## 5 MPM-Based Interference Assessment  
### Mpm  
Assessing the inter-beam interference approximation accuracy by selected antenna pattern models requires considering the propagation environment's transmission properties. This is ensured using geometrical modeling of propagation phenomena in simulation studies. In this case, a geometric structure is a semi-ellipsoid set representing the scattering elements' potential locations. A power delay profile (PDP) is the basis for constructing this geometric structure, which maps the propagation conditions of electromagnetic waves in a real environment. In the PDP, the occurrence of local extremes shows that the received signal can be treated as a superposition of components that reach the receiver (Rx) from different directions. The lengths of the propagation paths of all the components that form the individual extremes are the same. This is due to the same propagation time (i.e., delay). Thus, for individual delays, the potential locations of the scattering elements, from which  
Figure 6: Patterns of non-modified and modified Sinc antennas in the azimuth plane for different directions (\(\Phi_{0}=0^{\circ}\), \(30^{\circ}\), and \(60^{\circ}\)).  
Figure 7: RMSE between 3 GPP and Gaussian or Sinc antenna patterns versus \(\Phi_{0}\).  
the signal components reach the Rx, are determined by the surfaces of the semi-ellipsoids. Hence, this propagation model is called the MPM. The focal points of individual half-ellipsoids are determined by the locations of the transmitter (Tx) and Rx. The major \(a_{n}\), and minor \(b_{n}\), \(c_{n}\), axes corresponding to the \(n\)th delays \(\tau_{n}\) are described by the dependencies:  
\[a_{n}=\frac{1}{2}(c\tau_{n}+D),b_{n}=c_{n}=\frac{1}{2}\sqrt{c\tau_{n}(c\tau_{n} +2D)}, \tag{12}\]  
where \(c\) denotes the speed of light and \(D\) is the Tx-Rx distance.  
The spatial structure of the MPM is shown in Figure 8.  
Using the geometric model makes it possible to assess the interference between spatially differentiated channels, which are constructed based on multi-beam antenna systems. The MPM considers the influence of the spatial location of antenna beams in shaping the power distribution of the electromagnetic field that occurs in the vicinity of the receiving antenna. To determine this distribution, a similar method to the ray-tracing method is used. However, in this case, we did use not rays but radio wave propagation paths that represent the directions of energy flow of the transmitted signal. In the simulation process, the intensity of path generation in particular directions is proportional to the power radiation pattern of the transmitting antenna. In contrast, the intensity of the paths that arrive from the scattering elements \(\mathbf{S_{n}}\)\((\forall n=1,2,\dots,N)\) to the Rx, represents the power angular spectrum (PAS), \(p_{0}(\theta,\phi)\), around the receiving antenna. A detailed description of the PAS determination procedure that uses the MPM is included in [31; 32]. The product of \(p_{0}(\theta,\phi)\) and the antenna radiation pattern can be interpreted as the angular distribution of the signal power \(p(\theta,\phi)\) that is \"seen\" at the output of the receiving antenna. When the radiation direction of the transmitting antenna is equal to \(\Phi_{0}\), then this distribution is the basis for determining the power \(P(\Phi_{0})\) of the received signal in accordance with the relationship:  
\[P(\Phi_{0})=\frac{1}{4\pi}\iint_{\mathbf{\Omega}}p((\theta,\phi),\Phi_{0})\sin \theta d\theta d\phi. \tag{13}\]  
The relationship described by Equation (13) is the basis for determining both the power \(P_{S}(\Phi_{0S})\) of the serving signal and the power \(P_{I}(\Phi_{0I})\) of the interfering signal, which come from the main lobes in directions \(\Phi_{0S}\) and \(\Phi_{0I}\), respectively. Determining these powers makes it possible to evaluate the interference level as a function of the angular separation between the beams of the antenna system.  
Figure 8: Spatial structure of MPM [22].