{2}}[e^{it| \beta|^{2}\tilde{G}^{2}(\tilde{\rho}_{m},\Phi_{k})\text{sign}(\omega)}]}{t^{ \alpha+1}}\right]dt\right), \tag{14}\]  
where \(\tau_{m}=|\omega|/(h^{2}+\rho_{m}^{2})^{b}\). The CF of the aggregate interference statistically independent on all \(K\) wedges and \(M\) rings is \(\Psi_{I}(\omega)=\prod\limits_{m=0}^{M-1}\prod\limits_{k=0}^{K-1}\Psi_{I_{k,m }}(\omega)\). Given the solution of the integral \(\int_{0}^{|\omega|/h^{2b}}\left[\frac{1-e^{j\mu t}}{t^{\alpha+1}}\right]dt= \lim\limits_{\varepsilon\to 0}(-j\mu)^{\alpha}\Gamma(-\alpha,-j\mu t)-\frac{1}{ \alpha^{R}}\big{|}_{t=\varepsilon}^{t=|\omega|/h^{2b}}\), for any constant and real \(\mu\), and \(0<\alpha<1\), the CF of the aggregate interference can be shown to reduce to  
\[\Psi_{I}(\omega)=\exp(-\lambda 2\pi R_{max}^{2})\exp\biggl{(}-\frac{\pi \lambda}{C_{\alpha}}|\omega|^{\alpha}(1-j\text{sign}(\omega)\tan\frac{\pi \alpha}{2})P_{G}(\omega,\alpha,h)\biggr{)}, \tag{15}\]  
with \(C_{\alpha}^{-1}=\Gamma(1-\alpha)\cos(\pi\alpha/2)\) and  
\[P_{G}(\omega,\alpha,h)= \frac{\beta^{2\alpha}}{K}\sum\limits_{k=0}^{K-1}\sum\limits_{m=0} ^{M-1}\tilde{G}^{2\alpha}(\tilde{\rho}_{m},\Phi_{k})\] \[\biggl{[}P(-\alpha,-j\omega\xi(\tilde{\rho}_{m},\Phi_{k},h))-P(- \alpha,-j\omega\xi(\tilde{\rho}_{m+1},\Phi_{k},h))\biggr{]}. \tag{16}\]  
We used a compact notation for different moments of fading \(\tilde{\beta}^{c}=\mathbb{E}[|\beta|^{c}]\), and \(P(x,z)=\int_{0}^{z}t^{x-1}e^{-t}dt/\Gamma(x)\) in (16) is the normalized incomplete Gamma function ratio and  
\[\tilde{\mathfrak{G}}(R_{i},\phi_{i})=G_{c}^{2}(\phi_{i})G_{v}^{2}(\arctan(h/ R_{i})) \tag{9}\]0The relationship (16) for \(M\to\infty\) and \(R_{i}\leq R_{max}<\infty\), reduces after some calculus, to  
\[\tilde{\mathfrak{G}}(R_{i},\phi_{i})=G_{c}^{2}(\phi_{i})G_{v}^{2}(\arctan(h/ R_{i})) \tag{9}\]1  
This relation (18) completes the CF of aggregated interference (15). The PDF and the CDF of the interference \(I\) are obtained by numeric inversion. For \(h\to 0\), the UcyIA degenerates into UCA on the ground, and the distribution of \(I\) is alpha-stable for \(R_{max}\to\infty\) (Section 3.2). As can be observed, the general amplitude fluctuations \(\beta_{i}\) of the interferers are embodied in the derivation of the CF (whose values depend on the specific fading model chosen); however, for the numerical calculations throughout the paper, the fading power \(\bar{\beta}^{2}\) is neglected because it is averaged out in the SG over the summation of the interference for all UEs. This assumption holds as long as the average fading is \(\mathbb{E}[|\beta|]=1\)[48] (the fading power for the user of interest \(|\beta_{0}|^{2}\) does not average out, and the results depend on the choice of fading model). **Remark 2**. : _Although the CF of aggregated interference (15) for UcyIA depends on (18), its numerical computation has some trade-offs.