**2. Modeling the propagation channel**

Many approaches have been developed to theoretically study electromagnetic wave propagation inside a tunnel, the most well known being those based either on the ray theory or on the modal theory (Mahmoud, 1988; Dudley et al., 2007). The transmitting frequency range must be chosen such that the attenuation per unit length is not prohibitive. To fulfill this requirement, the tunnel must behave as an oversized waveguide. Consequently, the wavelength must be much smaller than the transverse dimensions of the tunnel, which leads to transmitting frequencies greater than few hundred MHz in usual road or train tunnels. The objective of this section is to conduct an overview of the techniques to treat tunnels of simple geometry, such as rectangular or circular straight tunnels, by using either the ray theory or the modal theory. Studying wave propagation along such structures will allow simple explanation and interpretation of the experimental results obtained in real tunnels, even of more complicated shapes. Theoretical approaches to treat tunnels of an arbitrary cross-section and/ or presenting a series of curves will also be briefly presented.

#### **2.1 Propagation in a straight rectangular tunnel**

Let us consider a rectangular tunnel along the z-axis, as shown in Fig. 1, the width and the height of the tunnel being equal to a and *b,* respectively. The coordinate origin is in the centre of the cross-section, at z = 0, which defines the excitation plane. The walls are either characterized by their complex permittivity s; or by an equivalent conductivity o and real permittivity i:,.

Fig. 1. Geometry of the rectangular tunnel.

#### **2.1.1 Ray theory**

Ray theory combined with image theory leads to a set of virtual transmitting (Tx) antennas. If the tunnel is of rectangular cross-section, the determination of the location of these virtual antennas is straightforward, and it is independent of the location of the receiving point. The total field is obtained by summing the contribution of all rays connecting the Tx images and the receiving point (Rx), whilst considering the reflection coefficients on the tunnel walls. However, even by assuming ray propagation, the summation of the contribution of the rays at the Rx point must take into account the vector nature of the electric field. Before each reflection on a wall, the electric field vector must be expressed as the sum of two components: one perpendicular to the incidence plane *Eperp* and one parallel to this plane *Epara•* To each of these components, reflection coefficients RrM and RTE are respectively applied, mathematical expressions for which can be found in any book treating

electromagnetic wave propagation (Wait, 1962; Dudley, 1994). After each reflection, one can thus obtain the new orientation of the electric field vector. The same approach is successively applied by following the rays and by finding the successive incidence planes. Nevertheless, it has been shown (Lienard et al., 1997) that if the distance between Tx and Rx becomes greater than three times the largest dimension of the tunnel cross-section, the waves remain nearly linearly polarized. In this case, the vector summation of the electric field radiated by an antenna and its images becomes a simple scalar summation as in:

$$E(\mathbf{x}, y, z) = \sum\_{m} \sum\_{n} \left( R\_{TM} \right)^{m} \left( R\_{TE} \right)^{n} E\_{d} \left( S\_{mn} \right) \tag{1}$$

where *Ed* ( *Smn)* is the electric field radiated in free space by the image source *Smn* and corresponding to rays having m reflections on the walls perpendicular to the Tx dipole axis and *n* reflections on the walls parallel to the dipole axis (Mahmoud & Wait, 1974). In the following examples the excitation by an electric dipole will be considered, but this is not a strong restriction in the ray approach since other kinds of antennas can be treated by introducing their free space radiation pattern into the model, i.e. by weighting the rays in a given direction by a factor proportional to the antenna gain in this direction.

Lastly, it must also be emphasized that the reflection coefficients on the walls tend to 1 if the angle of incidence on the reflecting plane tends to 90°. This means that, at large distances, only rays impinging the tunnel walls with a grazing angle of incidence play a leading part in the received power, thus the number of rays that fu1fill this condition is important. Typically, to predict the total electric field in standard tunnels and at distances of a few hundred meters, 20 to 30 rays are needed. One should note that if a base station is located outside the tunnel, and if a mobile moves inside the tunnel, the ray theory can still be applied by taking the diffraction in the aperture plane of the tunnel into account (Mariage et al., 1994).

