## 3 Simple Models of Antenna Patterns  
Simple antenna patterns can be modeled by mathematical formulas. Gaussian or circular functions are helpful [28]. To model the antenna power radiation patterns, we adopted the MATLAB function from the Phased Array System Toolbox [29]. The normalized Gaussian pattern is as follows [29]:  
\[g(\theta,\phi)=\exp\left(-2\ln 2\left(\frac{\phi-\Phi_{0}}{HPBW_{\phi}}\right)^{2} \right)\exp\left(-2\ln 2\left(\frac{\theta-\Theta_{0}}{HPBW_{\theta}}\right)^{2} \right). \tag{6}\]  
The normalized Cosine pattern is as follows [29]:  
\[g(\theta,\phi)=\cos^{m}(\phi-\Phi_{0})\cos^{n}(\theta-\Theta_{0}), \tag{7}\]  
where  
\[m=-\frac{\log_{10}\sqrt{2}}{\log_{10}(\cos\left(HPBW_{\phi}/2\right))},n=- \frac{\log_{10}\sqrt{2}}{\log_{10}(\cos\left(HPBW_{\theta}/2\right))}. \tag{8}\]  
The normalized Sinc pattern is as follows [29]:  
\[g(n(\theta),m(\phi))=\frac{\sin(m)}{m}\frac{\sin(n)}{n}, \tag{9}\]  
where  
\[m=\frac{x\sin(\phi-\Phi_{0})}{\sin\left(HPBW_{\phi}/2\right)},n=\frac{x\sin( \theta-\Theta_{0})}{\sin\left(HPBW_{\theta}/2\right)}, \tag{10}\]  
and \(x\) is a solution of \(\sin(x)=x/\sqrt{2}\). The numerators above \(HPBW_{\phi}\) and \(HPBW_{\theta}\) represent the HPBWs for the azimuth and elevation planes, respectively. Power pattern is defined as \(\left|g(\theta,\phi)\right|^{2}\). Moreover, we used the MATLAB function to determine the directivity of antennas [29]. Table 1 depicts the results. Additionally, we show the difference between the directivity of the simple models and 3 GPP pattern, \(\Delta G\). When modeling 3D patterns and calculating their directivities, the HPBWs of the antennas are considered in both the azimuth and elevation planes. Since, in further considerations, we analyze the change in the beam direction of antenna patterns only in the azimuth plane, this plane will significantly impact the shaping of SIR graphs. Figure 3: Directivity of 3 GPP beams versus maximum beam direction. The power patterns for the Gaussian and Cosine models are very similar. To assess the similarity of the two patterns for antenna direction \(\Phi_{0}\), we determine the root mean square error (RMSE), which is defined as:  
\[RMSE(\Phi_{0})\ (\mathrm{dB})=10\log_{10}\sqrt{\frac{1}{M}{\sum_{i=1}^{M} (g_{1}((\theta,\phi)_{i},\Phi_{0})-g_{2}((\theta,\phi)_{i},\Phi_{0}))^{2}}}, \tag{11}\]  
where \(g_{1}((\theta,\phi)_{i},\Phi_{0})\) and \(g_{2}((\theta,\phi)_{i},\Phi_{0})\) mean two analyzed patterns, \(M\) is a set of \(\left(\theta,\phi\right)_{i}\) elements. For the analyzed case, \(g_{1}((\theta,\phi),\Phi_{0})\) and \(g_{2}((\theta,\phi),\Phi_{0})\) are the Gaussian and Cosine patterns, respectively. Regardless of the choice of \(\Phi_{0}\), the RMSE equals \(-8.4\) dB. Due to this relatively small error value, we limited the further analyses to the Gaussian and Sinc models. Real antenna patterns are used at further research stages, which are based on simulations for deterministic scenarios. At the initial research stages, e.g., for the spatial design of mobile networks using a statistical approach, simple models are often used. Their main advantage is a shorter computation time and lower requirements for computational resources. In Table 2, the mean times for determining antenna patterns in relation to calculations for 1000 generated patterns with an angular resolution of 0.1\({}^{\circ}\) are presented. The presented times are small for all models, but calculations or simulations at the initial stage of research are often performed on millions of statistically independent cases. Referring to Table 2, the computation time needed to generate a non-modified simple model is, on average, 320 (Sinc) to 520 (Gaussian) times shorter than for the real 3 GPP model. The developed modification does not add significant computational costs, i.e., the modified