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Chapter 3 Propagation over Irregular Terrain 3.1 INTRODUCTION Land mobile radio systems are used in a wide variety of scenarios. At one extreme, county police and other emergency services operate over fairly large areas using frequencies in the lower part of the VHF band. The service area may be large enough to require several transmitters, operating in a quasi-synchronous mode, and is likely to include rural, suburban and urban areas. At the other extreme, in major cities, individual cells within a 900 or 1800 MHz cellular radio telephone system can be very small in size, possibly less than 1 km in radius, and service has to be provided to both vehicle-mounted installations and to hand-portables which can be taken inside buildings. It is clear that predicting the coverage area of any base station transmitter is a complicated problem involving knowledge of the frequency of operation, the nature of the terrain, the extent of urbanisation, the heights of the antennas and several other factors. Moreover, since in general the mobile moves in or among buildings which are randomly sited on irregular terrain, it is unrealistic to pursue an exact, deterministic analysis unless highly accurate and up-to-date terrain and environmental databases are available. Satellite imaging and similar techniques are helping to create such databases and their availability makes it feasible to use prediction methods such as ray tracing (see later). For the present, however, in most cases an approach via statistical communication theory remains the most realistic and pro®table. In predicting signal strength we seek methods which, among other things, will enable us to make a statement about the percentage of locations within a given, fairly small, area where the signal strength will exceed a speci®ed level. In practice, mobile radio channels rank among the worst in terrestrial radio communications. The path loss often exceeds the free space or plane earth path loss by several tens of decibels; it is highly variable and it ¯uctuates randomly as the receiver moves over irregular terrain and/or among buildings. The channel is also corrupted by ambient noise generated by electrical equipment of various kinds; this noise is impulsive in nature and is often termed man-made noise. All these factors will be considered in the chapters that follow; for now we will concentrate on methods of estimating the mean or average signal strength in a given small area. Several methods exist, some having speci®c applicability over irregular terrain, The Mobile Radio Propagation Channel. Second Edition. J. D. Parsons Copyright & 2000 John Wiley & Sons Ltd Print ISBN 0-471-98857-X Online ISBN 0-470-84152-4 others in built-up areas, etc. None of the simple equations derived in Chapter 2 are suitable in unmodi®ed form for predicting average signal strength in the mobile radio context, although as we will see, both the free space and plane earth equations are used as an underlying basis for several models that are used. Before going any further, we will deal with some further theoretical and analytical techniques that underpin many prediction methods. 3.2 HUYGENS' PRINCIPLE Discussions of re¯ection and refraction are usually based on the assumption that the re¯ecting surfaces or refracting regions are large compared with the wavelength of the radiation. When a wavefront encounters an obstacle or discontinuity that is not large then Huygens' principle, which can be deduced from Maxwell's equations, is often useful in giving an insight into the problem and in providing a solution. In simple terms, the principle suggests that each point on a wavefront acts as the source of a secondary wavelet and that these wavelets combine to produce a new wavefront in the direction of propagation. Figure 3.1 shows a plane wavefront that has reached the position AA'. Spherical wavelets originate from every point on AA' to form a new wavefront BB', drawn tangential to all wavelets with equal radii. As an illustration, Figure 3.1 shows how wavelets originating from three representative points on AA' reach the wavefront BB'. To explain the observable eect, i.e. that the wave propagates only in the forward direction from AA' to BB', it must be concluded that the secondary wavelets originating from points along AA' do not have a uniform amplitude in all directions and if a represents the angle between the direction of interest and the normal to the wavefront, then the amplitude of the secondary wave in a given direction is proportional to (1  cos a). Thus, the amplitude in the direction of propagation is proportional to 1 cos 02 and in any other direction it will be less than 2. In particular, the amplitude in the backward direction is 1  cos p0. Consideration of wavelets originating from all points on AA' leads to an expression for the ®eld at Propagation over Irregular Terrain 33 Figure 3.1 Huygens' principle applied to propagation of plane waves. any point on BB' in the form of an integral, the solution of which shows that the ®eld at any point on BB' is exactly the same as the ®eld at the nearest point on AA',with its phase retarded by 2pd=l. The waves therefore appear to propagate along straight lines normal to the wavefront. 