#### **2.1.2 Modal theory**

The natural modes propagating inside the tunnel are hybrid modes *EHmn,* the three components of the electric and magnetic fields that are present (Mahmoud, 2010; Dudley et al., 2007). Any electric field component E(x, *y,* z) can be expressed as a sum of the modal components:

$$E(\mathbf{x}, y, z) = \sum\_{m} \sum\_{n} A\_{mn}(\mathbf{0}) \, e\_{mn}(\mathbf{x}, y) \, e^{-\mathbf{y}\_{mn}z} \tag{2}$$

In this expression, *~n* (0) is the complex amplitude of the mode in the excitation plane, *emn* is the normalized modal eigenfunction, and y *mn* is the complex propagation constant, often written as *Ymn* = *amn* + *j/Jmn.* 

It is interesting to introduce the weight of the modes ~n ( z) at any abscissa z by stating:

$$\operatorname{E}(\mathbf{x}, \mathbf{y}, \mathbf{z}) = \sum\_{m} \sum\_{n} \operatorname{A}\_{mn}(\mathbf{z}) \, e\_{mn}(\mathbf{x}, \mathbf{y}) \quad \text{where} \quad \operatorname{A}\_{mn}(\mathbf{z}) = \operatorname{A}\_{mn}(\mathbf{0}) \text{ } e^{-\gamma\_{mn}z} \tag{3}$$

The analytical expressions of the modal eigenfunctions are usually obtained by writing the boundary conditions on the internal surface of the guiding structure. However, in the case

of lossy dielectric walls, as in the case of a tunnel, approximations are needed and they are detailed in (Emslie et al., 1975; Laakman & Steier, 1976). We have previously outlined that rays remain polarized if the distance between Tx and Rx is larger than a few times the transverse dimensions of the tunnel. In the modal theory, we have the same kind of approximation. If the tunnel is excited by a vertical (y-directed) dipole, the hybrid modes *EH~n* are such that the vertical electric field is dominant. For an x-directed dipole, the modes are denoted *EH!n.* The expressions of the modal functions *e~n(x,y)* for the *y*polarized modes, and *e!;n(x,y)* for the x-polarized modes can be found in (Mahmoud, 2010; Dudley et al., 2007). The solution of the modal equation leads to expressions for the phase and attenuation constants for the y-polarized modes:

$$\alpha\_{\rm inv} = \frac{2}{a} \left(\frac{m\lambda}{2a}\right)^2 \text{Re}\left[\frac{1}{\sqrt{\varepsilon\_r^\* - 1}}\right] + \frac{2}{b} \left(\frac{n\lambda}{2b}\right)^2 \text{Re}\left[\frac{\varepsilon\_r^\*}{\sqrt{\varepsilon\_r^\* - 1}}\right] \text{and } \beta\_{\rm inv} = \frac{2\pi}{\lambda} \left[1 - \frac{1}{2} \left(\frac{m\lambda}{2a}\right)^2 - \frac{1}{2} \left(\frac{n\lambda}{2b}\right)^2\right] \tag{4}$$

From (4), we see that the attenuation is inversely proportional to the waveguide dimension cubed and the frequency squared. It must be stressed that, given the finite conductivity of the tunnel walls, the modes are not precisely orthogonal. Nevertheless, numerical applications indicate that, when considering the first 60 modes with orders ~ 11 and m;; 7, the modes can be considered as practically orthogonal (Lienard et al., 2006; Molina-Garcia-Pardo et al., 2008a). As an example, let us consider a tunnel whose width and height are equal to 4.5 m and 4 m, respectively, and whose walls are characterized by an equivalent conductivity o = 10-2 S/m and a relative real permittivity Er = 10. Table 1 gives the attenuation, expressed in dB /km, of the first *EHm,n* hybrid modes and for two frequencies 2.4 GHz and 10 GHz.


Table 1. Attenuation along 1 km of the various *EHm,n* hybrid modes

At 10 GHz, the fundamental mode exhibits a negligible attenuation, less than 0.2 dB/km, while at 2.4 GHz it is in the order of 5 dB/km. Two other points must be outlined. First, if we consider, for example, a frequency of 2.4 GHz, and a distance of 1 km, one can expect that only 2 or 3 modes will play a leading part in the total received signal, while at 10 GHz, a large number of modes will still be present, the attenuation constant being rather low. This leads to the concept of the number *Na* of "active modes" significantly contributing to the total power at the receiver, and which will be extensively used in 4.1 to predict the capacity of multiantenna systems such as MIMO. To simply show the importance of introducing these active modes, let us recall that the phase velocity of the waves differs from one mode to another. Interference between modes occurs, giving rise to fluctuations in the signal both along the z-axis and in the transverse plane of the tunnel. One can expect that at 1 km, the fluctuations at 10 GHz will be more significant and rapid than at 2.4 GHz, taking the large number of "active modes" into account. Similarly, for a given frequency *Na* continuously decreases with the distance between Tx and Rx. Variations in the field components will thus

be more pronounced in the vicinity of Tx. This phenomenon will have a strong impact on the correlation between array elements used in MIMO systems.

