The Mobile Radio Propagation Channel P2

16 The Mobile Radio Propagation Channel So, given PT and G it is possible to calculate the power density at any point in the far ®eld that lies in the direction of maximum radiation.A knowledge of the radiation pattern is necessary to determine the power density at other points. The power gain is unity for an isotropic antenna, i.e. one which radiates uniformly in all directions, and an alternative de®nition of power gain is therefore the ratio of power density, from the speci®ed antenna, at a given distance in the direction of maximum radiation, to the power density at the same point, from an isotropic antenna which radiates the same power.As an example, the power gain of a half-wave dipole is 1.64 2.15dB) in a direction normal to the dipole and is the same whether the antenna is used for transmission or reception. There is a concept known as e€ective area which is useful when dealing with antennas in the receiving mode.If an antenna is irradiated by an electromagnetic wave, the received power available at its terminals is the power per unit area carried by the wave6the e€ective area, i.e. PWA.It can be shown [1, Ch.11] that the e€ective area of an antenna and its power gain are related by A l2G 2:1 2.2 PROPAGATION IN FREE SPACE Radio propagation is a subject where deterministic analysis can only be applied in a few rather simple cases.The extent to which these cases represent practical conditions is a matter for individual interpretation, but they do give an insight into the basic propagation mechanisms and establish bounds. If a transmitting antenna is located in free space, i.e. remote from the Earth or any obstructions, then if it has a gain GT in the direction to a receiving antenna, the power density i.e. power per unit area) at a distance range) d in the chosen direction is PTGT 4pd2 2:2 The available power at the receiving antenna, which has an e€ective area A is therefore PR 4pdT A PTGT l2GR 4pd2 4p where GR is the gain of the receiving antenna. Thus, we obtain 2 PT GTGR 4pd 2:3 Fundamentals of VHF and UHF Propagation 17 which is a fundamental relationship known as the free space or Friis equation [2].The well-known relationship between wavelength l, frequency f and velocity of propagation c cfl) can be used to write this equation in the alternative form 2 PT GTGR 4pfd 2:4 The propagation loss or path loss) is conveniently expressed as a positive quantity and from eqn.24.) we can write LF dB 10log10PT=PR 10log10 GT 10log10 GR 20log10 f 20log10 d k 2:5 where k 20log10 3 108 147:56 It is often useful to compare path loss with the basic path loss LB between isotropic antennas, which is LB dB 32:44 20log10 fMHz 20log10 dkm 2:6 If the receiving antenna is connected to a matched receiver, then the available signal power at the receiver input is PR.It is well known that the available noise power is kTB, so the input signal-to-noise ratio is PR PTGTGR c 2 i kTB kTB 4pfd If the noise ®gure of the matched receiver is F, then the output signal-to-noise ratio is given by SNRo SNRi=F or, more usefully, SNRodB SNRidB FdB Equation 2.4) shows that free space propagation obeys an inverse square law with range d, so the received power falls by 6dB when the range is doubled or reduces by 20dB per decade).Similarly, the path loss increases with the square of the transmission frequency, so losses also increase by 6dB if the frequency is doubled. High-gain antennas can be used to make up for this loss, and fortunately they are relatively easily designed at frequencies in and above the VHF band.This provides a solution for ®xed point-to-point) links, but not for VHF and UHF mobile links where omnidirectional coverage is required. Sometimes it is convenient to write an expression for the electric ®eld strength at a known distance from a transmitting antenna rather than the power density.This can be done by noting that the relationship between ®eld strength and power density is 18 The Mobile Radio Propagation Channel 2 W Z where Z is the characteristic wave impedance of free space.Its value is 120 p 377O) and so eqn.22.) can be written E2 PTGT 120p 4pd2 giving  E dT T 2:7 Finally, we note that the maximum useful power that can be delivered to the terminals of a matched receiver is E2A E2 l2GR El2 GR Z 120p 4p 2p 120 2:8 2.3 PROPAGATION OVER A REFLECTING