Figure 10.6 Ducting of a radio wave.

10.4 LOS Path

In an LOS path the receiving antenna is above the radio horizon of the transmitting station. Links between two satellites and between an earth station and a satellite are LOS paths, and a path between two terrestrial stations may be such. In the two latter cases the tropospheric effects discussed previously must be taken into account.

According to the ray theory, it is enough that just a ray can propagate unhindered from the transmitting antenna to the receiving antenna. In reality a radio wave requires much more space in order to propagate without extra loss. The free space must be the size of the so-called first Fresnel ellipsoid, shown in Figure 10.7. It is characterized by

where r 1 and r 2 are distances of a point on the ellipsoid from the transmitting and receiving points, and r 0 is their direct distance. The radius of the first Fresnel ellipsoid is

Figure 10.7 The first Fresnel ellipsoid.

256 Radio Engineering for Wireless Communication and Sensor Applications

We observe that the radius of the ellipsoid in a given path is the smaller the higher the frequency; that is, the ray theory holds better at higher frequencies.

Figure 10.8 shows the extra attenuation due to a knife-edge obstacle in the first Fresnel ellipsoid. According to the ray theory, an obstacle hindering the ray causes an infinite attenuation, and an obstacle just below the ray does not have any effect on the attenuation. However, the result shown in Figure 10.8 is reality and is better explained by Huygens’ principle: Every point of the wavefront above the obstacle is a source point of a new spherical wave. This explains diffraction (or bending) of the wave due to an obstacle. Diffraction also helps the wave to propagate behind the obstacle to a space, which is not seen from the original point of transmission. When the knifeedge obstacle is just on the LOS path (h = 0), it causes an extra attenuation of 6 dB. When the obstacle reaches just to the lower boundary of the first Fresnel ellipsoid, the wave arriving at the receiving point may be even stronger than that in a fully obstacle-free case. If there is a hill in the propagation path that cannot be considered a knife-edge, extra attenuation is even higher than in case of a knife-edge obstacle.

When terrestrial radio-link hops are designed, bending of the Earth’s surface as well as bending of the ray in the troposphere must be taken into

Figure 10.8 Extra attenuation due to the knife-edge diffraction.

account. The radio horizon is at the distance of r H if the transmitting antenna is at the height of

Example 10.1

Consider a 50-km radio-link hop at 10 GHz, when the antenna heights are the same. What are the required antenna heights?

Solution

Taking K = 4/3 we get from (10.13) that the radio horizon of the antennas is in the middle point of the path between the antennas, when hH = 36.8m. By introducing l = 0.03m and r 1 = r 2 = 25 km into (10.12) we get the radius of the first Fresnel ellipsoid in the middle point as hF = 19.4m. The antenna heights must be then at least h = hH + hF = 56.2m. In addition, if there are woods in the path, the height of the trees must be taken into account.

At low frequencies, fulfillment of the requirement of leaving the first Fresnel ellipsoid empty is difficult. Often we must compromise. For example, at 100 MHz in the 50-km hop of the example, the maximum radius of the first Fresnel ellipsoid is 194m.

A radio-link hop design is aided by using a profile diagram of the terrain, which takes into account the bending of the radio ray in standard conditions. In the diagram, the height scale is much larger than the horizontal scale, and the terrain along the hop is drawn so that perpendicular directions against the Earth’s surface are parallel. The middle point of the hop is placed in the middle of the diagram. After drawing the terrain, the antenna heights are selected so that the first Fresnel ellipsoid is fully in free space, if possible. Figure 10.9 presents a profile diagram between Korppoo and Turku in the Finnish archipelago.

10.5 Reflection from Ground

In an LOS path, the receiving antenna often receives, besides a direct wave, waves that are reflected from the ground or from obstacles such as buildings. This is called multipath propagation.

258 Radio Engineering for Wireless Communication and Sensor Applications

Figure 10.9 Profile diagram for radio-link hop planning.

Let us consider a simple situation, where there is a flat, smooth ground surface between the transmitting and receiving antenna masts, which are located so that their distance is d , and the distance between the transmitting and receiving points is r 0 , as presented in Figure 10.10. In practice this situation may be quite typical in VHF and UHF radio broadcasting, where an antenna mast with a height h1 is at the broadcasting station. Let us further assume that the transmitting antenna radiates isotropically. The electric field strength due to the direct wave at the distance r 0 from the transmitter is E0 . Taking also the reflected wave into account, the total electric field strength at the receiving point is

where r is the reflection coefficient of the ground surface, and we have assumed that r 1 + r 2 ≈ r 0 . The reflection coefficient depends on the electric

Figure 10.10 A direct and reflected wave.

properties of the surface and on the polarization of the wave. At frequencies above 30 MHz the reflection coefficient can be assumed to be −1 if the polarization is horizontal (or perpendicular) and the antenna heights are much less than the distance d ; that is, the grazing angle g is small. For the vertical or parallel polarization the amplitude and phase angle of the reflection coefficient vary rapidly at small angles g [5].

In the following we assume a horizontal polarization. In practice the antenna heights h1 and h2 are often small compared to the distance d , and therefore the angle g is small and r 1 + r 2 − r 0 ≈ 2h1 h2 /d . Now the field strength is

The actual radiation pattern corresponds to that of the array formed by the transmitting antenna and its mirror image. The electric field strength E varies as a function of distance between values 0 and 2E0 , as shown in Figure 10.11(a). Note that E0 decreases as 1/d . The nulls of the field strength (maxima for vertical polarization in case of an ideal conducting surface) are at heights h2 ≈ nl d /(2h1 ), where n = 0, 1, 2, . . . , as shown in Figure 10.11(b). The receiving antenna height must be selected correctly. In order to reduce the effects of reflections, it is possible to use a transmitting antenna that has radiation nulls at directions of the reflection points. Also, a forest at the theoretical reflection points helps because it effectively eliminates reflections.

Figure 10.11 Field strength (a) as a function of the distance d when the receiving antenna height is h2 = 10m and (b) as a function of the receiving antenna height h2 when d = 10 km. The situation resembles Figure 10.10: r = −1, f = 500 MHz, h1 = 300m.