For a vertical transmitting elementary dipole situated at *(x1.x,Y1.x),* the total electric field at the receiving point *(x, y,* z) can be determined from (Molina-Garcia-Pardo et al., 2008c):

$$E(\mathbf{x}, y, z) = \sum\_{m} \sum\_{n} e^{V}\_{m,n} \left( \mathbf{x}\_{tx}, y\_{tx} \right) e^{V}\_{m,n} \left( \mathbf{x}, y \right) e^{-\gamma\_{m,n}z} \tag{5}$$

To treat the more general case of a radiating structure presenting a radiation pattern which is assumed to be known in free space, the easiest solution to determine the weight of the modes in the Rx plane is to proceed in two steps. First, the E field in the tunnel is calculated numerically using, for example, the ray theory, and then the weight *Am,n(z)* of any mode *m,n*  for this abscissa can be obtained by projecting the electric field on the basis *ofe~n(x,y)*  (Molina-Garcia-Pardo et al., 2008a). This leads to:

$$A\_{mn}(z) = \int\_{-a/2-b/2}^{a/2} \int\_{-b/2}^{b/2} E(x, y, z) \cdot e\_{mn}^{\vee}(x, y) \, dx dy \tag{6}$$

In the following numerical application a 8 m-wide, 4.5 m-high tunnel is considered, the electrical parameters of the walls being o = 10-2 S/m and Er= 5. The radiating element is a vertical elementary dipole situated 50 cm from the ceiling, at 1/ 4 of the tunnel width, and the transmission frequency is 900 MHz. This configuration could correspond to a practical location of a base station antenna in a real tunnel. The six modes with the highest energies at a distance of 300 m and 600 m are provided in Table 2.


Table 2. Relative weights of the modes at 300 m and 600 m.

Due the non-centered position of the Tx dipole, the most energetic mode is mode 2,1, although mode 1,1 is the lowest attenuated. The other columns in Table 2 show the relative weight of the other modes, normalized for each distance in terms of the highest mode weight, i.e. mode 2,1. At 300 m, numerous other modes are still significantly contributing to the total field, since the 6th mode only presents a relative attenuation of 5 dB in relation to the strongest mode. On the other hand, at 600 m only a few higher-order modes remain. The application of such an approach for predicting the performance of MIMO systems will be used in 4.1 and 4.2.

#### **2.2 Propagation in a straight circular tunnel**

Even if a perfectly circular tunnel is less usual, it is interesting to outline some specific features related to the modes and polarization of the waves propagating in such a guiding structure. A detailed analysis (Dudley et al., 2007) shows that an elementary electric dipole produces a large set of modes, the possible modes being *TE0m, TM0m* and the hybrid modes *EHnm* and *HEnm•* 

Fig. 2. Configuration of a cylindrical tunnel and location of a transmitting antenna.

An important feature of the propagation phenomena for MIMO communication systems based on polarization diversity is the cross polarization discrimination factor, XPD, defined as the ratio of the co-polarized to the cross-polarized average received power. Indeed, as will be outlined in 3.1.2, XPD quantifies the separation between two transmission channels that use different polarization orientations. In a circular tunnel and assuming ax-oriented dipole Jx at (b,¢o,0), it can be shown (Dudley et al., 2007) that the cross-polar fields at an observation point(p,~,z) vanish at ¢o = 0,Jr / 2 but are maximum at ¢o =Jr/ 4. In this last case, if the observation point is at the same circumferential location as the source point, XPD becomes equal to 1. The polarization of the waves in a circular tunnel is thus quite different than in a rectangular tunnel, where, at large distances, the co-polarized field component is always dominant.

In practice, the shape of a tunnel is often neither perfectly circular nor rectangular. Consequently, numerous measurement campaigns have been carried out in different tunnel configurations and the results, compared to the theoretical approach based on simplified shapes of tunnel cross-sections, are presented in 3.2. However, before presenting narrow band and wideband channel characteristics in a real tunnel, the various methods for numerically treating propagation in tunnels of arbitrary shape will be briefly described.

