# Performance of Dense Wireless Networks in 5G and beyond Using Stochastic Geometry
## 4 Statistical Characterization of the Aggregated Interference Power
Having gained insight into the behavior vs. \(\omega\) and \(\bar{\omega}\), which is explained further on in the text, \(\Xi(\alpha,\omega)\) can be approximated as:  
\[\Xi(\alpha,\omega)=W_{1}(\omega)\times a(\omega)+W_{2}(\omega)\times(1-a( \omega)), \tag{37}\]  
where \(a(\omega)\) is a Heaviside step function, i.e., \(a(\omega)=1\) for \(\omega<\bar{\omega}\) and 0 otherwise (or some function with smoother transition), which acts as a switch between two cases with different behavior:  
\[W_{1}(\omega) \approx \frac{(\frac{1}{R^{2k}})^{1-\alpha}}{1-\alpha}j\omega\tilde{\beta }^{2}\,G_{avg}^{(1)}-\frac{(\frac{1}{R^{2k}})^{2-\alpha}}{2(2-\alpha)}\omega^{ 2}\tilde{\beta}^{4}\,G_{avg}^{(2)}, \tag{38}\] \[W_{2}(\omega) = (-j\omega)^{\alpha}\Gamma(-\alpha)\tilde{\beta}^{2\alpha}\,G_{avg }^{(\alpha)}, \tag{39}\]  
where \(W_{2}\) coincides with the skewed-stable distribution (23). Relation (37) means that the CF is can be decomposed as  
\[\Psi_{I}(\omega)=\exp(\pi\alpha\lambda\,W_{1}(\omega)\,a(\omega))\,.\,\exp(\pi \alpha\lambda\,W_{2}(\omega)\,(1-a(\omega))). \tag{40}\]  
Figure 10 demonstrates the behavior of the real part of \(\Xi(\alpha,\omega)\) vs. \(\omega\) for a single isotropic antenna and set of AP heights \(h=\{0,2,5,20,100\}\) m. It can be seen that by increasing the AP height, the breaking frequency \(\bar{\omega}\) increases, while for extremely large AP heights, it tends to infinite, which is Gaussian distribution behavior. Figure 11 is the same analysis, comparing an isotropic antenna with a UCA consisting of \(N_{c}=16\) isotropic antennas for the set of AP heights \(h=\{0,5\}\) m. It is observed that the slopes of the curves are maintained, while the breaking frequency is increased when using a UCA. One can approximate the CF for a UCA in every arbitrary height, given the knowledge on the transition frequency \(\bar{\omega}\). Knowing the transition point \(\bar{\omega}\), based on height \(h\), one may characterize different \(\omega\) with different statistical distributions. The numerical formulation of \(\bar{\omega}\) vs. antenna height and other parameters can be investigated further. A rule of thumb for \(\bar{\omega}\) can be achieved as follows. Let \(S_{i}(\omega)\) denote the \(i\)-th term of the series (34) as:  
\[S_{i}(\omega)=\frac{(\frac{1}{k^{28}})^{i-\alpha}}{(i-\alpha)!i}\Big{(}j\omega \bar{B}^{2}\,G^{(i)}_{avg}\Big{)}^{i}. \tag{41}\]  
It is empirically observed that \(\bar{\omega}\) can be approximately achieved by imposing the condition \(S_{2}(\omega)=\,S_{3}(\omega)/\bar{B}^{2}\), and by solving for \(\omega\), the \(\bar{\omega}\) can be achieved. Please note that this formula holds for normalized array gain. The rationale behind these conditions lies in the fact that for an \(\alpha\)-stable distribution and for height \(h=0\) m, all the terms for \(i>1\) tend  
\[\Psi_{I}(\omega)=\exp(\pi\alpha\lambda\Xi(\alpha,\omega)), \tag{33}\]0 Real part of the \(\Xi(\alpha,\omega)^{\prime}\) vs. \(\omega\) for \(2b=2.6\) and \(\lambda=1\,\)m\({}^{-2}\), comparing the UCA with single antenna. The two guidelines are parallel with \(\omega^{\alpha}\) and \(\omega^{2}\): Solid lines correspond to a point antenna while dashed lines correspond to a UCA with \(N_{c}=16\) isotropic antennas on a ring. \[\Psi_{I}(\omega)=\exp(\pi\alpha\lambda\Xi(\alpha,\omega)), \tag{33}\]1 Real part of the \(\Xi(\alpha,\omega)^{\prime}\) vs. \(\omega\) for \(2b=2.6\) and \(\lambda=1\,\)m\({}^{-2}\), with a single isotropic antenna for different heights of the array. The two guidelines are parallel to \(\omega^{\alpha}\) and \(\omega^{2}\). For large heights, \(\Xi^{\prime}(\alpha,\omega)\) vs. \(\omega\) behaves as \(\omega^{2}\), and for very small heights it behaves as \(\omega^{\alpha}\), while for medium heights, it has two different slopes based on \(\omega\). to infinite, while for Gaussian distribution, only the first three terms exist. In the mixture case, for \(\bar{\omega}<\omega\) behavior resembles \(\alpha\)-stable.