# A Novel 3D Analytical Scattering Model
## 4 Shape Factors  
The SFs proposed in [28] are the favourable parameters for the quantification of spatial spread. The joint PDF of AoA, the marginal PDF of azimuth AoA and the marginal PDF of elevation AoA can be represented by \(p(\phi_{m},\beta_{m})\), \(p(\phi_{m})\), \(p(\beta_{m})\), where the subscript \(m\) may be replaced by \(a\) or \(b\) for the statistics of the AS and GS sides, respectively. The joint and marginal PDFs of AoA for such A2G/G2A scenarios w.r.t. the azimuth and elevation angles observed from both ends of the link are derived in [21]. The methodology used for computing the SFs is consistent with the methods proposed in [28,29]. Fourier coefficients or trigonometric moments are used for analysing the PDF of AoA. These two methods are similar in definition; however, the approach given in [29] is applicable for discrete measurements of angular data. The \(n\)-th complex trigonometric moment of any angular distribution is \(\bar{R}_{n}\). For example, for \(p(\varphi)\) (i.e., where \(\varphi\) may be \(\phi_{m}\) or \(\beta_{m}\) for the azimuthal and elevational AoA), whose total power is equal to \(P_{o}=\int_{0}^{2\pi}p(\varphi)\mathrm{d}\varphi\), \(\bar{R}_{n}\) is defined as,  
\[\bar{R}_{n}=\bar{C}_{n}+j\bar{S}_{n}, \tag{36}\]  
where,  
\[\bar{C}_{n}=\frac{1}{P_{o}}\int_{0}^{2\pi}p(\varphi)\ \cos(n\varphi)\mathrm{d}\varphi \tag{37}\]  
and,  
\[\bar{S}_{n}=\frac{1}{P_{o}}\int_{0}^{2\pi}p(\varphi)\ \sin(n\varphi)\mathrm{d}\varphi. \tag{38}\]  
In the case of discrete data, the definition for trigonometric moments can be modified. Only first and second moments are used in the characterization of the SFs. \(|\bar{R}_{1}|\) is the magnitude of the first trigonometric moment. It can take values between zero and one. A close to zero value means the receiver is receiving signals from a wider range of angles, and close to one means a small angular width of AoA. The SF angular spread, \(\Lambda_{\varphi_{m}}\), is given as below,  
\[\Lambda_{\varphi_{m}}=\sqrt{1-|\bar{R}_{1}|^{2}}. \tag{39}\]  
Figure 4: (**a**) Spatial distribution of the scattering objects used for simulation; (**b**) simulation and analytical results of the PDF of the normalized Doppler spectrum; (**c**) simulation and analytical results of the CDF of the normalized Doppler spectrum. The other two SFs, angular constriction \(\gamma_{\varphi_{m}}\) and orientation parameter \(\varphi_{m}^{MF}\), are defined by [PERSON] et al. in [28]. \(\gamma_{\varphi_{m}}\) gives the concentration of the multipath about two directions, and \(\varphi_{m}^{MF}\) provides the direction of maximum fading. Their relation with trigonometric moments is given below,  
\[\gamma_{\varphi_{m}}=\frac{|\bar{R}_{2}-\bar{R}_{1}^{2}|}{1-|\bar{R}_{1}|^{2}}, \tag{40}\]  
and:  
\[\varphi_{m}^{MF}=\frac{1}{2}\text{phase}\{\bar{R}_{2}-\bar{R}_{1}^{2}\}. \tag{41}\]  
A thorough analysis on the results of the SFs for the A2G/G2A communication channel is presented. The SF angular spread \(\Lambda_{\varphi_{m}}\) gives the measurement of the concentration of multipaths around a single direction by using the range zero to one. The azimuthal and elevational angular spread observed from AS with respect to the height of AS for different delays of the longest propagation path is shown in Figure 5a,b, respectively. The azimuthal angular spread decreases with an increase in the height of AS. However, the rate of decrease increases rapidly with respect to the increase in the height of AS by reducing the \(\tau_{\text{max}}\). The elevational angular spread observed from AS increases gradually with the increase in height of AS up to the height when the projection of AS on the \(x\)-\(y\) plane lies within the scattering region, and it decreases with the further increase in the height of AS. Moreover, the elevational angular spread shows a converse trend with respect to \(\tau_{\text{max}}\). The elevational angular spread with an increase in \(\tau_{\text{max}}\) decreases up to a certain height of AS: whereas, with the further increase in the height of AS, it increases.