## 4 Distributions of Path Delay, Path Attenuation, Normalized Doppler Shift of the Scattering Path
### Simulations of CIR, TF, ACF, and Normalized Doppler PSD  
In our simulations of the CIRs, tapped delay lines model is adopted in this paper with \(\Delta t=\)0.1 \(\upmu\)s. Figure 4 shows a snapshot of the impulse responses of the presented model at the moment \(t=1\) s of the rising stage. Six scattering taps are illustrated in this figure.  
As shown in Figure 4, the LOS path, which is also the path with the minimum delay, can be distinguished from other paths easily, the specular reflection path is adjacent to the scattering paths. Closer scattering components to the specular component would be observed if nearer scatterers to the specular reflection point were distributed.  
The time-frequency transfer function of the scattering path during \(t\in(1,1.01)\) s of the rising stage is shown in Figure 5 which depicts the time variant feature of the \(H_{\text{DIF}}\left(t,f\right)\). Although \(H_{\text{DIF}}\left(t,f\right)\) is only a sample of the spectrum of the random process \(h_{\text{DIF}}\left(t,\tau\right)\), we can observe its time variant and random feature.  
The normalized ACFs of the scattering path in different flight states and at different moments, i.e., \(t=300\) s, \(t=450\) s, \(t=650\) s, and \(t=800\) s of the rising state, \(t=400\) s, \(t=500\) s, \(t=600\) s, and \(t=700\) s of the cruising state, and \(t=450\) s, \(t=600\) s, \(t=750\) s, and \(t=900\) s of the descending state,  
\begin{table}
\begin{tabular}{l l} \hline \hline
**Parameter** & **Value** \\ \hline Altitude of the transmitter & 36,000 km \\ Initial altitude of the receiver (Rising, \(t=0\) s) & 300 m \\ Initial altitude of the receiver (Cruising, \(t=0\) s) & 11 km \\ Initial altitude of the receiver (Descending, \(t=0\) s) & 11 km \\ Initial horizontal distance between the transmitter and the receiver & 400 km \\ \(\tau_{\text{max}}\) & \(\tau_{\text{SPE}}+0.7\upmu\text{s}-\tau_{\text{LOS}}\) \\ Central frequency & 2 GHz \\ Ground velocity vector of the receiver (Rising) & \((0,2222,5.94)\) m/s \\ Ground velocity vector of the receiver (Cruising) & \((0,2222,0)\) m/s \\ Ground velocity vector of the receiver (Descending) & \((0,222,-5.94)\) m/s \\ Tap interval \(\Delta t\) & 0.1 \(\upmu\)s \\ Number of taps & 7 \\ number of scatterers \(L\) in CIR simulation & 100 \\ number of scatterers \(L\) in CDF/PDF simulation & \(1\times 10^{9}\) \\ \(G_{\text{R}}\), \(G_{\text{T}}\) & 15 dBi omini-directional \\ \hline \hline \end{tabular}
\end{table}
Table 2: Simulation Parameters.  
Figure 3: The Trajectories of the Aircraft  
Simulations of the Distributions of Path Delay, Path Attenuation, and Normalized Doppler Shift of the Scattering Path  
In this subsection, simulation results of the distributions of \(\tau_{\text{DIF}}\) are illustrated and compared with the derived CDFs. For the distributions of \(\alpha_{\text{DIF}}\) and \(\gamma_{\text{DIF}}\), since there are no available closed form expression for the presented scenario, for more convenient application, the fitting functions are presented. For better observation of the symmetries of the Doppler shift, the PDFs are observed instead of CDFs. It's to be noted that, different from the trajectories in Section 5.2, the trajectory of the aircraft in descending state is the return route of the trajectory of the rising state, i.e., 1800 s were simulated for both states.