# Performance of Dense Wireless Networks in 5G and beyond Using Stochastic Geometry
## 3 CF of the Aggregated Interference in LOS
### Numerical Validation on Aggregated Interference  
The CF derived for each of the arrays is validated here by numerically computing the PDF and CDF using inverse Fourier methods tailored for statistical distributions to be accurate on the tails of the distributions [50; 51]. The service probability (6) is the comparison metric adopted here for the validation of the CFs in the previous sections, by considering a LOS system with \(\left|\beta_{o}\right|^{2}=1\) and a threshold of \(T=1\) (or 0 dB). The amplitudes are normalized as \(\left|\beta_{o}\right|=\mathbb{E}[\left|\beta\right|]=1\), since in the analysis of the SIR, the transmit power and the path-loss at the reference distance are not important as they are equal for the interferers and user of interest. **Remark 4**. : _Since the CF (15) is based on \(R_{max}\rightarrow\infty\), it is impossible to validate it numerically. Instead, a maximum numerical radius \(R_{max}^{(num)}\) is used. To match the simulation results with analytical ones, we limit the integral in (12) to \(R_{max}=R_{max}^{(num)}\). The choice of a proper \(R_{max}\) for a numerical example to explore the behavior of an infinite area is discussed in [48; 52]._  
Figure 2 demonstrates the comparative analysis of UcyIA and UCA, showing the \(P_{s}(R_{o})\) vs. radius \(R_{o}\) using \(\Psi_{I}(\omega)\) for UcyIA in (15) and for UCA in (19) where the total number of antennas is preserved in all cases (\(N_{c}\times N_{v}=128\)) and the SIR threshold \([T]_{dB}=0\). The transmitting interferers are numerically generated as random PPP with a maximum radius \(R_{max}^{(num)}\), as specified below for every Monte Carlo iteration, and are affected by the array gain \(G(\phi,\theta)=G_{\mathcal{E}}(\phi)G_{v}(\theta)\) (see (1) and (2)) keeping the radial position fixed at \(R_{o}\) for the UE of interest as aggregated interference is isotropic vs. azimuth. The analytical curves (shown in solid lines) are for \(R_{max}=300\) m (unless otherwise mentioned), which are validated by simulation curves (shown in dashed lines with markers). Here, for  
the purpose of illustration, we have also shown the analytical curve (in yellow color) with \(R_{max}\to\infty\), for the case of UCA with \(\lambda=10^{-3}\), that lower bounds a simulation curve with \(R_{max}^{(num)}=2000\) m. Although, in case of UCA, the analytical curves are a lower bound of \(P_{s}(R_{o})\) when compared to the numerical simulation with \(R_{max}^{(num)}\ll R_{max}\), this margin is not considerable in the case of UcylA, since this array automatically cancels out the effect of far-away UEs. Service probability \(P_{s}(R_{o})\) decreases for increasing served UE position \(R_{o}\) as interference from \(\lambda\)-density interferers dominates. For smaller density (here \(\lambda=10^{-3}\) m\({}^{-2}\)), the service probability \(P_{s}(R_{o})>0.5\), up to \(70\) m for UCA and around \(50\) m for UcylA. It can be observed that in the given scenario with given parameters, the UCA seems to surpass the UcylA from the service probability point of view. Increasing the path-loss exponent (\(2b=3.6\) in Figure 3) affects the service probability as aggregated interference is more attenuated for far-away interferers, and it is more effective for denser users (i.e., for larger \(\lambda\), the increase in the path-loss is more beneficial for \(P_{s}(R_{o})\), while it is detrimental for small \(\lambda\)). The rest of the figures in the paper are based on analytical formulas, and simulation results are not reported for the sake of the readability of the figures. As an illustrative example of the optimum height based on relations (26) to (31), Figure 4 shows the received signal power and the mean aggregated interference power, where the transmit power of all the UEs is \([P_{tx}]_{dBm}=20\), and the normalized array gain for the user of interest located at distance \(R_{0}\) is maximum (i.e., \(G_{c}(\phi_{0})=1\)).