# A Non-WSSUS Channel Simulator for V2X Communication Systems
## 5 The Proposed Vehicular Channel Simulator  
For the simulation of the reference channel model given by the transfer function \(H[m,k]\) in (2), we follow the principle of deterministic channel modeling [52]. According to this principle, the simulation of a non-realizable stochastic reference channel model can be accomplished by generating a single realization or sample function in each simulation run. Therefore, a realization of the reference channel can be computed by following the transfer function [53; 54]:  
\[\widetilde{H}[m,k]=\Pi_{T_{0}}[k]\sum_{\ell=1}^{\mathcal{L}}\widetilde{g}_{ \ell}\times\exp\left\{-j\Big{(}\widetilde{\theta}_{\ell}+2\pi\widetilde{ \widetilde{\tau}}_{\ell}[k](f_{c}+m\Delta f)\Big{)}\right\} \tag{11}\]  
where \(\widetilde{g}_{\ell}=\sqrt{\frac{1}{\mathcal{L}}}\) for all \(\ell\) is an observation of the amplitude, \(\widetilde{\theta}_{\ell}\in[-\pi,\pi)\) is a single realization of the phase shift that follows a uniform circular distribution, and the time-varying propagation delays \(\widetilde{\tau}_{\ell}[k]\) are given by:  
\[\widetilde{\tau}_{\ell}[k]=\frac{\mathcal{G}_{T}(\widetilde{\phi}_{\ell}^{T}) +\mathcal{G}_{R}(\widetilde{\phi}_{\ell}^{R})}{C}-\frac{kT_{s}}{f_{c}}\left[ \widetilde{f}_{\ell}^{S}+\frac{\widetilde{f}_{\ell}^{A}[k]}{2}\right] \tag{12}\]  
with:  
\[\widetilde{f}_{\ell}^{S}=f_{\max}^{T}\cos(\widetilde{\phi}_{\ell}^{T}-\gamma_ {T})+f_{\max}^{R}\cos(\widetilde{\phi}_{\ell}^{R}-\gamma_{R}) \tag{13}\]  
\[\widetilde{f}_{\ell}^{A}[k]=\tilde{f}_{\max}^{T}\cos(\widetilde{\phi}_{\ell}^ {T}-\rho_{T})+\tilde{f}_{\max}^{R}\cos(\widetilde{\phi}_{\ell}^{R}-\rho_{R}) \tag{14}\]  
where \(\widetilde{\phi}_{\ell}^{T},\widetilde{\phi}_{\ell}^{R}\in[-\pi,\pi)\) are realizations of the AOD and AOA, respectively, that follow the distributions of the circular random variables \(\phi_{\ell}^{T}\) and \(\phi_{\ell}^{R}\), as described in Section 4.2. The rest of the parameters \(\mathcal{L}\), \(f_{c}\), \(D\), \(d\), \(v_{q}\), \(a_{q}\), \(\gamma_{q}\), and \(\rho_{q}\) for \(q\in\{T,R\}\), as well as \(\mu_{R}\) and \(\kappa\) for the distribution of \(\phi_{\ell}^{R}\) are deterministic quantities introduced by the user in order to generate arbitrary propagation environments. This simulation model can be classified as an SOC model of Class XII according to the classification described in [55]. The described channel simulator is suitable for the performance analysis of V2X communication systems based on multicarrier transmission schemes, such as specified in the current V2X communication standards. For this purpose, we consider that the system uses \(N_{p}\) subcarriers to transmit \(K\) symbols. Then, the simulation model in (11) can be used as \(\langle\widetilde{\mathbf{H}}\rangle_{mk}=\widetilde{H}[m-N_{p}/2,k-1]\) for \(m=1,\ldots,N_{p}\) and \(k=1,\ldots,K\) to generate the channel transfer function matrix. This expression allows us to start the transmission at time \(t=0\) and generate a spectrum centered at the carrier frequency \(f_{c}\). A realization of the channel model requires us to specify the bandwidth \(B\) and the carrier frequency \(f_{c}\), as well as the spacing in time and frequency according to the system specifications (e.g., IEEE 802.11p or LTE-V2X) for a total of \(K\) symbols and \(N_{p}\) subcarriers. Furthermore, the propagation scenario should be initialized by specifying the number of IOs, the scattering parameters, and the components for the dynamics of the MSs. Besides, for the plane wave propagation approach to be valid in the simulation channel model, the far-field condition must be met. Therefore, we consider defining arbitrary distance values such that \(D\gg d\). Table 3 shows the pseudocode for the function _GenerateChannel_, which produces a single realization of the channel based on the proposed simulation model.