# Performance of Dense Wireless Networks in 5G and beyond Using Stochastic Geometry
## 3 CF of the Aggregated Interference in LOS
### Uniform Cylindrical Arrays  
\(N_{c}\times N_{v}\) UcylA is composed of \(N_{v}\) uniformly spaced rings consisting of \(N_{c}\) antennas each arranged in a cylinder shape (Figure 1). The interference power \(I\) originated from a coverage radius \(R_{max}\rightarrow\infty\) represents the largest possible interference for a density \(\lambda\) and thus it is the upper bound of the interference \(I\) when \(R_{max}<\infty\), the service probability (6) depends on \(R_{max}\), and it is lower bounded: \(P_{s}(R_{o}|R_{max})\geq P_{s}(R_{o}|R_{max}\rightarrow\infty)\). For the computation of the CF of the aggregate interference, one should consider the entire gain pattern of the UcylA (1) and (2). Let \(R_{max}\rightarrow\infty\), the aggregate interference power is  
\[I=\sum_{i=1}^{\infty}\frac{|\beta_{i}|^{2}}{\left(R_{i}^{2}+h^{2}\right)^{b}} \tilde{\mathfrak{G}}^{2}(R_{i},\phi_{i}), \tag{8}\]  
where the served UE is in \((\theta_{o},\phi_{o})=(0,0)\) for analytical notation convenience, and the beamforming gains are reformulated in terms of azimuth (\(\phi_{i}\)) and elevation (\(\theta_{i}=\arctan(h/R_{i})\)) angles  
\[\tilde{\mathfrak{G}}(R_{i},\phi_{i})=G_{c}^{2}(\phi_{i})G_{v}^{2}(\arctan(h/ R_{i})) \tag{9}\]  
The fluctuation power \(|\beta_{i}|^{2}\) is independent of interfering users and identically distributed (iid). The randomly distributed interfering UEs in \(\phi_{i}\in[0,2\pi)\) can be partitioned into a set of \(K\) disjoint angular sectors \(\Phi_{1},\Phi_{2},\ldots\) such that \(\cup_{k}\Phi_{k}\equiv[0,2\pi)\), where \(K\) is large enough so that the array gain \(G_{c}(\Phi_{k})\) in each sector can be considered as constant. The aggregate interference power (8) is  
\[I=\sum_{k=1}^{K}\sum_{\phi_{i}\in\Phi_{k}}\frac{|\beta_{i}|^{2}}{\left(R_{i}^ {2}+h^{2}\right)^{b}}\tilde{\mathfrak{G}}^{2}(R_{i},\phi_{i})\simeq\sum_{k}I_ {k}, \tag{10}\]  
where  
\[I_{k}=\sum_{i\in\Phi_{k}}\frac{|\beta_{i}|^{2}}{\left(R_{i}^{2}+h^{2}\right)^{ b}}\tilde{\mathfrak{G}}^{2}(R_{i},\Phi_{k}). \tag{11}\]  
The asymptotic equality in (10) is due to the fact that every user \(i\) with azimuth \(\phi_{i}\) that is located within the \(k\)-th angular sector is assumed to have the array gain \(\tilde{\mathfrak{G}}^{2}(R_{i},\Phi_{k})\), as defined in (11). This approximation asymptotically converges to the true aggregated interference power for \(k\rightarrow\infty\), while for \(k\) that is large enough (around twice the number of horizontal antennas), it is quite accurate. The CF for the interference within the \(k\)th angular sector follows on from [PERSON]'s theorem as in [11]\[\Psi_{I_{k}}(\omega)=\exp\Biggl{(}-\frac{2\pi}{K}\lambda\int_{0}^{\infty}\Biggl{[}1 -\Psi_{|\beta|^{2}}\left(\omega\frac{\tilde{G}^{2}(r,\tilde{\phi}_{k})}{(r^{2}+h ^{2})^{b}}\right)\Biggr{]}rdr\Biggr{)}, \tag{12}\]  
where \(\Psi_{|\beta|^{2}}(\omega)=\mathbb{E}[e^{i\omega|\beta|^{2}}]\) is the CF of \(|\beta|^{2}\). In order to solve the integral (12), it is more convenient to define \(\omega\left(r^{2}+h^{2}\right)^{-b}=t\). By replacing the variable r with t and changing the boundaries of the integral accordingly, i.e., \(t\to 0\) for \(r\rightarrow\infty\) and \(t\rightarrow|\omega|/h^{2b}\) for \(r\to 0\), after some mathematical manipulations, the (12) becomes (for \(\alpha=1/b\))  
\[\Psi_{I_{k}}(\omega)=\exp\left(-\frac{\pi}{K}\lambda\alpha|\omega|^{\alpha} \int_{0}^{|\omega|/h^{2}}\frac{1-\mathbb{E}_{|\beta|^{2}}[e^{it|\beta|^{2} \tilde{G}^{2}(f(t),\Phi_{k})\text{sign}(\omega)}]}{t^{\alpha+1}}dt\right). \tag{13}\]  
where \(f(t)=\left(|\omega|^{1/b}t^{-1/b}-h^{2}\right)^{1/2}\) follows on from the conversion from variable \(r\) to \(t\). (13) is based on radius \(r\rightarrow\infty\); if the radius is limited, the lower bound of the integral of (13) would be \(t\rightarrow|\omega|/(h^{2}+R_{max}^{2})^{b}\) for \(r\to R_{max}\). Since the array gain is a function of \(t\), this expression can only be solved numerically.