# A Novel 3D Analytical Scattering Model
## 5 Second Order Fading Statistics  
SFs can further be used to study the second order fading statistics, like LCR, AFD, auto-covariance and the coherence distance of the propagation channel. These statistics are defined in terms of angular spread, angular constriction and the direction of maximum fading, as in [28; 35]. Azimuthal SFs are used in the analysis of the second order fading statistics. The LCR, \(N_{R}\), quantifies how often a threshold is crossed by a fading and is defined as,  
\[N_{R}=\sqrt{2\ \pi}\ \Lambda_{\phi_{m}}f_{m}\rho\sqrt{1+\gamma_{\phi_{m}}\ \cos(2\ (\phi_{p}-\phi_{m}^{MF}))\ \exp(-\rho^{2})}, \tag{42}\]  
where \(\rho\) is the normalised threshold level. The AFD, \(\bar{\tau}\), quantifies the time spent by a signal below the threshold level and can be defined by the relation given below,  
Figure 10: The behaviour of the elevational angular spread and the angular constriction against the elevation LoS angle observed at the AS side of the link: (**a**) angular spread in azimuth plane; (**b**) angular constriction in elevation plane (\(h_{\text{AS}}=400\) m, \(h_{\text{GS}}=50\) m and \(\sigma_{c}=\sigma_{\text{t}}=30\) m). Figure 9: The behaviour of the azimuthal angular spread and angular constriction against the elevation LoS angle observed at the AS side of the link: (**a**) angular spread in the azimuth plane; (**b**) angular constriction in the azimuth plane (\(h_{\text{AS}}=400\) m, \(h_{\text{GS}}=50\) m and \(\sigma_{c}=\sigma_{\text{t}}=30\) m). \[\bar{\tau}=\frac{(\exp(-\rho^{2})-1)}{\sqrt{2\ \pi\ \Lambda_{\phi_{m}}f_{m}}\rho \sqrt{1+\gamma_{\phi_{m}}\ \cos(2\ (\phi_{v}-\phi_{m}^{MF}))}}. \tag{43}\]  
Auto-covariance is another important statistic, which determines the correlation of the received voltage envelope as a function of the change in the receiver position and is useful for studies in spatial diversity [28; 42]. An expression of auto-covariance based on SFs is given in [43], which can be approximated as below,  
\[p(r,\phi_{v})\approx\exp\Big{[}-23\ \Lambda_{\phi_{m}}^{2}\big{(}1+\gamma_{ \phi_{m}}\ \cos(2\ (\phi_{v}-\phi_{m}^{MF}))\big{)}\ \Big{(}\frac{r}{\lambda}\Big{)}^{2}\ \Big{]}. \tag{44}\]  
This equation provides the envelop correlation between two points separated by distance \(r\). The coherence distance, \(D_{c}\), is the separation in space over which channel response is unchanged. It is important for spatial diversity in the design of wireless receivers. The coherence distance can be defined as,  
\[D_{c}\approx\frac{\lambda\ \sqrt{\ln(2)}}{\Lambda_{\phi_{m}}\sqrt{23\ \big{(}1+\gamma_{\phi_{m}}\ \cos(2\ (\phi_{v}-\phi_{m}^{MF}))\big{)}}}. \tag{45}\]  
A detailed analysis on the second order statistics of the A2G channel is presented. The impact of major parameters, such as the height of AS, the longest propagation path's delay and maximum Doppler frequency on these fading statistics, is analysed. The plots of the normalised LCR against the threshold level observed from AS and GS for different heights of AS are shown in Figure 11a,b, respectively. The LCR observed from AS increases gradually with the increase in the threshold level up to a certain level, and then, it decreases with the further increase in the threshold level. However, LCR decreases by increasing the height of AS, because LCR is sensitive to the density of scatterers, and AS is assumed as a scattering-free region. The LCR observed from GS is showing the same behaviour with respect to the threshold level as observed from AS, but it shows a converse trend with respect to the height of AS. The plots of the normalised LCR against the threshold level observed from AS and GS for different values of \(\tau_{\max}\) are shown in Figure 12a,b, respectively. The LCR observed from AS increases gradually with the increase in the threshold level up to a certain level, and then, it decreases with the further increase in the threshold level. However, LCR decreases by reducing \(\tau_{\max}\), and the rate of decrease is changed rapidly by reducing the \(\tau_{\max}\), because AS moves away from GS.