## 4 Performance Evaluation of Millimeter-Wave V2V
### Millimeter-Wave V2V Communications with Antenna Height Diversity
In particular, there were sharp drops at 18 and 35 m, and there was a sharp rise at 84 m. These sharp changes were due to antenna vibration, which made the fading point at 57 m move \(\pm 21\) m and at 28 m move \(\pm 10\) m. Considering that a shift from \(-3\sigma\) to \(+3\sigma\) happened at around 99.7% and the shift of 0.01% outage capacity was larger than the shift of \(\pm 3\sigma\), this shift was a reasonable result. The red solid line shows the outage capacity with height diversity. Although it was improved from no diversity, the outage capacity gradually changed up and down. The main reason for this change was that there were some places where both receivers encountered strong fading. To improve this outage capacity with height diversity, we had to solve the changing up and down of the outage capacity. The following analysis consisted of two parts. The first analysis focused on the best receiving antenna space for the second receiving antenna. In this analysis, carrier frequency, inter-vehicle distance, the height of the transmitting antenna, and the height of the first receiving antenna were given. The second analysis focused on the relation between the communication range and the number of receiving antennas. In the analysis of the receiving antenna space, firstly, we focused on the difference of the phase difference between the direct wave path and the reflected wave path. When the height diversity worked well, this phase difference was around \(\pi\), as the following formula. Figure 14: The channel capacity under the two-ray ground reflection model with and without the antenna vibration and the 0.01% outage capacity with and without height diversity at 60 GHz. \[\frac{2\pi}{\lambda}\{(r_{\rm d2}-r_{\rm r2})-(r_{\rm d1}-r_{\rm r1})\}\equiv\pi \pmod{2\pi} \tag{15}\]  
where \(r_{\rm d1}\) (\(r_{\rm d2}\)), \(r_{\rm r1}\) (\(r_{\rm r2}\)) are the lengths of the direct and reflected paths from the transmitter whose height is \(h_{\rm trv}\) to the lower (upper) receiver whose height is \(h_{\rm rv1}\) (\(h_{\rm rv2}\)). Using the approximation of \(\sqrt{1+x}\approx 1+x/2\quad(1\gg|x|)\), Equation (15) can be described as follows. \[\frac{2\pi}{\lambda}\{(r_{\rm d2}-r_{\rm r2})-(r_{\rm d1}-r_{\rm r 1})\}\] Figure 13.0 Figure 13.1  
where \(d\) is the inter-vehicle distance. From the above approximation, it was shown that variable in the difference was only \(\delta_{\rm t}\). We expressed the difference as \(\Phi(\delta_{\rm t})\). When there is no vibration, that is \(\delta=0\), we describe the solutions \(h_{\rm n}\) of Equation (16) as follows. Figure 13.2  
To choose the best \(h_{\rm n}^{\rm best}\) among the solutions \(h_{\rm n}\), we estimated \(\Phi\) where \(\delta_{\rm t}\) ranged from \(-3\sigma_{0}\) to \(3\sigma\), which fell within around 99.73%. This was because, recalling that the goal was to solve the changing up and down of the outage capacity due to the antenna vibration, it was necessary to select one solution for which the phase difference \(\Phi\) did not change more than \(\pi/2\) with a large probability. Since \(\Phi(0\mid h_{\rm n})-\Phi(-3\sigma_{0}\mid h_{\rm n})\) and \(\Phi(3\sigma_{0}\mid h_{\rm n})-\Phi(0\mid h_{\rm n})\) were the same, we focused on \(\Phi(0\mid h_{\rm n})-\Phi(-3\sigma_{0}\mid h_{\rm n})\). Moreover, we restricted the maximum phase difference to \(65^{\circ}\), which ensured more than half power at the other receiver. The second analysis was about the communication range. In the first analysis, we analyzed the best receiving antenna space under the fixed inter-vehicle distance. However, it was not realistic to equip receiving antennas for each inter-vehicle distance, which would require too many antennas. In order to solve this problem, we derived the range of the inter-vehicle distance where the height diversity worked well and minimized the number of receiving antennas.