## 2 The Delay Spread Theory in Tunnels and Long Corridors
The average (mean) delay of all the paths is defined as \(\overline{\tau}\)[18]:  
\[R(m,n)=\sqrt{[x_{t}(m)-x_{r}]^{2}+[y_{t}(n)-y_{r}]^{2}+z^{2}} \tag{2}\]0  
where  
\[R(m,n)=\sqrt{[x_{t}(m)-x_{r}]^{2}+[y_{t}(n)-y_{r}]^{2}+z^{2}} \tag{2}\]1  
Substituting \(\overline{\tau}\) and \(\langle\tau^{2}\rangle\) into (12), the expression for \(\sigma_{\tau}\) results in (14):  
\[R(m,n)=\sqrt{[x_{t}(m)-x_{r}]^{2}+[y_{t}(n)-y_{r}]^{2}+z^{2}} \tag{2}\]2  
Since we have described the propagation of the signal in space as rays sent in all directions, it is necessary to specify which rays we will refer to. On the one hand, the number of rays tends to infinity, and on the other hand in terms of mathematical model calculations it is certainly not possible to reach such accuracy. The number of relevant rays depends very much on the parameters of the communication (wavelength, tunnel size, distance). In an earlier study we conducted [23], we examined the error as a function of the number of rays. We have seen that there is certainly some improvement with the results compared to the experiment as more rays are considered. However, even with a small number of rays the results are satisfactory. To avoid mistakes and inaccuracies, in this study, we took into consideration rays that bounced off the walls up to 30 times and off the ceilings or floor up to 30 times as well, before arriving at the receiver, which is above and beyond any required level of accuracy. As described, the reflected rays propagate along the tunnel at different times, thus not reaching the receiver simultaneously. In order to paint a clearer picture of this phenomenon, we will demonstrate this by using an example of a tunnel's impulse response from the model. The model was run in a case that represents the dimensions of the tunnel that will be used in the experiment, as detailed in \[R(m,n)=\sqrt{[x_{t}(m)-x_{r}]^{2}+[y_{t}(n)-y_{r}]^{2}+z^{2}} \tag{2}\]3 The signal was transmitted at the beginning of the tunnel and recorded at the end of the tunnel. The plot, as shown in Figure 1, visualizes each ray arriving at different times at the receiver. The first ray to arrive represents the LOS ray which has the shortest flight time, \begin{table}
\begin{tabular}{l c} \hline \hline
**Parameters** & **Value** \\ \hline Frequency & 1.5 GHz \\ Power transmitted & 20 dBm \\ a (half of the tunnels width) & 0.5 m \\ b (half of the tunnels height) & 0.925 m \\ Length & 20 m \\ Material & concrete \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Model example parameters. and gradually thereafter all the rest of the rays which did not have a direct path will appear. The Y axis represents normalized values of the received power in relation to the transmitted power. To present normalized time values of the received arrays in relation to the LOS array, we define (15):  
\[t_{normalized}=\frac{L_{m,n}-d}{d}=\frac{\Delta t}{t_{r}} \tag{15}\]  
where \(L_{m,n}\) is the distance that the \(m,n\) ray travels, \(d\) is the length between the transmitter and receiver which the LOS ray travels, in this case \(d=20\) m, \(\Delta t\) is the additional time the \(m,n\) ray travels, and \(t_{r}\) is the LOS the ray travels. In the following sections, the analytically calculated and experimental outcome of, inter alia, \(S_{21}\), time domain results and RMS delay spread in a long pedestrian tunnel are presented.