#### **2.3 Propagation in a tunnel of arbitrary shape**

If we first consider a bent tunnel of rectangular cross-section whose radius of curvature is much larger than the transverse dimensions of its cross-section, approximate solutions of the modal propagation constants based on Airy function representation of the fields have been obtained (Mahmoud, 2010). For an arc-shaped tunnel, the deviation of the attenuation and the phase velocity of the dominant modes from those in a perfectly rectangular tunnel are treated in (Mahmoud, 2008) by applying a perturbation theory. Lastly, let us mention that for treating the propagation in tunnels of arbitrary shapes, various approaches have recently been proposed, despite the fact that they are more complicated to implement and that the computation time may become prohibitive for long-range communication. Ray launching techniques and ray-tube tracing methods are described in (Didascalou et al., 2001) and in (Wang & Yang, 2006), the masking effect of vehicles or trains being treated by introducing additional reflection/ diffraction on the obstacle. A resolution of the full wave Maxwell equations in the time domain through a high order vector finite element discretization is proposed in (Arshad et al., 2008), while solutions based on the parabolic equation and spectral modeling are detailed in (Popov & Zhu, 2000). We will not describe in detail all these methods, since the objective of this chapter is to present the general behavior of the propagation in tunnels, rather than to emphasize solutions to specific problems.

#### **3. Narrow band and wideband channel characterization**

As previously outlined, a number of propagation measurements in tunnels have been taken over the last 20 years, and it is not within the scope of this chapter to make an extensive overview of what has been done and published. Let us simply mention that the results cover a wide area of environment and applications, starting from mine galleries (Lienard & Degauque, 2000a; Zhang et al., 2001; Boutin et al. 2008), road and railway tunnels (Lienard et al., 2003; Lienard et al., 2004), pedestrian tunnels (Molina-Garcia-Pardo et al., 2004), etc. We have thus preferred to select one or two scenarios throughout this chapter in order to clearly identify the main features of the propagation phenomena and their impact on the optimization of the transmission scheme.

#### **3.1 Narrow band channel characteristics**

We first consider the case of a rectangular tunnel for applying the simple approach described in the previous sections and for giving an example of path loss versus the distance between Tx and Rx. Then, results of the experiments carried out in an arched tunnel will be presented and we will study the possibility of interpreting the measurements from a simple propagation model, i.e. by means of an equivalent rectangular tunnel.

#### **3.1.1 Path loss determined from a propagation model in a rectangular tunnel**

Let us consider a wide rectangular, 13 m wide and 8 m high, corresponding to the transverse dimensions of a high-speed train tunnel. Curve (a) in Fig. 3 represents the variation of the field amplitude, expressed in dB and referred to an arbitrary value, versus the distanced, which is determined from the theoretical approach based on the ray theory. The transmitting frequency is 2.1 GHz and the Tx antenna is supposed to be a half-wave dipole situated at a height of 2 m and at a distance of 2 m along a vertical wall.

Fig. 3. a) Amplitude of the received signal, referred to an arbitrary value, determined from the ray theory in a rectangular tunnel. b) Amplitude of the signal for a free space condition (Lienard & Degauque, 1998).

The walls have a conductivity of 10-2 S/m and a relative permittivity of 10. As a comparison, curve (b) corresponds to the case of free space propagation . Along the first 50 m the field amplitude decreases rapidly, in the order of 5 dB/100 m, and fast fluctuations can be observed . They are due to the interference between the numerous paths relating Tx and Rx or, from the modal theory point of view, to interference between modes . Beyond this distance *d1* of 50 m, we note a change in the slope of the path loss which becomes equal to 1 dB/100 m. A twoslope model is thus well suited for such a tunnel and is also detailed in (Marti Pallares et al., 2001; Molina-Garcia-Pardo et al., 2003). In general, the abscissa of the break point *d1* depends on the tunnel excitation conditions and hence, on the position of the Tx antenna in the transverse plane, and on its radiation pattern. For a longer transmission range, a three/fourslope model has also been proposed (Hrovat et al., 2010).

After having determined the regression line in each of the two intervals and subtracted the effect of the average attenuation, one can study the fading statistics. Usually, as in an urban environment, the fading is divided into large scale and small scale fading, by considering a running mean on a few tens of wavelengths (Rappaport, 1996). However, in a straight tunnel, making this distinction between fading cannot be related to any physical phenomena. Furthermore, as it appears from curve 3a, the fading width and occurrence depend on the distance. An analysis of the fading characteristics is given in (Lienard and Degauque, 1998). Let us mention that the masking effect due to traffic in road tunnels or to trains in railway tunnels are detailed in (Yamaguchi et al., 1989; Lienard et al., 2000b; Chen et al., 1996, 2004).

