# A Novel GBSM for Non-Stationary V2V Channels Allowing 3D Velocity Variations
## 6 Simulation and Analysis of the Results
The simulation results of the PDFs and ACFs of this method were consistent with the theoretical results. Thorough the different moments of time, the evaluated outcomes of the PDFs and ACFs showed a good agreement with the relevant theoretical outcomes, which proved that the simulated outcomes were relevant with respect to the theoretical ones. According to Table 1, all the simulation parameters are described for every different scenario, i.e., for the Tx and Rx in Scenario I, which have the same direction, where the velocity parameter \(\left\|\mathbf{v}\right\|^{i}=0.6,0.8\) (ms\({}^{-1}\)), the acceleration parameter \(\left\|\mathbf{a}\right\|^{i}=0.8,0.6\) (ms\({}^{-2}\)), and the azimuth and elevation angle \(\varphi_{v}^{i}=\pi/2\,,\pi/2\,\) and \(\alpha_{v}^{i}=0,0\), respectively. Similarly, for all the other scenarios, the Doppler frequency changed over time accordingly. Besides, the hypothetical and evaluated DPSDs at various moments of time, i.e., \(t=0\) s, \(2\) s, and \(5\) s were analyzed and are derived in Figure 5. The DPSDs of the introduced model and the model in [31] were calculated and are compared in Figure 5. In this paper, we considered the 3D velocity variation for different time instants, while the model in [31] merely considered 2D velocity variations. Thus, the DPSDs of the proposed model were more precise than the model in [31] compared to the theoretical results. Figure 4: Theoretical and simulated ACF at different time instants. In addition, the velocity parameters \(\mathbf{v}^{Tx},\mathbf{v}^{Rx}\) and the acceleration parameters \(\mathbf{a}^{Tx},\mathbf{a}^{Rx}\) affected the Doppler frequency, which showed the perfect comprehension among the theoretical and simulated consequences over time. Figure 5 also demonstrates the comparison between the reference model [31] and our model, respectively. The observation of the Doppler frequency changed over time where \(\varphi_{v}^{i}\) and \(\alpha_{v}^{i}\) affected the amplitude of the Doppler frequency. Figure 5 shows that the analytical and evaluated outcomes of the DPSDs of the mentioned scenarios illustrated velocity variation with traveling scatterers, which was a confirmation of the non-isotropic scattering scenarios. It can also be seen that using (12), the channel parameters influenced the Doppler frequency at different moments of time, such as t = 0s, 2s, and 5s. The overall Doppler frequency in Figure 5d is much bigger than that in Figure 5c; similarly, that in Figure 5b is comparatively bigger than that in Figure 5a, which indicates that the Doppler frequency was affected by the initial value of the movement direction. By comparing the Doppler frequencies at various time intervals in various scenarios, the effect of the velocity variations of the Tx and Rx can clearly be seen. Ultimately, the good agreement of the theoretical and experimental outcomes in Figure 5 assures the exactness of the introduced model. \begin{table}
\begin{tabular}{c c c c c} \hline \hline _Scenarios_ & \(\|\mathbf{v}\|^{i}\) & \(\|\mathbf{a}\|^{i}\) & \(\boldsymbol{\varphi}_{v}^{i}\) & \(\boldsymbol{\alpha}_{v}^{i}\) \\ \hline \(I\) & 0.6,0.8 (ms\({}^{-1}\)) & 0.8,0.6 (ms\({}^{-2}\)) & \(\frac{\pi}{2},\frac{\pi}{2}\) & 0,0 \\ _II_ & 0.5,0.6 (ms\({}^{-1}\)) & 0.8,1 (ms\({}^{-2}\)) & \(\frac{\pi}{2},\frac{\pi}{2}\) & 0,\(\frac{\pi}{2}\) \\ _III_ & 0.9,1 (ms\({}^{-1}\)) & 1.0,8 (ms\({}^{-2}\)) & \(\frac{\pi}{2},\frac{\pi}{2}\) & \(\frac{\pi}{18},0\) \\ _IV_ & 1,0.7 (ms\({}^{-1}\)) & 0.7,1 (ms\({}^{-2}\)) & \(\frac{\pi}{2},-\frac{\pi}{2}\) & \(\frac{\pi}{12},\frac{\pi}{18}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Velocity parameters of the Tx and Rx. Figure 5: Theoretical and simulated DPSD at different time instants.