# Research on Vehicle-to-Vehicle MIMO Wireless Channels in Various Tunnels
## 3 Channel Analysis for V2V MIMO in Tunnels  
### V2V MIMO Channel Model  
In the proposed V2V channel model, we define the reference scatter on the tunnel inter-surfaces as \(S^{(m,n,\rho)}\), where the variables \(m\), \(n\) and \(o\) are infinite in the 3D coordinate system. Then the V2V MIMO channel can be described by an \(M_{R}\times M_{T}\) channel matrix, \(H_{kl}(\tau^{\prime},t)=\left[h_{kl}(\tau^{\prime},t)\right]_{M_{R}\times M_{T}}\), where \(h_{kl}(\tau^{\prime},t)\) presents the time-variant impulse response. The TVTF indicated by \(H_{kl}(f^{\prime},t)\) can be simply analyzed, and is the Fourier transform of \(h_{kl}(\tau^{\prime},t)\); then the \(H_{kl}(f^{\prime},t)\) of the reference model can be expressed as [27; 28]  
\[H_{kl}(f^{\prime},t)=H_{kl}^{Dif}(f^{\prime},t)+H_{kl}^{LoS}(f^{\prime},t)+H_{ kl}^{Spe}(f^{\prime},t)\,, \tag{10}\]  
where \(H_{kl}^{Dif}(f^{\prime},t)\), \(H_{kl}^{LoS}(f^{\prime},t)\), and \(H_{kl}^{Spe}(f^{\prime},t)\) are assumed to be independent, and stand for the impulse responses of the diffuse, LoS, and specular propagation components, respectively. Based on the assumptions that only considering single-bounce scattering components, the scattered signal from the transmitter from the MIMO \(l\)-th transmitter antenna unit \(A_{T}(l)\) travels over the scatter \(S^{(m,n,\rho)}\) before being received at the MIMO \(l\)-th receiver antenna unit \(A_{R}(k)\), shown in Figure 4, then \(H_{kl}^{Dif}(f^{\prime},t)\) in Equation (10) from MIMO antenna element \(A_{T}(l)\) to \(A_{R}(k)\) can be expressed as  
\[H_{kl}^{Dif}(f^{\prime},t) = \lim_{ \begin{subarray}{c}M\rightarrow\infty\\ \Omega\rightarrow\infty\end{subarray} }\frac{1}{\sqrt{(c_{R}+1)MNO}}\sum_{m,n, \rho=1}^{M,N,O} \tag{11}\] \[\times e^{-j\frac{2\pi}{\Delta}P_{kl}^{(mmo)}+j\left[2\pi(f_{T}^{(mmo)}+f _{R}^{(mmo)})+\vartheta^{(mmo)}-2\pi r_{kl}^{(mmo)}f^{\prime}\right]}\,,\]where  
\[D_{kl}^{(mno)}=D_{T}^{(L,mno)}+D_{R}^{(mno,k)}\,, \tag{12}\]  
\[D_{T}^{(L,mno)} = \sqrt{(x_{m}-x_{T})^{2}+(y_{n}-y_{T})^{2}+(z_{o}-z_{T})^{2}} \tag{13}\] \[- (M_{T}-2l+1)\] \[\times \frac{\delta_{T}}{2}\big{[}\,\cos\phi_{T}\cos\beta_{T}^{(mno)}\cos \left(\gamma_{T}-\alpha_{T}^{(mno)}\right)+\sin\phi_{T}\sin\beta_{T}^{(mno)} \,\big{]}\,,\]  
\[D_{R}^{(mno,k)} = \sqrt{(x_{m}-x_{R})^{2}+(y_{n}-y_{R})^{2}+(z_{o}-z_{R})^{2}} \tag{14}\] \[- (M_{R}-2k+1)\] \[\times \frac{\delta_{R}}{2}\big{[}\,\cos\phi_{R}\cos\beta_{R}^{(mno)} \cos\left(\gamma_{R}-\alpha_{R}^{(mno)}\right)+\sin\phi_{R}\sin\beta_{R}^{(mno )}\,\big{]}\,,\]  
\[f_{T}^{(mno)}=f_{T_{max}}\cos\left(\alpha_{T}^{(mno)}-\varphi_{v}^{T}\right) \cos\beta_{T}^{mno}\,, \tag{15}\]  
\[f_{R}^{(mno)}=f_{R_{max}}\cos\left(\alpha_{R}^{(mno)}-\varphi_{v}^{R}\right) \cos\beta_{R}^{mno}\,. \tag{16}\]  
In Equation (11), the phase \(\theta^{(mno)}\) is related to the plane wave and the scatter, which is assumed to be independently randomly distributed in the interval \([\,0,2\pi\,)\). The \(\tau_{kl}^{(mno)}\) stands for the propagation delay of the diffuse reflection components, which can be derived by \(\tau_{kl}^{(mno)}=D_{kl}^{(mno)}/c_{o}\), where \(c_{o}\) is the speed of light. In Equations (15) and (16), \(f_{T_{max}}\) and \(f_{R_{max}}\) represent the maximum Doppler associated with the transmitter and receiver, which can be, respectively, calculated by \(f_{T_{max}}=v_{T}/\lambda\) and \(f_{R_{max}}=v_{R}/\lambda\), with \(\lambda\) representing the wavelength. For the LoS propagation links, the complex channel response can be written by  
\[D_{kl}^{(mno)}=D_{T}^{(L,mno)}+D_{R}^{(mno,k)}\,, \tag{12}\]0  
where  
\[D_{kl}^{(mno)}=D_{T}^{(L,mno)}+D_{R}^{(mno,k)}\,, \tag{12}\]1 \[D_{kl}^{(mno)}=D_{T}^{(L,mno)}+D_{R}^{(mno,k)}\,, \tag{12}\]2  
\[D_{kl}^{(mno)}=D_{T}^{(L,mno)}+D_{R}^{(mno,k)}\,, \tag{12}\]3  
\[D_{kl}^{(mno)}=D_{T}^{(L,mno)}+D_{R}^{(mno,k)}\,, \tag{12}\]4  
where \(\alpha_{T}^{(0)}\), \(\beta_{T}^{(0)}\), \(\alpha_{R}^{(0)}\), and \(\beta_{R}^{(0)}\), respectively, stand for the AAOD, EAOD, AAOA, and EAOD of the NLoS propagation links in the proposed V2V channel model.