3.3 DIFFRACTION OVER TERRAIN OBSTACLES The analysis in Section 3.2 applies only if the wavefront extends to in®nity in both directions; in practice it applies if AA' is large compared to a wavelength. But suppose the wavefront encounters an obstacle so that this requirement is violated. It is clear from Figure 3.2 that beyond the obstacle (which is assumed to be impenetrable or perfectly absorbing) only a semi-in®nite wavefront CC' exists. Simple ray theory would suggest that no electromagnetic ®eld exists in the shadow region below the dotted line BC, but Huygens' principle states that wavelets originating from all points on BB', e.g. P, propagate into the shadow region and the ®eld at any point in this region will be the resultant of the interference of all these wavelets. The apparent bending of radio waves around the edge of an obstruction is known as diraction. 34 The Mobile Radio Propagation Channel Figure 3.2 Diraction at the edge of an obstacle. To introduce some concepts associated with diraction we consider a transmitter T and a receiver R in free space as in Figure 3.3. We also consider a plane normal to the line-of-sight path at a point between T and R. On this plane we construct concentric circles of arbitrary radius and it is apparent that any wave which has propagated from T to R via a point on any of these circles has traversed a longer path than TOR. In terms of the geometry of Figure 3.4 , the `excess' path length is given by D 9 h 2 2  d 1  d 2 d 1 d 2  3:1 assuming h ( d 1 , d 2 . The corresponding phase dierence is f  2pD l  2p l  h 2 2  d 1  d 2 d 1 d 2 3:2 This is often written in terms of a parameter v,as f  p 2 v 2 3:3 where v  h  2d 1  d 2  ld 1 d 2 s 3:4 and is known as the Fresnel±Kirchho diraction parameter. Propagation over Irregular Terrain 35 Figure 3.3 Family of circles de®ning the limits of the Fresnel zones at a given point on the radio propagation path. Figure 3.4 The geometry of knife-edge diraction. Alternatively, using the same approximation we can obtain f  pa 2 l  d 1 d 2 d 1  d 2  3:5 and v  a  2d 1 d 2 ld 1  d 2  s 3:6 There is a need to keep a region known as the ®rst Fresnel zone substantially free of obstructions, in order to obtain transmission under free space conditions (see Section 1.3.1). In practice this usually involves raising the antenna heights until the necessary clearance over terrain obstacles is obtained. However, if the terminals of a radio link path for which line-of-sight (LOS) clearance over obstacles exists, are low enough for the direct path to pass close to the surface of the Earth at some intermediate point, then there may well be a path loss considerably in excess of the free space loss, even though the LOS path is not actually blocked. Clearly we need a quantitative measure of the required clearance over any terrain obstruction and this may be obtained in terms of Fresnel zone ellipsoids drawn around the path terminals. 3.3.1 Fresnel-zone ellipsoids If we return to Figure 3.3 then it is clear that on the plane passing through the point O, we could construct a family of circles having the speci®c property that the total path length from T to R via each circle is nl=2 longer than TOR, where n is an integer. The innermost circle would represent the case n  1, so the excess path length is l=2. Other circles could be drawn for l,3l=2, etc. Clearly the radii of the individual circles depend on the location of the imaginary plane with respect to the path terminals. The radii are largest midway between the terminals and become smaller as the terminals are approached. The loci of the points for which the `excess' path length is an integer number of half-wavelengths de®ne a family of ellipsoids (Figure 3.5). The radius of any speci®c member of the family can be expressed in terms of n and the dimensions of Figure 3.4 as [1, Ch. 4]: h  r n   nld 1 d 2 d 1  d 2 s 3:7 36 The Mobile Radio Propagation Channel Figure 3.5 Family of ellipsoids de®ning the ®rst three Fresnel zones around the terminals of a radio path. and hence, v n   2n p This is an approximation which is valid provided d 1 , d 2 ) r n and is therefore realistic except in the immediate vicinity of the terminals. The volume enclosed by the ellipsoid de®ned by n  1 is known as the ®rst Fresnel zone. The volume between this ellipsoid and the ellipsoid de®ned by n  2 is the second Fresnel zone, etc. It is clear that contributions from successive Fresnel zones to the ®eld at the receiving point tend to be in phase opposition and therefore interfere destructively rather than constructively. If an obstructing screen were actually placed at a point between T and R and if the radius of the aperture were increased from the value that produces the ®rst Fresnel zone to the value that produces the second Fresnel zone, the third Fresnel zone, etc., then the ®eld at R would oscillate. The amplitude of the oscillation would gradually decrease since smaller amounts of energy propagate via the outer zones. 