## 2 The Delay Spread Theory in Tunnels and Long Corridors  
Electromagnetic wave propagation inside tunnels suffers from multi-path interference, which distorts the received signal due to constructive and destructive waves arriving simultaneously at the receiver. The propagation depends on different parameters such as: Construction materials, antenna polarization, transmitter and receiver location in the cross-sectional area, the separation between the transmitter and receiver, and so on. It is described using (1): [22; 23]  
\[\bar{H}_{norm}(jf)=\frac{\bar{E}_{r}(Tunnel)}{\bar{E}_{r}(LOS)}=\sum_{m=- \infty}^{+\infty}\sum_{n=-\infty}^{+\infty}\underbrace{\frac{R_{0}}{R(m,n)} \cdot\Gamma_{\alpha}^{|m|}\cdot\Gamma_{\beta}^{|n|}\cdot\frac{G(\theta,\phi)}{ G_{LOS}}}_{A_{mut}}\cdot e^{-j2\pi f/\Delta\tau(m,n)} \tag{1}\]  
\(\bar{H}_{norm}(jf)\) is normalized to the LOS ray for convenience. \(R_{0}=R(0,0)\) is the line of sight (LOS) ray length and \(R(m,n)\) is the length of the path that then'th and m'th order of reflected ray has traveled, as described in (2):  
\[R(m,n)=\sqrt{[x_{t}(m)-x_{r}]^{2}+[y_{t}(n)-y_{r}]^{2}+z^{2}} \tag{2}\]  
The image coordinates of the transmitter are defined in (3):  
\[x_{t}(m)=2 ma+(-1)^{m}\cdot x_{t}\quad\text{and}\quad x_{t}(m)=2 ma+(-1)^{m} \cdot x_{t} \tag{3}\]when \(a\) represents half of the width of the tunnel and \(b\) represents half of the tunnel's height. The reflection coefficient, \(\Gamma\), is calculated using [PERSON]'s equations. Since one reflection coefficient will be transverse electric (TE) and the other transverse magnetic (TM) depending on the polarization of the antenna [23; 24], they were marked \(\Gamma_{\alpha}^{[m]}\) and \(\Gamma_{\beta}^{[n]}\). \(m\) represents the number of reflections in the horizontal dimension, and \(n\) is the same but in the vertical dimension. \(f\) is the frequency of the wave. For a directive link, \(G(\theta,\phi)\) is the geometrical mean gain of both the transmitter and receiver antennas, \(G(\theta,\phi)=\sqrt{G_{t}(\theta,\phi)\cdot G_{r}(\theta,\phi)}\), and \(G_{LOS}\) is the gain of the antenna for the LOS ray. For simplicity, we define \(A_{mn}\) as the following (4):  
\[\tilde{A}_{mn}=\frac{R_{0}}{R(m,n)}\cdot\Gamma_{\alpha}^{[m]}\cdot\Gamma_{ \beta}^{[n]}\cdot\frac{G(\theta,\phi)}{G_{LOS}} \tag{4}\]  
The propagation time for the LOS ray from the transmitter to the receiver is defined in (5):  
\[\tau_{0}=\frac{R_{0}}{c} \tag{5}\]  
The propagation time delay of each of the other rays is defined as follows (6):  
\[\tau(m,n)=\frac{R(m,n)}{c}=\tau_{0}+\Delta\tau(m,n) \tag{6}\]  
where \(\Delta t(m,n)\) is the additional time that the \((m,n)\) ray travels more then the LOS ray travels. Assuming the antenna's gain is not dependent on the frequency, the impulse response is:  
\[h(t)=\mathcal{F}^{-1}\{\tilde{H}_{norm}(jf)\}=Re\{\int_{0}^{\infty}\tilde{H}_ {norm}(jf)\cdot e^{+\beta/2\pi ft}df\}=Re\{\sum_{m=-\infty}^{+\infty}\sum_{n=- \infty}^{+\infty}\tilde{A}_{mn}\cdot\delta[t-\Delta\tau(m,n)]\} \tag{7}\]  
The Power Delay Profile (PDP) of the channel, \(P_{h}(\tau)\), describes how much power (from a transmitted delta pulse with unit energy) is transferred to the receiver with a delay of \(\tau\) from the moment of transmission. The Power Delay Profile is acquired from the normalized complex impulse response as described in (7) and defined as follows [25]:  
\[P_{h}(\tau)=\frac{|h(\tau)|^{2}}{\|h(\tau)\|^{2}} \tag{8}\]  
where the norm of \(h(\tau)\) is defined by (9):  
\[\|h(\tau)\|^{2}=\int_{-\infty}^{+\infty}|h(\tau)|^{2}d\tau=\int_{0}^{\infty}| \tilde{H}_{norm}(jf)|^{2}df=\sum_{m=-\infty}^{+\infty}\sum_{n=-\infty}^{+ \infty}|\tilde{A}_{mn}|^{2} \tag{9}\]  
From the last definition, it follows that:  
\[\int_{-\infty}^{+\infty}P_{h}(\tau)d\tau=1 \tag{10}\]  
The PDP is the probability density function (PDF) of the delay \(\tau\) of an impulse, with dimensions of \([\frac{1}{sec}]\).