# Performance of Dense Wireless Networks in 5G and beyond Using Stochastic Geometry
## 4 Statistical Characterization of the Aggregated Interference Power  
The aggregate interference \(I\) in Section 3.1 for UcylA and arbitrary height \(h>0\) is complex to be computed analytically in a closed-form. Herein, we propose a method for the approximation of the aggregate interference CF. We show that the aggregated interference power for an array of antennas located at an arbitrary height can be approximated using a weighted mixture of two stable distributions, and we detail herein the equivalent CF. In order to gain a deeper insight into the distribution at an arbitrary height, one can start from the Taylor series of the argument of the CF (\(\Psi_{I}(\omega)\)). For example, the series for a UCA with arbitrary height and \(|\beta_{i}|=1\), follows from the CF (19) that, with some simplifications, yields:  
Figure 9: Average number of the users served vs. antenna ratio \(\log_{2}(N_{c}/N_{v})\), within \(\bar{R}=100\) m while keeping constant the total number of antennas \((N_{c}N_{v}=256)\) for different interferer density \(\lambda=\{1,5,10,20,50\}\times 10^{-3}\) m\({}^{-2}\), SIR threshold \([T]_{dB}=5\), AP height \(h=5\) m, \(R_{max}=400\) m, normalized amplitude \(|\beta_{0}|=1\): Solid lines correspond to \(2b=2\) and dashed lines corresponding to \(2b=3.6\). \[\Psi_{I}(\omega)=\exp(\pi\alpha\lambda\Xi(\alpha,\omega)), \tag{33}\]  
where the Taylor series is  
\[\Xi(\alpha,\omega)=\frac{1}{K}\sum_{k=1}^{K}\sum_{i=1}^{\infty}\frac{(\frac{1}{ \frac{1}{R^{2k}}})^{j-\alpha}}{(i-\alpha)i!}\Big{(}j\omega\tilde{\beta}^{2}\,G _{c}^{2}(\Phi_{k})\Big{)}^{i}. \tag{34}\]  
Let us define \(G_{avg}^{(z)}=\frac{1}{K}\sum_{k=1}^{K}G_{c}^{2\,z}(\Phi_{k})\), which for \(h=0\) relation (34) simplifies to  
\[\Xi(\alpha,\omega)=(-j\omega)^{\alpha}\Gamma(-\alpha)\tilde{\beta}^{2\alpha} \,G_{avg}^{(\alpha)}. \tag{35}\]  
For very large heights \(h\gg 0\) in (33), the terms with a higher index \(i\) are negligible, and it can approximated with only the first two terms, which would make the CF (\(\Psi_{I}(\omega)\)) have a Gaussian distribution (the CF of a Gaussian distribution is \(\exp\Big{(}j\mu\omega-\frac{\sigma^{2}}{2}\omega^{2}\Big{)}\) with \(\mu\) being the shift parameter and \(\sigma\) is the standard deviation). In order to evaluate the appropriate CF for any \(h>0\), one isolates the behavior vs. \(j\omega\) from \(\Xi(\alpha,\omega)\) as:  
\[\Xi(\alpha,\omega)=\Xi^{\prime}(\alpha,\omega)+\frac{(\frac{1}{h^{2k}})^{1- \alpha}}{1-\alpha}j\omega\tilde{\beta}^{2}\,G_{avg}^{(1)}, \tag{36}\]  
where the second term (corresponding to \(i=1\) in (33)) is a shift or location parameters. The behavior of the real part of \(\Xi(\alpha,\omega)^{\prime}\) reveals the corresponding exponent of \(\omega\), for every defined \(\omega\). The reason behind separating \(\Xi(\alpha,\omega)\) into two parts is that we need to omit the shift, to be able to visualize the exponent of the \(\omega\) within the distribution. For large h, \(\Xi^{\prime}(\alpha,\omega)\) vs. \(\omega\) behaves as \(\omega^{2}\), and for very small h, behaves as \(\omega^{\alpha}\), while for medium heights, it has two different slopes based on \(\omega\). The transition \(\omega\) where the behavior changes is \(\bar{\omega}\), which depends on the height, the path-loss exponent, UE density and array gain.