# A Non-WSSUS Channel Simulator for V2X Communication Systems
## 4 Reference Channel Model
### Mathematical Model of the Channel Transfer Function  
Following [31], the transfer function for non-WSSUS channels with Rayleigh fadings is represented as follows:  
\[H[m,k] = \Pi_{T_{0}}[k]\sum_{\ell=1}^{\mathcal{L}}g_{\ell}\times\exp\left\{ -j\Big{(}\theta_{\ell}+2\pi\tau_{\ell}[k](f_{\ell}+m\Delta f)\Big{)}\right\} \tag{2}\] \begin{table}
\begin{tabular}{l c} \hline \hline
**Description** & **Parameter** \\ \hline Velocity vector of \(T_{X}\) & \(\mathbf{v}_{T}=v_{T}\angle\gamma_{T}\) \\ \hline Velocity vector of \(R_{X}\) & \(\mathbf{v}_{R}=v_{R}\angle\gamma_{R}\) \\ \hline Acceleration vector of \(T_{X}\) & \(\mathbf{a}_{T}=a_{T}\angle\rho_{T}\) \\ \hline Acceleration vector of \(R_{X}\) & \(\mathbf{a}_{R}=a_{R}\angle\rho_{R}\) \\ \hline Position of the \(\ell\)-th IO with respect to the reference point \(\mathbf{\mathcal{O}}\) & \(\mathbf{p}_{\ell}^{T}=d_{\ell}^{T}\angle\phi_{\ell}^{T}\) \\ \hline \(R_{X}\) position seen from the \(\ell\)-th IO & \(\mathbf{p}_{\ell}^{R}=d_{\ell}^{R}\angle\phi_{\ell}^{R}\) \\ \hline Initial distance between \(T_{X}\) and \(R_{X}\) & \(D\) \\ \hline Radius of the ring of IOs around \(R_{X}\) & \(d\) \\ \hline Mean of the AOA distribution & \(\mu_{R}\) \\ \hline Concentration parameter of the AOA distribution & \(\kappa\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: List of parameters that describe the propagation scenario (\(\mathbf{x}=M_{x}\angle\Theta_{x}\) denotes a vector \(\mathbf{x}\) with magnitude \(M_{x}\) and direction \(\Theta_{x}\)). Figure 3: Reference propagation environment at \(t_{0}=0\). where \(j^{2}=-1\), \(m\) corresponds to the frequency subcarrier index, \(k\) is the data symbol index, \(\Delta f\) denotes the separation between subcarriers, \(\mathcal{L}\) represents the total number of IOs around \(R_{X}\), \(g_{\ell}\) and \(\tau_{\ell}[k]\) are the amplitude and the time-varying propagation delay of the \(\ell\)-th replica of the transmitted signal, and \(\theta_{\ell}\) stands for the phase shift caused by the interaction of the transmitted signal with the \(\ell\)-th IO. The carrier frequency is given by \(f_{c}=c/\lambda\), with \(c\) denoting the speed of light and \(\lambda\) the wavelength of the transmitted signal. Furthermore, \(\Pi_{T_{0}}[k]\) is defined as:  
\[\Pi_{T_{0}}[k]\triangleq \begin{cases}1,&0\leq kT_{s}\leq T_{0}\\ 0,&\text{others}\end{cases}  \tag{3}\]  
where \(T_{s}\) is the duration of a single symbol. The function \(\Pi_{T_{0}}[k]\) denotes a rectangular windowing function introduced to constrain \(H[m,k]\) within a time interval \(T_{0}\) where the plane wave approach is valid. The gains \(g_{\ell}\) are assumed positive random variables, whereas the phase shifts \(\theta_{\ell}\) are independent and identically distributed (i.i.d.) random variables with a circular uniform distribution in \([-\pi,\pi)\). The time-varying propagation delays \(\tau_{\ell}[k]\) in (2) are defined by:  
\[\tau_{\ell}[k]=\frac{d_{\ell}^{T}+d_{\ell}^{R}}{c}-\frac{kT_{s}}{f_{c}}\left[ f_{\ell}^{S}+\frac{\dot{f}_{\ell}^{A}[k]}{2}\right] \tag{4}\]  
where \(f_{\ell}^{S}\) and \(\dot{f}_{\ell}^{A}[k]\) characterize the Doppler frequency shifts caused by the initial speed and the acceleration of the MSs, respectively, and they are given by:  
\[f_{\ell}^{S}=f_{\max}^{T}\cos(\phi_{\ell}^{T}-\gamma_{T})+f_{\max}^{R}\cos( \phi_{\ell}^{R}-\gamma_{R}) \tag{5}\]  
\[\dot{f}_{\ell}^{A}[k]=\dot{f}_{\max}^{T}\cos(\phi_{\ell}^{T}-\rho_{T})+\dot{f} _{\max}^{R}\cos(\phi_{\ell}^{R}-\rho_{R}) \tag{6}\]  
with \(f_{\max}^{q}=\
u_{q}/\lambda\) and \(\dot{f}_{\max}^{q}[k]=(kT_{s}a_{q})/\lambda\) for \(q\in\{T,R\}\), which correspond to the maximum Doppler frequency shifts due to the initial speed and the acceleration of \(T_{X}\) and \(R_{X}\), respectively. The parameters \(\
u_{q}\) and \(a_{q}\) for \(q\in\{T,R\}\) represent the initial speeds and accelerations of the MSs.