#### **3.1.2 Experimental approach to propagation in an arched tunnel and the concept of an equivalent rectangular tunnel**

Numerous measurements have been carried out in the tunnel shown in Fig. 4a and 4b. The straight tunnel, 3 km-long, was closed to traffic during the experiments . The walls are made of large blocks of smooth stones. It is difficult to estimate the roughness accurately, but it is in the order of a few millimeters . In a first series of experiments, the transmitting power was 34 dBm

Fig. 4. a) Photo of the tunnel where measurements took place; b) Cross-section of the tunnel.

and the Tx and Rx half-wave dipole antennas were at the same height (2 m) and centered in the tunnel. Curves in Fig. 5 show the variation in the received power at 510 MHz, versus the axial distance *d* between Tx and Rx, the Tx and Rx antennas being both horizontally (HH) or vertically (VV) polarized (Molina-Garcia-Pardo et al., 2008b).

Fig. 5. Received power (dBm) in the arched tunnel for two polarizations at 510 MHz (Molina-Garcia-Pardo et al., 2008b).

We first note that the slope of the path loss is much higher for the VV polarization than for HH. Furthermore, if *d* becomes greater than 500 m, the spatial fading becomes periodic, but the periodicity is not the same for HH (160 m) and for VV (106 m). Since such results, showing a lack of cylindrical symmetry, cannot be interpreted with a simple model based on a cylindrical structure, we have tried to find the transverse dimensions of a rectangular tunnel, equivalent to the actual arched tunnel, which minimize the difference between the theoretical and experimental values. The choice of a rectangular cross-section whose surface is nearly equal to the surface of the actual tunnel seemed to be relevant. It appears that the best results were obtained for a rectangle 7.8 m wide and 5.3 m high, as shown in Fig. 4 b, the electrical characteristics of the walls being o = 10-2 S/m and Er= 5. To be useful, this equivalent rectangle must still be valid for other frequencies or configurations . As an example, let us now consider a frequency of 900 MHz, a vertical polarization (VV), both Tx and Rx vertical antennas being either centered in the tunnel (position noted C) or not centered (NC), i.e. situated at ¾ of the tunnel width (Molina-Garcia-Pardo et al., 2008c). Curves in Fig. 6 show the relative received power in dB (normalized to an arbitrary value) at 900 MHz for the two positions of the antennas ("C" and "NC") either measured or determined from the modal theory. In the theoretical modeling, only the first two dominant modes have been taken into account *(EH11* and *EH13* for "C", *EH11* and *EH12* for "NC") .

A rather good agreement is obtained at large distance from Tx, where high order modes are strongly attenuated. The last point which can be checked before using the concept of equivalent rectangular tunnel, deals with the polarization of the waves at a large distance

from Tx, cross-polar components appearing for critical positions of the Tx and Rx antennas in a circular tunnel, while in a rectangular tunnel the waves always remain polarized. Other measurements were made by placing the Tx and Rx antennas in the transverse plane at the critical location where XPD=l in a circular tunnel ( </Jo = *1r* / 4 in Fig. 2). As detailed in Molina-Garcia-Pardo et al. (2008c), the experiments show that the waves remain polarized. The simple model of the propagation in an equivalent rectangular tunnel seems thus quite suitable to predict and/ or justify the performances of communication systems using, for example, MIMO techniques, as described in 4.1 and 4.2.

Fig. 6. Relative received power ( dB) at 900 MHz for two configurations either measured or determined from the modal theory taking only two modes into account (Molina-Garcia-Pardo et al., 2008c).

#### **3.2 Wideband and double directional channel characteristics**

For optimizing the transmission scheme and predicting the performance of the communication link, the channel is characterized in the frequency domain by the coherence bandwidth *Be*  defined as the bandwidth over which the frequency correlation function is above a given threshold (Rappaport, 1996; Molisch, 2005), and in time domain by the delay spread Ds determined from the power delay profile (PDP). In the new generation of transmission schemes as in MIMO, double directional channel characteristics, such as the direction of arrival (DOA) and the direction of departure (DOD) of the rays are interesting to analyze since they have a strong impact on the optimization of the antenna arrays . We will successively present these characteristics established either from a propagation model or from measurements .
