# Modeling of Downlink Interference in Massive MIMO 5G Macro-Cell +
## 3 Interference Evaluation in Downlink  
### Fundamentals of the Multi-Elliptical Propagation Model  
Dispersion in the angular domain is characteristic of areas where multipath propagation occurs, e.g., urbanized areas with non-line-of-sight (NLOS) or even line-of-sight (LOS) conditions [10]. In such propagation environments, the basis for power assessment is a power angular spectrum (PAS), \(p(\theta,\phi,D)\), where \(\theta\) and \(\phi\) are the angles of arrival (AOA) in the elevation and azimuth planes, respectively, and \(D\) is the distance between a transmitter (Tx) and receiver (Rx). This function allows the received power \(P_{R}(D)\) to be determined according to the relationship [33]  
\[P_{R}(D)=\iint\limits_{\Omega}p(\theta,\phi,D)d\theta d\phi. \tag{1}\]  
where \(\Omega=\left\{(\theta,\phi):\theta\in\left[0^{\circ},90^{\circ}\right),\phi\in \left[-180^{\circ},180^{\circ}\right)\right\}\). Thus, knowing the PAS for signals for useful (i.e., reference, serving) and unwanted (i.e., interfering) beams allows the energy relation between them to be assessed. In this paper, we analyze the transmission of signals in a frequency range from 3 to 4 GHz and with receiving point distances at 100, 200, and 500 m. For these conditions, we can assume that the dispersion phenomenon of the received power dominates in the azimuth plane. This fact is shown in [27; 33]. In this case, the SIR between the useful signal strength \(P_{R0}(D)\)and the power of the interfering signal \(P_{RI}(D)\) that comes from the unwanted beam has the following form [5]:  
\[SIR(D)(\text{dB})=10\log_{10}\frac{p_{R0}(D)}{p_{RI}(D)}=10\log_{10}\frac{\int\limits _{-180^{\circ}}^{180^{\circ}}p_{0}(\phi,D)d\phi}{\int\limits_{-180^{\circ}}^{180^ {\circ}}p_{I}(\phi,D)d\phi}, \tag{2}\]  
where \(p_{0}(\phi,D)=\int\limits_{0}^{90^{\circ}}p_{0}(\theta,\phi,D)d\theta\) and \(p_{I}(\phi,D)=\int\limits_{0}^{90^{\circ}}p_{I}(\theta,\phi,D)d\theta\) represent the PASs of the serving and interfering signals in the azimuth plane, respectively. Equation (2) reduces the SIR evaluation to determine \(p_{0}(\phi,D)\) and \(p_{I}(\phi,D)\) in the case when the MPM is used. The geometry of this model describes the most probable locations of scatterers. Its structure consists of a set of confocal ellipses whose foci determine the positions of the Tx and Rx, i.e., the gNodeB and UE for the DL scenario, respectively. The scattering geometry of the MPM in the azimuth plane is illustrated in Figure 1[27], whereas Figure 2 depicts the simplified MPM simulation procedure. Based on considered assumptions, i.e., the Tx-Rx distance--spatial scenario (step 1) and a chosen power delay profile (PDP) for LOS/NLOS conditions (step 2), in step 3, we calculate parameters of scattering geometry structure. For the \(n\)th ellipse (time-cluster), the major, \(a_{xn}\), and minor, \(b_{yn}\), axes are defined based on the PDP according to the following relationships [27; 34]:  
\[a_{xn}=\frac{1}{2}(c\tau_{n}+D), \tag{3}\]  
\[b_{yn}=\frac{1}{2}\sqrt{c\tau_{n}(c\tau_{n}+2D)}, \tag{4}\]  
where \(c\) denotes the speed of light, \(\tau_{n}\) is a delay for which the PDP takes the \(n\)th local extreme, \(n=1,2,\ldots,N\), and \(N\) is the number of time-clusters (i.e., the local extremes) in the PDP. Figure 1: Scattering geometry of the multi-elliptical propagation model (MPM) in the azimuth plane. The adopted way of creating the MPM geometric structure enables mapping of the transmission properties of propagation environments. Detailed descriptions of this structure are provided in [27; 33; 34; 35]. In step 4, we choose the Tx and Rx antenna parameters, i.e., their pattern shapes, gains, directions of maximum radiation/reception, and half-power-beamwidths (HPBWs). In the simulation testing procedure, the mapping of directional antennas is obtained using their normalized radiation pattern [35], which is realized in step 5.