# Research on Vehicle-to-Vehicle MIMO Wireless Channels in Various Tunnels
## 5 Numerical Results and Discussions
### Channel Characteristics of SET with and without Ceilings  
Figure 5 shows the effects of the cross-sectional size of the semi-elliptical on the correlation properties. It can be seen that when we increase the values of the antenna element spacings \(\delta_{T}\) and \(\delta_{R}\) from \(0\) to \(3\), the space correlations of the proposed channel model gradually decrease from \(1\) to \(0\), in agreements with the results in [30], verifying the correctness of the simulations and derivations in (33). Furthermore, we notice that when the distances between antenna elements are \(\delta_{T}=\delta_{R}=0\), the spatial CCFs of SET reaches the maximum value \(1\). However, the absolute value of the SET without ceiling is \(0.37\). Compared with Figure 5 in the LoS scenario, the Rice distribution factor is \(c_{R}=1\), the SCCF is given in Figure 6 for a LoS propagation scenario. As in Figure 6, the TVTF \(H(f,t)\) and \(H(f+v,t+\tau)\) are highly correlated; the same results are shown even for relatively large spacing between the antenna elements. It is worth mentioning that when the transmitter and receiver move in the same direction in V2V channels, the temporal ACFs of the proposed channel model for different Rice factors \(c_{R}\) are as shown in Figure 7a. It can be observed that when we increase the  
Figure 5: Spatial cross-correlation functions of SET and without ceiling for NLoS scenario (\(c_{R}=0\)). (**a**) SCCF of SET. (**b**) SCCF of SET without ceiling. Figure 6: Spatial cross-correlation functions of SET and without ceiling for LoS scenario (\(c_{R}=1\)). (**a**) SCCF of SET. (**b**) SCCF of SET without ceiling. motion time, the temporal correlations of the proposed channel model gradually decrease from 1 to 0, and are in agreement with the results in [11; 12], verifying the correctness of the simulations and derivations. It can also be seen that when we set \(\tau\) as zero, the temporal ACFs of normal SET is 1, while the others are one below 1, as shown in Figure 7b, which demonstrates that SET without ceilings fluctuates more obviously. Figure 8 shows the frequency CCFs of the proposed channel model in different communication situations. It can be seen that there is almost no serious fluctuation as to without ceiling, because of more scatters by the tunnels with the ceiling. Overall, the simulation results in Figures 7 and 8 are consistent with the corresponding analytical results, thereby validating the accuracy of the analysis. Figure 7: ACF of SET and without ceiling for different Rice factors \(c_{R}\). (**a**) ACF of SET. (**b**) ACF of SET without ceiling. Figure 9 shows the frequency CCFs of the proposed V2V channel model for tunnel scenarios with respect to the tunnel size. It is not difficult to see that the resulting values gradually decrease as the scattering range increases. Furthermore, if the cross-sectional area of the tunnel increases, the coherence bandwidth of the proposed channel also decreases. In Figure 9, when the tunnel width \(R_{a}\) is increased from 10 to 16 m, the coherence bandwidth is reduced from 40 to 20 MHz. The coherence bandwidth is more susceptible to frequency fluctuations due to more severe interference from scattering paths in the tunnel. Other verification results are not possible due to the lack of required V2V channel measurement data in other literature studies. However, validating the proposed channel against other channel statistics may be the subject of future research. Figure 10 depicts the Doppler PSDs of the proposed V2V channel model with respect to the different heights of the tunnel and different relative motion directions of the transmitter and receiver. It can be found that when the transmitter and receiver move in the same directions, the values of the Doppler PSDs behave according to U-shaped distribution. On the other hand, when the transmitter and receiver move in the other directions, the Doppler PSDs decrease gradually. Furthermore, as the tunnel height rises, the value of Doppler PSDs rises as well.