# A Novel 3D Analytical Scattering Model
## 3 PDF of the Normalized Doppler Spectrum
\begin{split}& p(p_{r},\phi_{a},\beta_{a})=\frac{1}{\left(8\;V \sec\beta_{a}\;n\;p_{r}\right)\left(-d_{\text{LoS}}+d\;k\cos\beta_{a}\cos \phi_{a}+h_{\epsilon}\;k\;\sin\beta_{a}\right)^{4}}\times\\ &\left\{-d_{\text{LoS}}^{6}\;k^{-3}\left(-1+k^{2}\right)^{2} \left(d_{\text{LoS}}\left(1+k^{2}\right)-2\;d\;k\cos\beta_{a}\cos\phi_{a}-2\; h_{\epsilon}\;k\sin\beta_{a}\right)\right\}\end{split}  \tag{26}\]  
The secant function represents \(\sec(.)=1/\cos(.)\). The joint density function \(p(p_{r},\gamma,\beta_{a})\), can be obtained by transforming the density function \(p(p_{r},\phi_{a},\beta_{a})\) using the relationship given in (17),  
\[p(p_{r},\gamma,\beta_{a})=\left.\frac{p(p_{r},\phi_{a},\beta_{a})}{|J(p_{r}, \phi_{a},\beta_{a})|}\right|_{\phi_{a}=\phi_{v}-\cos^{-1}\left(\frac{\gamma}{ \cos(\beta_{v}-\beta_{a})}\right)}. \tag{27}\]  
The Jacobian transformation shown in (27) can be derived as,  
\[J(p_{r},\phi_{a},\beta_{a})=\left|\frac{\partial\phi_{a}}{\partial\gamma} \right|^{-1}=\sqrt{\cos^{2}(\beta_{a}-\beta_{v})-\gamma^{2}}. \tag{28}\]  
Substituting (26) and (28) in (27), the joint density function in (27) can be given in simplified form as below. \[ \begin{split}& p(p_{r},\gamma,\beta_{a})=\Big{\{}\Big{(}\frac{-d_{ \text{LoS}}^{6}\;\left(-1+k^{2}\right)^{2}}{8V\;k^{3}\sec\beta_{a}\;\cos(\beta _{a}-\beta_{v})\,n\;p_{r}\sqrt{(1-\gamma^{2}\sec^{2}(\beta_{a}-\beta_{v}))} \Big{)}\Big{.}\\ &\times\Big{(}-d_{\text{LoS}}+d\;k\cos\beta_{a}\cos(\phi_{v}- \arccos(\gamma\sec(\beta_{a}-\beta_{v})))+h_{\epsilon}\;k\;\sin\beta_{a}\Big{)} ^{-4}\\ &\times\Big{(}d_{\text{LoS}}\left(1+k^{2}\right)-\left(2\;d\;k \cos\beta_{a}\cos(\phi_{v}-\arccos(\gamma\sec(\beta_{a}-\beta_{v}))\right)\\ &\qquad\qquad-2\;h_{\epsilon}\;k\sin\beta_{a}\Big{)}\Big{\}}. \end{split}  \tag{29}\]  
The joint PDF for the Doppler and power can thus be derived after integrating (29) over \(\beta_{a}\) for its appropriate range. \[p(p_{r},\gamma)=\int_{\beta_{l}(\gamma)}^{\beta_{u}(\gamma)}p(p_{r},\gamma, \beta_{a})\;\mathrm{d}\beta_{a}. \tag{30}\]  
It is obvious from (16), that \(|\cos\theta_{a}\leq 1|\); we may have \(|\gamma|\leq\cos(\beta_{a})\). Therefore, here, \(\beta_{u}(\gamma)\) and \(\beta_{l}(\gamma)\) can be defined by the same approach used in [40]. \[\beta_{u}(\gamma)=\left\{ \begin{array}{ll}\beta_{\text{max}}&,\quad 0 \leq|\gamma|\leq\cos\beta_{\text{max}},\\ \arccos(|\gamma|)&,\quad\cos\beta_{\text{max}}\leq|\gamma|\leq 1.\end{array} \right. \tag{31}\]\[\beta_{l}(\gamma)=\left\{ \begin{array}{ll}-\beta_{\min}&,&0\leq|\gamma|\leq\cos \beta_{\min},\\ -\arccos(|\gamma|)&,&\cos\beta_{\min}\leq|\gamma|\leq 1.\end{array} \right. \tag{32}\]  
The marginal PDF for the Doppler shift for the proposed model can thus be derived by integrating (30) over \(p_{r}\),  
\[p(\gamma)=\int_{p_{l}}^{p_{u}}p(p_{r},\gamma)\;\mathrm{d}p_{r}. \tag{33}\]  
\(p_{u}\) and \(p_{l}\) are the upper and lower limits of the received power. The power received from the shortest propagation path is the maximum power. The shortest path in this model is \(d_{\mathrm{LoS}}\); so from (22), it is evident that \(p_{u}=p_{o}\). For a particular azimuth and elevation angle, the multipath component received from the boundary of the scattering region is the longest propagation path. The power received from the longest propagation path is the lowest power received. We know that the longest delay for the reflected wave is \(\tau_{\max}\), which is assumed to be a known value on the receiver. Therefore, the lower limit for the received power can thus be obtained as:  
\[p_{l}=p_{o}\left(\frac{\tau_{\max}}{c\;d_{\mathrm{LoS}}}\right)^{-n}. \tag{34}\]  
The definition of the cumulative distribution function (CDF) of the Doppler shift is:  
\[F(\gamma)=\int_{-1}^{\gamma}p(\Gamma)\mathrm{d}\Gamma. \tag{35}\]  
The results for the marginal PDF of the Doppler spectrum are obtained by numerically integrating the trivariate density function in (29) over elevation AoA and power for appropriate limits.