## 3 Number of Dominant Rays: Simulation Results
### Directive Link  
Millimeter wave antennas are usually directive (see for example the radiation pattern of a horn antenna as shown in Figure 6). Moreover, 5G links will utilize beam steering antennas that direct the transmission towards the receiver side. Thus, the radiation pattern of the antenna needs to be considered. Each ray arrives at the receiver at a different spatial angle and has its own intensity and phase. Figure 4: Overview of different numbers of reflected rays at 28 GHz. Figure 5: Root mean squared error at 28 GHz inside the tunnel. Figure 6: Millimeter wave horn antenna radiation pattern. The patterns of transmitting and receiving antennas play a role in the link budget via their respective gains \(G_{t}(\theta,\phi)\) and \(G_{r}(\theta,\phi)\). The resulting overall gain \(G(\theta,\phi)=\sqrt{G_{t}(\theta,\phi)\cdot G_{r}(\theta,\phi)}\) is given in terms of the azimuthal \(\phi\) and elevation \(\theta\) angles, which are determined by the geometry:  
\[ \begin{split}\theta=\arccos\left[\frac{z}{R(m,n)}\right]\\ \phi=\arctan\left[\frac{y_{t}(m)-y_{r}}{x_{t}(m)-x_{r}}\right] \end{split}  \tag{14}\]  
Knowing the effective azimuthal and elevation beam widths \(\theta_{H}\) and \(\theta_{V}\), respectively, a criterion estimating the number of relevant multi-path rays is required for each direction, \(M\) for the vertical and \(N\) for the horizontal directions, respectively. Figure 7 illustrates the utilization of the image method for the \(N^{\text{th}}\) transmitter image location. This image transmits the ray that will be reflected the most, corresponding to the beam angle, \(0.5\theta_{H}\). The distance between the transmitter position \(y_{t}\) and its most distant image location \(y_{t}^{\prime}\) is equal to \(R_{0}tan(0.5\theta)\). For N reflections, the ray crosses the tunnel \(N-1\) times, so we get \(R_{0}tan(0.5\theta)=(N-1)\cdot 2b+y_{r}+y_{t}\). Considering a scenario where the transmitter and receiver are located at the center of the tunnel (e.g., \(y_{r}=y_{t}=b\)), one can find an expression for the maximum number of horizontal reflections, \(N\) (the same way for finding the maximum number of vertical reflections, \(M\)), as follows:  
\[ \begin{split} M&=int\left[\frac{R_{0}\tan(0.5\theta _{H})}{2a}\right]\\ N&=int\left[\frac{R_{0}\tan(0.5\theta_{V})}{2b} \right]\end{split}  \tag{15}\]  
where \(int(x)\) is the nearest integer number. This is an important result, which is useful for evaluating the maximum number of rays required in the simulation for obtaining a reliable estimation of the link budget for different tunnel scenarios. Figure 8a shows the number of rays that play a role in a directive link for which \(\theta_{H}=10^{\circ}\) and \(\theta_{V}=15^{\circ}\). In this case, propagation along the first 140 m of the 2.4 km long Lincoln Tunnel was studied with the parameters summarized in Table 3. Figure 8a shows the number of multi-path rays N for the vertical direction and M for the horizontal direction that play a role in the power budget of a 28 GHz wireless link operating along the tunnel. Inspection of Figure 8a reveals that, up to the first 30 m, only the direct line-of-sight link was dominant, and no multi-path rays (M = N = 0) contributed to the link  
Figure 7: Demonstration of the image method used to derive the number of maximum rays criterion. budget. From 30 m to 60 m, one additional elevated multi-path ray from the tunnel ceiling and floor was added (M = 0, N = 1). The first multi-path ray in the horizontal direction appeared at a distance of 75.5 m, where a single reflection (M = 1) from the walls and two reflections (N = 2) from the ceiling and floor contributed to the link budget. Figure 8b shows the path loss normalized to the LOS direct link. Up to the first 30 m, only the LOS played a role, resulting in 0 dB normalized path loss. The constructive and destructive contributions of the multi-path began after the first 30 m. The resulting incremented group delay is plotted in Figure 8c.