# A Novel GBSM for Non-Stationary V2V Channels Allowing 3D Velocity Variations
## 6 Simulation and Analysis of the Results  
This section describes the simulated method focused on the suggested model and verifies the results with the simulated, theoretical, and measured ones. In this simulation, the AoA and AoD are described to follow the VMF distribution with \(\kappa=1\), and the carrier frequency is 2.48 GHz. We assumed that the shifting of scatterers is evenly divided around two terminals. The arbitrary variables are as follows, \(\left\|\mathbf{v}^{C_{n}^{i}}\right\|\)\(\left\|\cdot\)\(U(0,1)\) m/s, where \(U\) denotes the uniform distribution, \(\varphi_{v}^{C_{n}^{Tx}}\), \(\varphi_{v}^{C_{n}^{Rx}}\), and \(\alpha_{v}^{C_{n}^{Tx}}\), \(\alpha_{v}^{Rx}\) are evenly allocated over \((-\pi,\pi)\). In addition, the Tx and Rx shift over various trajectories, and the specific velocity parameters can be found as follows:  
To verify the simulation results with the theoretical results, four different trajectories, which are related to realistic V2V communication scenarios, were considered. Figure 2 shows four different trajectories of the Tx and Rx as the same direction, straight-right turn, left-straight turn, and left-right turn in the opposite direction, respectively. All the variables, i.e., the 3D relative velocity speed \(\|\mathbf{v}\|^{i}\), the acceleration \(\|\mathbf{a}\|^{i}\), and the angle variables \(\alpha_{\
u}^{i}\), affect the amplitude and varying patterns of the channel statistical characteristics. In this paper, we considered the envelope PDFs, ACFs, and DPSDs for a few moments of time in order to verify the simulated output along with the theoretical output. Therefore, in (28), the PDF result of this introduced model at three different moments of time where \(t=0\) s, 2 s, and 5 s differentiates along the analytical output in Figure 3. Because of the time-variant channel parameters, the simulated PDFs change with the corresponding theoretical PDFs. The channel parameters in (28) indicate the variation of time through the simulation process, which is random. The simulated PDF of Figure 3 clarifies the Rayleigh channel fading where all the intervals are considered to have normalized power. The figure shows a good similarity of the simulated PDF with the theoretical Rayleigh PDF. By comparing the theoretical and calculated outcomes of the ACFs, the proposed model was validated by considering four distinct standard V2V scenarios with trajectory variations of the Tx and Rx. All of the scenarios in Figure 2 were considered at various speeds with initial directions. The theoretical and simulated ACFs at three different moments were derived and shown in Figure 4. By comparing the ACFs of this model and the reference model in [31] at various time instants, it was obvious that the velocity parameters had a significant impact on the ACFs. Due to the fact that the model in [31]  
Figure 3: Theoretical and simulated PDF at different time instants. Figure 2: Different trajectories of the Tx and Rx. only considered 2D velocity variations, the proposed model provided better simulated outcomes with respect to the theoretical outcomes when the velocity changed arbitrarily. The measurement result from [27] was considered, which came into consideration to compare with this proposed model, and likewise, the measurement result of [27] was also considered in this paper. Figure 4a-d demonstrates the theoretical and simulated ACFs of our proposed model at various time instants under four different scenarios using (29)-(34). Through the comparison of Figure 4a,b the ACFs under straight-right turn changed faster than in the same direction. Besides that, in Figure 4c,d it is easy to obtain that the left-right opposite turn changed faster than the left-straight turn for the different initial values of the channel parameters. As the ACFs were shifting at a faster rate, it was fair to assume that the larger the parameter value, the more complex the communication scenarios would be and that the ACFs would change faster.