# Inter-Beam Co-Channel Downlink and Uplink Interference for 5G New Radio in mm-Wave Bands +
## 2 Multi-Ellipsoidal Propagation Model  
In 5G networks, designing wireless links with narrow beams and beamforming technology enforces the use of GBMs. It is particularly important in relation to urbanized areas. In this case, there is a large directional variation in the received powers. The use of GBMs ensures the spatial power distribution in the vicinity of the receiving antenna. In combination with the narrow beam patterns, this approach gives the possibility of a statistical evaluation of the transmission properties of the wireless link. The MPM is one of the GBMs. The set of confocal ellipsoids forms its geometrical structure representing the potential locations of the scattering elements for an emitted radio wave. In the foci of the ellipsoids, a transmitter (Tx) and receiver (Rx) are located. In propagation scenarios, where the Tx and Rx are on the Earth's surface, the MPM structure is represented by a set of semi-ellipsoids, as shown in Figure 1[31].  
Figure 1: MPM geometrical structure considering Earth surface.  
The MPM geometrical structure is closely related to the transmission properties of the propagation environment, which may be described by the PDP or TDL. In multipath environments, we can observe the occurrence of several or a dozen taps or local extremes in the TDLs or PDPs, respectively. It means that as a result of scattering on terrain obstacles, the electromagnetic wave reaches the Rx by different propagation paths but with the same delays \(\tau_{n}\) for \(n=1,2,\ldots,N\)[33]. Thus, based on the properties of geometrical figures, the most probable locations of the scattering elements form an ellipsoid. Of course, the number of ellipsoids is equal to the number \(N\) of time-clusters representing the taps or local extremes in the TDL or PDP, respectively. If the distance between the Tx and Rx is equal to \(D\), then the parameters of individual ellipsoids such as a major, \(a_{xn}\), and minor semi-axes \(b_{yn}\), \(c_{zn}\) describe the following relationships:  
\[a_{xn}=\frac{1}{2}(c\tau_{n}+D), \tag{1}\]  
\[b_{yn}=c_{zn}=\frac{1}{2}\sqrt{c\tau_{n}(c\tau_{n}+2D)}, \tag{2}\]  
where \(c\) denotes the speed of light.  
The geometrical structure of the MPM is described in detail in [29; 30; 31]. The 3D MPM may be reduced to the 2D multi-elliptical model, where propagation phenomena in the azimuth plane are dominant [31]. For this modeling procedure in relation to other GBMs, minimizing the estimation error of the PAS is shown in [34]. In the MPM, the phenomenon of local scattering around the transmitting and receiving antennas is also included. In this case, the AOAs of propagation paths are generated based on the 2D von Mises distribution [29; 35]:  
\[\begin{split}& f_{0}(\theta,\phi)=C_{0}\frac{\exp(\gamma_{\theta }\cos(90^{\circ}-\theta))}{2\pi\mathrm{I}_{0}(\gamma_{\theta})}\cdot\frac{\exp (\gamma_{\phi}\cos\phi)}{2\pi\mathrm{I}_{0}(\gamma_{\phi})}\\ &\quad\text{for }\theta\in\langle 0,90^{\circ}\rangle\text{ and }\phi\in \langle-180^{\circ},180^{\circ}\rangle,\end{split} \tag{3}\]  
where \((\theta,\phi)\) is AOA in the elevation and azimuth planes, respectively, \(\gamma_{\theta}\) and \(\gamma_{\phi}\) define the angular dispersion of the components in the elevation and azimuth planes, respectively, \(\mathrm{I}_{0}(\cdot)\) is the zero-order modified Bessel function of the imaginary argument, and \(C_{0}\) represents the normalizing constant such that \(\frac{C_{0}}{2\pi\mathrm{I}_{0}(\gamma_{\theta})}\int\limits_{0}^{90^{\circ} }\exp(\gamma_{\theta}\cos(90^{\circ}-\theta))d\theta=1\).