## 5 Power Delay Profile Measurements  
The IFFT function was applied to the frequency domain results in the time domain. By doing so we obtained the Power Delay Profile (PDP) as described in (8). The PDP was normalized and plotted in Figure 8, for 5, 10, 20 and 45 m of distance between the transmitter and receiver. This visualization of the experiment and simulation data can offer a clearer understanding of the multi-path components that reflect off the walls on their way to the receiver. The multi-path components can challenge the use of high data rates [28; 29]. The PDP plot demonstrates the different rays received at the receiver. For example, in the case of a 5 m gap between the transmitter and the receiver, the LOS ray reaches the receiver after a flight time of \(t_{\text{{LOS}}}=5/c\approx 17\) ns where \(c\) is the speed of light. This phenomenon can be observed in Figure 8a. The LOS ray arrives first with the strongest power at 17 ns, and subsequently the rest of the rays arrive with decreasing power. Although there is an excellent fit between the experiment and the ray-tracing model throughout the entire pulse, as can be seen in Figure 8a, as the distance between the transmitter and receiver increases, the fit is greater in the main area of the pulse and there is  
Figure 8: The time domain response of the transmission scattering parameter \(S_{21}\) for various distances. less fit at the beginning and end. This phenomenon is due to the fact that since the received power was obviously decreasing along the tunnel, the experiment setup was nearing the noise threshold. In addition, although the tunnel is almost completely flat, it still has a certain roughness. The effect of the roughness intensifies as you progress along the tunnel, and each ray may disperse several more times beyond what is calculated in the model. Despite all this, there is still a perfect match in the main and important part of the results, and the differences only occur with a difference of over 30 dB. In order to obtain a broad observation of the tunnels' influence on the wide-band communication, a 3D plot of the PDP throughout the tunnel is shown in Figure 9.  
As can be observed in Figure 9, the LOS ray is absorbed at the receiver with a time suitable to the speed of light. This means that the LOS ray is received first at the receiver, followed by the rest of the rays which have been reflected multiple times off the walls of the tunnels. It can be recognized that the main phenomena remain commensurate along the tunnel, and with high similarity between the experiment and simulation. As described in Section 2, the average delay and RMS delay spread were calculated with the data obtained from the experiment and simulation. A comparison between the experiment and simulation data is shown in Table 7.  
The mean delay, \(\overline{\tau}\), remains almost at the same value along the tunnel. The differences that can be noticed between the experiment and the model are within the resolution range of the measurement setup, as mentioned previously. A parameter with a significant value in wide-band communication is the RMS delay spread, as defined in Equation (12). RMS delay spread can accurately convey the message of whether a communication link is possible in the current medium and distance [28]. The RMS delay spread is presented as a function of the distance between the Tx and Rx, as visible in Figure 10. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Distance (m)** & \multicolumn{2}{c}{**Mean Delay Spread \(\overline{\tau}\) (ns)**} & \multicolumn{2}{c}{**RMS Delay Spread \(\sigma_{\tau}\) (ns)**} \\ \hline  & **Experiment** & **Simulation** & **Experiment** & **Simulation** \\ \hline
5 & 3.4 & 3.2 & 0.78 & 0.85 \\
10 & 2.4 & 2.65 & 0.62 & 0.63 \\
20 & 2.6 & 2.85 & 0.66 & 0.67 \\
45 & 2.3 & 2.4 & 0.41 & 0.44 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Average and RMS delay spread for various distances. Figure 9: 3D plot of the PDP throughout the tunnel.