# A Novel 3D Analytical Scattering Model
## 5 Second Order Fading Statistics
The LCR observed from GS is showing the same behaviour with respect to the threshold level, as well as with respect to \(\tau_{\max}\). However, when the projection of AS on the \(x\)-\(y\) plane lies inside the TESR, LCR increases up to a level because of the density of scatterers and then almost remains constant for the further increase in \(\tau_{\max}\). The plots of the normalised AFD against the threshold level observed from AS and GS for different values of \(\tau_{\max}\) are shown in Figure 13a,b, respectively. When observed from AS, AFD increases gradually with the increase in the threshold level up to a certain level, and then, it slightly decreases with the further increase in the threshold level. However, AFD increases by reducing \(\tau_{\max}\), and the rate of increase is changed rapidly by reducing the \(\tau_{\max}\) because AS moves away from GS. The AFD, when observed from GS, is showing the same behaviour with respect to the threshold level when observed from AS. However, when the projection of AS on the \(x\)-\(y\) plane lies inside the TESR, AFD increases with the increase in \(\tau_{\max}\) up to a level because of the density of scatterers and then almost remains the same for the further increase in \(\tau_{\max}\). The the plots of the normalised AFD against different values of the maximum Doppler frequency observed from AS and GS showing the effects of the height of AS are shown in Figure 14a,b, respectively. When observed from AS, AFD decreases smoothly with the increase in Doppler frequency. However, AFD increases by increasing the height of AS, and the rate of increase is higher with respect to the increase in the height of AS. Therefore, by increasing the height of AS, the projection of AS on the \(x\)-\(y\) plane moves out of the scattering region for a given delay; therefore, the number of multipaths is reduced, which results in the increase of AFD. The AFD, when observed from GS, is showing the same behaviour with respect  
The results for auto-covariance against \(r/\lambda\) observed from AS and GS for different heights of AS are plotted in Figure 15a,b, respectively. The auto-covariance decreases from one to zero with the increase in \(r/\lambda\) distance, and it increases by increasing the height of AS because AS is moving away from GS for a given delay. The rate of increase changes rapidly with the increase in height of AS. For the height of AS at 8 km, it decrease from one to 0.9 only because for such an environment, the multipath components are received over a small angular range, which results in high correlation. However, for lower heights of AS, the projection of AS on the \(x\)-\(y\) plane lies inside the scattering region; multipath components arrive from a larger area, which results in lower correlation. When observed from GS, auto-covariance is showing the same behaviour as observed from AS. The results for auto-covariance against \(r/\lambda\) observed from AS and GS for different values of \(\tau_{\text{max}}\) are plotted in Figure 16a,b, respectively. The auto-covariance decreases from one to zero with the increase in \(r/\lambda\) distance and \(\tau_{\text{max}}\) because of the projection of AS on the \(x\)-\(y\) plane moves inside the TESR for a given height of AS. The rate of decrease changes rapidly with the increase in \(\tau_{\text{max}}\). When observed from GS, auto-covariance is showing the same behaviour as observed from AS. The results for coherence distance (\(D_{c}\)), against \(\lambda\) observed from AS and GS for different heights of AS, are plotted in Figure 17a,b, respectively. The coherence distance gives information about the stability of the channel. The \(D_{c}\) increases linearly with the increase in \(\lambda\), and it is higher for high values of the height of AS because by increasing the height of AS, AS moves away from GS for a given delay. When observed from GS, \(D_{c}\) is showing the same behaviour as observed from AS.