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FIGURE 4.6. A free-space pattern of the rst Fresnel zone covering both terminals, the transmitter TX and the receiver RX .
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visibility or clearance of the propagation channel by using the right-hand term in (4.24) to evaluate the radius of the rst Fresnel zone on the basis of the knowledge of the wavelength of the radiated wave and the range between the two terminal antennas. This is very important for link budget design in atmospheric channels where the terminal antennas are far from the Earth s surface. This aspect will be discussed in 6 when we deal with atmospheric communication links. 4.1.5. Polarization of Radio Waves To understand the aspect of wave polarization, let us de ne this phenomenon. The alignment of the electric eld vector E of a plane wave relative to the direction of propagation k de nes the polarization of the wave (see 2). If E is transverse to the direction of wave propagation k then the wave is said to be TE-wave or vertically polarized. Conversely, when H is transverse to k the wave is said to be TM-wave or horizontally polarized. Both of these waves are linearly polarized, as the electric eld vector E has a single direction along the entire propagation axis (vector k). If two plane linearly polarized waves of equal amplitude and orthogonal polarization (vertical and horizontal) are combined with a 90 phase difference, the resulting wave will be a circularly polarized (CP) wave, in which the motion of the electric eld vector will describe a circle around the propagation vector. The eld vector will rotate by 360 for every wavelength traveled. Circularly polarized waves are most commonly used in land cellular and satellite communications, as they can be generated and received using antennas that are oriented in any direction around their axis without loss of power [1 3]. They may be generated as either right-hand circularly polarized or left-hand circularly polarized, depending on the direction of vector E rotation (see Fig. 4.7). In the most general case, the components of the combining waves could be of unequal amplitude, or their phase difference could be other than 90 . This combination result is an elliptically polarized wave, where vector E still rotates at the same rate as for circular polarized wave, but varies in amplitude with time. In the case of elliptical polarization, the axial ratio, AR Emaj =Emin , is usually introduced (see Fig. 4.7). AR is de ned to be positive for left-hand polarization and negative for right-hand polarization. Now let us turn our attention to the wave eld polarization in the case of a free-space propagation channel.
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FIGURE 4.7. Different kinds of eld polarization.
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4.2. PATH LOSS IN FREE SPACE Let us consider a non isotropic source placed in free space as a transmitter antenna with PT watts and a directivity gain GT . At an arbitrary large distance r ( r ) l, where l cT c=f is a wavelength) from the source, the radiated power is uniformly distributed over a surface area of a sphere of radius r. If PR is the power at the receiving antenna, which is located at distance r from the transmitter antenna and has a directivity gain GR , then the path loss, in decibels, is given by PT L 10 log 10 log PR "  , #   4pr 2 1 GT GR L0 10 log l GT GR
Here L0 is the path loss for an isotropic point source (with GR GT 1) in free space and can be presented in decibels as L0 10 log     4pfr 2 4pfr 32:44 20 log r 20 log f 20 log c c 4:28
where the value 32.44 is obtained from     4p 103 m 106 1=s 40p 32:44 20 log 20 log 3 108 m=s 3 Notice that all the above formulas are related to the well-known Friis formula obtained in 2. In expression (4.28) the distance r is in kilometers (km), and frequency f is in megahertz (MHz). As the result, the path loss between the two directive antennas (receiver and transmitter) is given by LF 34:44 20 log d km 20 log f MHz 10 log GT 10 log GR It can be presented in a straight line form as LF L0 10 g log d where L0 34:44 20 log f 10 log GT 10 log GR and g 2. 4.3. RADIO PROPAGATION ABOVE FLAT TERRAIN The simplest case of radio wave propagation over a terrain is one where the ground surface can be assumed to be at and perfectly conductive. The assumption of at terrain is valid for radio links between subscribers up to 10 20 km [4 8]. The second condition of a perfectly conductive soil medium can be satis ed only for some special cases, because the combination of conductivity s and frequency o such as 4ps=o, that appears in total formula of permittivity e ero i4ps=o play important role for high frequencies (VHF/L-band, usually used for terrain communication channel design) and nite sub-soil conductivity, as well as for small grazing angles of incident waves [1 8]. To introduce the reader to the subject of re ection from the terrain, we start with the simplest case of a perfectly conductive at terrain. 4.3.1. Boundary Conditions at the Perfectly Conductive Surface For a perfectly conductive ground surface the total tangential electric eld vector is equal to zero, that is, Et 0. Consequently, from r E r ioH r the normal 4:30 4:29