# A Novel 3D Analytical Scattering Model
## 3 PDF of the Normalized Doppler Spectrum
These results are obtained in relation to the direction of motion in the azimuth and the elevation plane. The impact of the important parameters of the proposed channel model, such as the height of AS and the longest path's delay, is also studied on the Doppler spectrum. The scatterers are assumed as uniformly distributed for obtaining these results. The effect of the direction of the motion of AS in the azimuth plane (\(\phi_{v}\)) on the marginal PDF of the Doppler spectrum is shown in Figure 2a,b. When the geometric scenario is such that the projection of AS on the \(x\)-\(y\) plane lies within the scattering region, the probability of the arrival of multipath waves at AS is the same from all of the directions in the azimuth plane. It can be seen that for \(\phi_{v}=90^{\circ}\), the PDF is observed as U-shaped and symmetric around \(\gamma=0\). This is due to the fact that multipath components contributing to the maximum and minimum Doppler shift are present in an equal amount. For \(\phi_{v}=0^{\circ}\) and \(\phi_{v}=180^{\circ}\), the PDF is seen tilted towards the lower and higher frequencies, respectively. Moreover, when the image of AS on the \(x\)-\(y\) plane is away from the scattering region and the elevation of AS is set as \(h_{\mathrm{AS}}=7\) km, this can be an en-route scenario; the spatial distribution of the scatterers is non-isotropic. This leads to the selection of only a part of the band from the whole U-shaped PDF of the Doppler shift depending on the direction of the AS's mobility, which is evident in Figure 2b. The effect of the change in the elevation of AS on the CDF of the Doppler spectrum is shown in Figure 2c, plotted for the different values of \(h_{\mathrm{AS}}\). It can be observed that with an increase in the height of AS, the angular spread decreases, which further leads to an increase in the slope of CDF. The effect of the direction of the motion of AS in the elevation plane (\(\beta_{v}\)) on the CDF of the Doppler spectrum is shown in Figure 3a,b. When the projection of AS on the \(x\)-\(y\) plane lies inside the scattering region, the CDFs are symmetric around \(\gamma=0\). It can be observed that with an increase in the angle for the elevational direction of motion, the slope of CDF slowly decreases up to \(\gamma=0\) and then increases slowly, and CDF approaches one when \(\gamma\) approaches one. When distance between nodes is increased, and the shadow of AS on the \(x\)-\(y\) plane is outside the scattering region; the spread in angular data observed at AS decreases to a smaller number from \(2\pi\). Therefore, CDFs are approaching one very quickly. The increase in delay span defines an increase in the horizontal dimensions of TESR. This effect of increasing the delays of the longest paths on the Doppler spectrum is demonstrated in Figure 3c. It can be observed that for large values of the delays of the longest paths, the dimension of the TESR is large enough to have the projection of AS on the \(x\)-\(y\) plane inside the scattering region; therefore, the signals are received from all around, so the CDFs of the Doppler shift are almost symmetric around \(\gamma=0\). The obtained analytical results are also validated through the comparison with the performed simulation results. The computer simulations are performed by adopting the approach presented in detail in [41]. The scatterers are generated in the defined TESR, as shown in Figure 4a. The positions of AS and GS are also shown in Figure 4a. The direction of the motion of AS in the azimuth plane is taken as \(\phi_{v}=90^{\circ}\). The heights of AS and GS are assumed such that AS is close to the ground level. Using the simulation setup, the simulated results for the PDF of the Doppler spectrum are compared to the proposed analytical results shown in Figure 4b, where PDF is U-shaped because the multipath components contributing in the positive and negative Doppler are in equal amounts. The simulated results for the CDF of the Doppler spectrum are compared to the proposed analytical results in Figure 4c.