# Research on Vehicle-to-Vehicle MIMO Wireless Channels in Various Tunnels
## 4 Channel Capacity and Doppler of V2V MIMO  
To improve the performances of wireless communications, the improvement of the spectral efficiency or the reduction of the bit error rate is obviously emphasized, so the path between the antenna elements of the transmitter and the receiver has a great influence on the transmission performance. While a large number of diffuse and specular components are distributed in the tunnel environment, one needs to know whether the number of active modes will bring enough spatial correlation and distribution of the singular values of the channel \(H\) matrix to significantly reduce the capacity and Doppler of V2V communication. The capacity can be used to fully characterize the MIMO, which is of great significance for designing the system. Past studies in the literature have addressed the Independent Identical Distribution (IID) complex Gauss channel model, which has laid the theoretical foundation for channel capacity. Here, we will use spatial CCF \(\rho_{kl,k^{\prime}l^{\prime}}(\delta_{T},\delta_{R})\) to explore their important impacts on the MIMO system and describe the statistical characteristics by complementary cumulative distribution function and Ergodic capacity. Usually, if the exact channel characteristics are not known to the system, the optimal method is to distribute the power equally to each antenna element, and the average channel capacity can be expressed as  
\[\bar{C} = E[\log 2 det(I_{N_{r}}+\frac{SNR}{N_{t}}H\cdot H^{T})] \tag{42}\] \[= E[\log 2 det(I_{N_{r}}+\frac{P}{N_{t}\sigma^{2}}R_{r}^{1/2}H_{W}R_{ t}H_{W}^{T}(R_{r}^{T})^{1/2})]\,.\]  
In Equation (42), \(N_{r}\) and \(N_{t}\) are the number of antenna elements at the receiving and transmitting stations, respectively. In the case of IID, channel matrix \(H\) can be expressed as \(H=R_{r}^{1/2}H_{W}(R_{t}^{1/2})^{T}\), where \(R_{r}\) and \(R_{t}\) denote the \(N_{r}\times N_{r}\) and \(N_{t}\times N_{t}\) correlation matrixes of the receiving and transmitting antenna arrays. Generally \(H_{W}\) is the independent and identically distributed \(R_{r}\times R_{t}\) complex Gauss random matrix and \(I_{N_{r}}\) is the unit matrix. When the noise is Gaussian, the signal-to-noise ratio of the channel can be expressed as \(SNR\). The robustness of MIMO in terms of coping with the high correlation among the antenna elements, strongly depends on the antenna array architecture. Here, use Maximum Ratio Combining (MRC) to measure the different response of the power spectrum and the standard deviation in the Binary-Phase-Shift-Keying (BPSK) modulation. Therefore, the BER of the multi-antenna arrays can be computed as  
\[P_{e}=\frac{1}{2}\sum_{k=1}^{K}\pi_{k}(1-\sqrt{\frac{\lambda_{k}}{\lambda_{k}+ 1}})\,, \tag{43}\]  
where \(\pi_{k}=\prod_{i=1}^{N}(\frac{\lambda_{k}}{\lambda_{k}-\lambda_{i}}),i\
eq k\). In Equation (43), \(\lambda_{k}\) is the \(k\)-th eigenvalue of spatial CCF matrix \(R_{r}\). So far, we can obtain the required eigenvalues by decomposition of the spatial CCF matrix, where the matrix elements have been derived in the previous sections. The eigenvalue decomposition formula can be expressed as  
\[R_{r}=U\beta V^{H}\,, \tag{44}\]  
where the matrices \(U\) and \(V\) denote the unit matrix and \(\beta\) is the diagonal matrix. We have  
\[R_{r}= \begin{bmatrix}1&\rho_{k^{\prime}l^{prime}}(\delta_{R})\\ \rho_{k^{\prime}l^{prime}}(\delta_{T})&1\end{bmatrix}