# A Novel GBSM for Non-Stationary V2V Channels Allowing 3D Velocity Variations
## 2 A GBSM for Non-Stationary V2V Channels  
Let us consider a typical NS V2V communication scenario in Figure 1 including the transmitter (Tx), receiver (Rx), and clusters, which is categorized by 3D random trajectories and 3D antenna arrays, where the local coordinate systems of the Tx and Rx are xyz and XYZ, respectively. It can be presumed that the direction of transport or the distance of the Tx or Rx at the initial time corresponds to their own x-axis. The velocity of the Tx or Rx, which is shifting over time, can be denoted by \(v^{i}(t)\) where \(i\in\{\text{Tx},\,\text{Rx}\}\). Multiple propagation pathways are used in the channel among each pair of antenna parts, and each route often includes many clusters. The V2V channel propagation consists of various propagation paths like the line of sight (LoS), single bounce, double bounce, and multiple-bounce paths. In addition, the single and double bounce can be considered as special forms of the NLoS. The first and last clusters are denoted under general NLoS scenarios as \(C_{n,m}^{Tx}\) and \(C_{n,m}^{Rx}\), including time-variant velocities, respectively.  
Based on the twin cluster idea, a virtual link derives the 3D location of the first cluster \(C_{n,m}^{Tx}\) or the last cluster \(C_{n,m}^{Rx}\) and defines an analogous path delay and power. Thus, the periodical diversion of the complex channel impulse response (CIR) for NLoS paths, as well as the LoS path can be represented as:  
\[h(t,\tau)=\sum_{n=1}^{N(t)}\sqrt{P_{n}(t)}h_{n}(t)\delta(\tau-\tau_{n}(t)) \tag{1}\]  
where \(N(t)\) refers to the number of directions that can be described by path power \(P_{n}(t)\), channel fading coefficient \(h_{n}(t)\), and path delay \(\tau_{n}(t)\). Through the usual scattering scenarios, the channel fading coefficient in this paper is modeled as:  
\[h_{n}(t)=\lim_{M\rightarrow\infty}\frac{1}{\sqrt{M}}\sum_{m=1}^{M}e^{i(2\pi \int_{0}^{t}f_{n,m}(t^{\prime})dt^{\prime}+\theta_{n,m})} \tag{2}\]  
where \(M,\theta_{n,m}\) represents the numbers of sub-paths and the random phase, respectively, and \(f_{n,m}(t)\) describes the Doppler frequency and can be represented as:  
\[\begin{split} f_{n,m}(t)&=f_{n,m}^{Tx,C_{n}^{Tx}}(t )+f_{n,m}^{Cx,Rx}(t)\\ &=\frac{1}{\lambda}\Big{(}\Psi^{Tx,C_{n,m}^{Tx}}(t)\cdot\textbf{ s}_{n,m}^{Tx}(t)+\psi^{Cx,Rx}(t)\cdot\textbf{s}_{n,m}^{Rx}(t)\Big{)}=\frac{1}{ \lambda}\sum_{i}^{Tx,Rx}\Big{(}\psi^{iC_{n,m}^{i}}(t)\cdot\textbf{s}_{n,m}^{i }(t)\Big{)}^{\cdot}\end{split} \tag{3}\]  
The relative velocity vector among the vehicles and clusters is represented by \(\textbf{v}^{iC_{n,m}^{i}}(t)\) where \(i\in[\text{Tx, Rx}]\), and \(\lambda=c/f_{0}\) is the wavelength, where \(c\) and \(f_{0}\) represent the speed of  
Figure 1: 3D GBSM-based V2V communication channel model.  
light and the carrier frequency, respectively. In (3), \(\mathbf{s}_{n,m}^{i}(t)\) indicates the unit vectors of the AoD and AoA of the mth sub-path, where \(i\in\{\text{Tx},\text{Rx}\}\).  
\[\mathbf{s}_{n,m}^{i}(t)=\begin{bmatrix}\cos\alpha_{n,m}^{i}(t)\cos\varphi_{n,m }^{i}(t)\\ \cos\alpha_{n,m}^{i}(t)\sin\varphi_{n,m}^{i}(t)\\ \sin\alpha_{n,m}^{i}(t)\end{bmatrix} \tag{4}\]  
where \(\alpha_{n,m}^{i}(t)\) and \(\varphi_{n,m}^{i}(t)\) represent the elevation and azimuth angle, respectively.