# Research on Vehicle-to-Vehicle MIMO Wireless Channels in Various Tunnels
## 2 Geometrical Tunnel Propagation Model  
It is worth mentioning that the transmitter and receiver are equipped with low elevation antennas in V2V channel model for tunnel communication scenarios, as shown in Figure 1. The V2V channel models in the prior works are mainly for open communication environments, which are not suitable for characterizing the high dynamic propagation  
Figure 1: Various geometrical tunnels. (**a**) Rectangular tunnel. (**b**) Rectangular tunnel without ceiling. properties of the V2V channel models, especially the tunnel communication scenarios. Furthermore, we notice that the structures of the tunnels are complicated and changeable, which bring us difficulty in building a general channel model to effectively describe the V2V communications in tunnel scenarios. To solve this issue, we conducted the research in this paper. In many studies in the literature, the diffuse, LoS, and SPE scattering components occupy the vast majority of the whole. Hence, the components all need to be considered. Otherwise, since different channel characteristics are for different geometries, they can describe the propagation as rectangle-like or ellipse-like tunnels. Here, the channel model describes the scattering environments inter-tunnel with a length of \(L\). It is assumed that the cross section is semi-oval with radius \(R_{a}\) and \(R_{b}\), shown in Figure 2; the rectangular or semi-rectangular tunnels with width \(W=2R_{a}\) and height \(H=R_{b}\), respectively, are shown in Figure 3.  
It is assumed that the scatters are randomly distributed on the tunnel inter-surfaces, as shown in Figure 4, where the coordinates of the reference scatter are expressed as \(m=1,2,\ldots,+\infty\), \(n=1,2,\ldots,+\infty\), and \(o=1,2,\ldots,+\infty\). For similarity, we define \(z=R_{b}\sqrt{1-(y/R_{a})^{2}}\) for SET, \(z=\sqrt{R^{2}-y^{2}}\) for SET and \(0\leq z\leq R_{b}\) for SRT. This allows us to present the position of the reference scatter \(S^{(m,n,o)}\) by \((x_{m},y_{n},z_{o})\), where \(x_{m}\) and \(y_{n}\) are random variables. Hence, the distribution of the scatters is completely determined by the distribution of \(x_{m}\) and \(y_{n}\). Figure 4 shows \(MS_{T}\) and \(MS_{R}\), the mean the vehicle transmitter and receiver, respectively. We assumed that the transceivers carry a uniform linear MIMO array, consisting of \(M_{T}\) and \(M_{R}\) antenna elements. Spacing of antenna units is indicated by \(\delta_{T}\) and \(\delta_{R}\), respectively. The moving direction of the transceivers in the \(xy\)-plane relative to the \(x\)-axis are denoted by the angles \(\gamma_{T}\) and \(\gamma_{R}\). Similarly, the elevation angles of the transceivers with respect to the \(xy\)-plane are described by \(\varphi_{T}\) and \(\varphi_{R}\), respectively. The coordinates of the MIMO arrays of \(MS_{T}\) and \(MS_{R}\) are indicated by \((x_{T},y_{T},z_{T})\) and \((x_{R},y_{R},z_{R})\). It is also supposed that \(MS_{T}\) and \(MS_{R}\) are inside the tunnels, such that \(0\leq x_{T}\leq x_{R}\leq L\) and \(-R_{a}\leq y_{T}\leq y_{R}\leq R_{a}\) for SET. This is considering the LoS component between \(MS_{T}\) and \(MS_{R}\). Similarly, \(\alpha_{T}^{(m,n,o)}\), \(\alpha_{R}^{(m,n,o)}\), \(\beta_{T}^{(m,n,o)}\) and \(\beta_{R}^{(m,n,o)}\) indicate AAOD, AAOA, EAOD, and EAOA angles, respectively. The transceivers move at a speed of \(v_{T}\) and \(v_{R}\) in an angle of \(\phi_{p}^{T}\) and \(\phi_{p}^{R}\) with respect to the \(x\)-axis. \(D_{T}^{(m,n,o)}\) represents the  
Figure 3: Propagation scenarios in RT and in SRT with an arch ceiling. (a) RT. (b) RT without ceiling. Figure 2: Propagation scenario in SET and SET without ceiling. (a) SET. (b) SET without ceiling. path distance between \(A_{T}^{l}\) and \(S^{(m,n,\alpha)}\); \(D_{R}^{(m,n,\alpha)}\) represents the path distances of \(S^{(m,n,\alpha)}\) to \(A_{R}^{k}\). The path of \(A_{T}^{l}\) through \(S^{(m,n,\alpha)}\) to \(A_{R}^{k}\) is NLoS propagation, and \(D_{LoS}^{(m,n,\alpha)}\) represents the LoS path distance of \(A_{T}^{l}\) to \(A_{R}^{k}\).