SURFACE The free space propagation equation applies only under very restricted conditions; in practical situations there are almost always obstructions in or near the propagation path or surfaces from which the radio waves can be re¯ected.A very simple case, but one of practical interest, is the propagation between two elevated antennas within line-of-sight of each other, above the surface of the Earth.We will consider two cases, ®rstly propagation over a spherical re¯ecting surface and secondly when the distance between the antennas is small enough for us to neglect curvature and assume the re¯ecting surface to be ¯at.In these cases, illustrated in Figures 21. and 2.4 the received signal is a combination of direct and ground-re¯ected waves. To determine the resultant, we need to know the re¯ection coecient. 2.3.1 The re¯ection coecient of the Earth The amplitude and phase of the ground-re¯ected wave depends on the re¯ection coecient of the Earth at the point of re¯ection and di€ers for horizontal and vertical polarisation.In practice the Earth is neither a perfect conductor nor a perfect dielectric, so the re¯ection coecient depends on the ground constants, in particular the dielectric constant e and the conductivity s. For a horizontally polarised wave incident on the surface of the Earth assumed to be perfectly smooth), the re¯ection coecient is given by [1, Ch.16]: p rh sinc p where o is the angular frequency of the transmission and e0 is the dielectric constant of free space.Writing er as the relative dielectric constant of the Earth yields Fundamentals of VHF and UHF Propagation 19 Figure 2.1 Two mutually visible antennas located above a smooth, spherical Earth of e€ective radius re. p rh sinc p 2:9 where s 18 109s oe0 f For vertical polarisation the corresponding expression is p rv r jxsinc p 2:10 The re¯ection coecients rh and rv are complex, so the re¯ected wave will di€er from the incident wave in both magnitude and phase.Examination of eqns 29. ) and 2.10) reveals some quite interesting di€erences. For horizontal polarisation the relative phase of the incident and re¯ected waves is nearly 1808 for all angles of incidence.For very small values of c near-grazing incidence), eqn.29.) shows that the re¯ected wave is equal in magnitude and 1808 out of phase with the incident wave forallfrequenciesandallgroundconductivities.Inotherwords,forgrazingincidence rh rh ejy 1ejp 1 2:11 As the angle of incidence is increased then rh and y change, but only by relatively small amounts.The change is greatest at higher frequencies and when the ground conductivity is poor. 20 The Mobile Radio Propagation Channel For vertical polarisation the results are quite di€erent.At grazing incidence there is no di€erence between horizontal and vertical polarisation and eqn.21.1) still applies.As c is increased, however, substantial di€erences appear.The magnitude and relative phase of the re¯ected wave decrease rapidly as c increases, and at an angle known as the pseudo-Brewster angle the magnitude becomes a minimum and the phase reaches 7908.At values of c greater than the Brewster angle, rv increases again and the phase tends towards zero.The very sharp changes that occur in these circumstances are illustrated by Figure 2.2, which shows the values of rv and y as functions of the angle of incidence c.The pseudo-Brewster angle is about 158 at frequencies of interest for mobile communications x er), although at lower frequencies and higher conductivities it becomes smaller, approaching zero if x er. Table 2.1 shows typical values for the ground constants that a€ect the value of r. The conductivity of ¯at, good ground is much higher than the conductivity of poorer ground found in mountainous areas, whereas the dielectric constant, typically 15, can be as low as 4 or as high as 30.Over lakes or seas the re¯ection properties are quite di€erent because of the high values of s and er.Equation 21.1) applies for horizontal polarisation, particularly over sea water, but r may be signi®cantly di€erent from 71 for vertical polarisation. Figure 2.2 Magnitude and phase of the plane wave re¯ection coecient for vertical polarisation.Curves drawn for s 12 103, e 15.Approximate results for other frequencies and conductivities can be obtained by calculating the value of x as 18 103s=fMHz. ... - tailieumienphi.vn