# A Novel 3D Analytical Scattering Model
## 3 PDF of the Normalized Doppler Spectrum  
The angles \(\phi_{v}\) and \(\beta_{v}\) are azimuth and elevation angles along the direction of the motion of AS, respectively. The azimuth and elevation angle of the motion of AS with a certain multipath signal are represented by \(\theta_{a}=\phi_{v}-\phi_{a}\) and \(\alpha_{a}=\beta_{v}-\beta_{a}\), respectively. The velocity of AS is denoted by \(v_{a}\). The Doppler shift of a certain multipath can be obtained as,  
\[f_{\text{DS}}=f_{m}\ \cos\theta_{a}\cos\alpha_{a}. \tag{15}\]  
However, the maximum shift in the Doppler spectrum can be given as \(f_{m}=(v_{a}/c)\ f_{c}\), where \(c\) is the velocity of light and \(f_{c}\) is the carrier frequency. The normalized Doppler spectrum [26; 27; 39] can be expressed as follows,  
\[\gamma=f_{\text{DS}}/f_{m}=\cos\theta_{a}\cos\alpha_{a}. \tag{16}\]  
The azimuth angle \(\phi_{a}\) can be expressed as a function of the direction of motion, elevation AoA and Doppler shift, which is as below,  
\[\phi_{a}=\phi_{v}-\cos^{-1}\left(\gamma/\cos(\beta_{v}-\beta_{a})\right) \tag{17}\]  
The joint density function of azimuth AoA, elevation AoA and propagation path distance \(p(l,\phi_{a},\beta_{a})\) can be obtained by transforming the density function \(p(r_{a},\phi_{a},\beta_{a})\) using the relationship given in (13) from the following equation. \[p(l,\phi_{a},\beta_{a})=\frac{p(r_{a},\phi_{a},\beta_{a})}{|J(l,\phi_{a},\beta _{a})|}. \tag{18}\]  
The joint density function given in (18), \(p(r_{a},\phi_{a},\beta_{a})\), can be expressed as, given in [21],  
\[p\left(r_{a},\phi_{a},\beta_{a}\right)=r_{a}^{2}\cos\beta_{a}f(x,y,z). \tag{19}\]  
where \(f(x,y,z)\) represents the spatial distribution of scatterers. This density can be written as \(f(x,y,z)=1/V\) for a uniform distribution of scatterers. The Jacobian transformation given in (18), \(J(l,\phi_{a},\beta_{a})\), can be derived as,  
\[J(l,\phi_{a},\beta_{a})=\left|\frac{\partial r_{a}}{\partial l}\right|^{-1}= \frac{2(l-d\cos\beta_{a}\cos\phi_{a}+(-h_{\text{AS}}+h_{\text{GS}})\sin\beta_ {a})^{2}}{l^{2}+d_{\text{LoS}}^{2}-2l(d\cos\beta_{a}\cos\phi_{a}+(h_{\text{AS }}-h_{\text{GS}})\sin\beta_{a})}. \tag{20}\]  
Substituting (19) and (20) in (18), the simplified solution for \(p(l,\phi_{a},\beta_{a})\) can be re-written as,  
\[p(l,\phi_{a},\beta_{a})=\frac{d_{\text{LoS}}^{2}+l^{2}-2l(d\cos\beta_{a}\cos \phi_{a}+(h_{\text{AS}}-h_{\text{GS}})\sin\beta_{a})}{8V\sec\beta_{a}(l-d\cos \beta_{a}\cos\phi_{a}+(-h_{\text{AS}}+h_{\text{GS}})\sin\beta_{a})^{4}}(d_{ \text{LoS}}^{2}-l^{2})^{2}. \tag{21}\]  
The power received from a certain multipath \(p\) can be obtained by using the path loss model given in [39] as,  
\[p_{r}(l_{p})=p_{o}\left(\frac{l_{p}}{d_{\text{LoS}}}\right)^{-n}, \tag{22}\]  
where \(d_{\text{LoS}}\) is the length of LoS from GS to AS, \(p_{o}\) is the power received corresponding to the LoS component, \(n\) is the path loss exponent and \(l_{p}\) is the length of the \(p\)-th multipath. By re-arranging this relation for \(l_{p}\), we get:  
\[l_{p}=d_{\text{LoS}}\left(\frac{p_{r}}{p_{o}}\right)^{-\frac{1}{n}}. \tag{23}\]The joint density function \(p(p_{r},\phi_{a},\beta_{a})\) can be obtained by transforming the density function \(p(l,\phi_{a},\beta_{a})\) using the relationship given in (23),  
\[p(p_{r},\phi_{a},\beta_{a})=\left.\frac{p(l_{p},\phi_{a},\beta_{a})}{|J(l_{p}, \phi_{a},\beta_{a})|}\right|_{l_{p}=d_{\text{LoS}}\left(\frac{p_{r}}{p_{o}} \right)^{-\frac{1}{h}}}. \tag{24}\]  
The Jacobian transformation shown in (24) can be derived as:  
\[\gamma=f_{\text{DS}}/f_{m}=\cos\theta_{a}\cos\alpha_{a}. \tag{16}\]0  
Substituting (21) and (25) in (24) and considering the following assumptions, \(k=\left(p_{r}/p_{o}\right)^{\frac{1}{h}},\)\(h_{\epsilon}=\left(h_{\text{AS}}-h_{\text{GS}}\right)\), the joint density function \(p(p_{r},\phi_{a},\beta_{a})\) can be given in simplified form as below,  
\[