3.3.2 Diraction losses If an ideal, straight, perfectly absorbing screen is interposed between T and R in Figure 3.4 then when the top of the screen is well below the LOS path it will have little eect and the ®eld at R will be the `free space' value E 0 . The ®eld at R will begin to oscillate as the height is increased, hence blocking more of the Fresnel zones below the line-of-sight path. The amplitude of the oscillation increases until the obstructing edge is just in line with T and R, at which point the ®eld strength is exactly half the unobstructed value, i.e. the loss is 6 dB. As the height is increased above this value, the oscillation ceases and the ®eld strength decreases steadily. To express this in a quantitative way, we use classical diraction theory and we replace any obstruction along the path by an absorbing plane placed at the same position. The plane is normal to the direct path and extends to in®nity in all directions except vertically, where it stops at the height of the original obstruction. Knife-edge diraction is the term used to describe this situation, all ground re¯ections being ignored. The ®eld strength at the point R in Figure 3.4 is determined as the sum of all the secondary Huygens sources in the plane above the obstruction and can be expressed as [2, Ch. 16]: E E 0  1 j 2  I v exp  À j p 2 t 2  dt 3:8 This is known as the complex Fresnel integral and v is the value given by eqn. (3.4) for the height of the obstruction under consideration. We note that if the obstruction lies below the line-of-sight then h, and hence v, is negative. If the path is actually obstructed then h and v are positive, as in Figure 3.6. An interesting and relevant insight into the evaluation of eqn. (3.8) can be obtained in the following way. We can write  I v exp  À j p 2 t 2  dt   I v cos  p 2 t 2  dt Àj  I v sin  p 2 t 2  dt and Propagation over Irregular Terrain 37  I v cos  p 2 t 2  dt  1 2 À  v 0 cos  p 2 t 2  dt which is usually written as 1 2 À Cv. Similarly,  I v sin  p 2 t 2  dt  1 2 À Sv: The complex Fresnel integral (3.8) can therefore be expressed as E E 0  1 j 2 f 1 2 À Cv Àj 1 2 À Svg 3:9 Let us now consider the integral C vÀjSv  v 0 exp  À j p 2 t 2  dt 3:10 Plotting this integral in the complex plane with C as the abscissa and S as the ordinate results in Figure 3.7, a curve known as Cornu's spiral. In this curve, positive values of v appear in the ®rst quadrant and negative values in the third quadrant. The spiral has the following properties: . A vector drawn from the origin to any point on the curve represents the magnitude and phase of eqn. (3.10). . The length of arc along the curve, measured from the origin, is equal to v.As v 3Ithe curve winds an in®nite number of times around the points ( 1 2 , 1 2  or À 1 2 , À 1 2 . 38 The Mobile Radio Propagation Channel Figure 3.6 Knife-edge diraction: (a) h and v positive, (b) h and v negative. It is clear that [ 1 2 À Cvand [ 1 2 À Svrepresent the real and imaginary parts of a vector drawn from the point ( 1 2 , 1 2 to a point on the spiral. Thus the value of jEjcorresponding to any particular value of v,sayv 0 , is proportional to the length of the vector joining ( 1 2 , 1 2 to the point on the spiral corresponding to v 0 . Thus Cornu's spiralgives a visual indication of how the magnitude and phase of E varies as a function of the Fresnel parameter v. Figure 3.8 shows the diraction loss in decibels relative to the free space loss, as given by eqn. (3.9). In the shadow zone below the LOS path the loss increases smoothly; above the LOS path the loss oscillates about its free space value, the amplitude of oscillation decreasing as v becomes more negative. When there is grazing incidence over the obstacle there is a 6 dB loss, i.e. the ®eld strength is 0.5E 0 ; but Figure 3.8 shows that this loss can be avoided if v %À0:8, which corresponds to about 56% of the ®rst Fresnel zone being clear of obstructions. In practice, therefore, designers of point-to-point links try to make the heights of antenna masts such that the majority of the ®rst Fresnel zone is unobstructed. As an alternative to using Figure 3.8, nomographs of the form shown in Figure 3.9 exist in the literature [3]. They enable the diraction loss to be calculated to within about 2 dB. Alternatively, various approximations are available that enable the loss to be evaluated in a fairly simple way. Modi®ed expressions as given by Lee [4] are L vdB À20 log0:5 À0:62 vÀ0:8 < v < 0 À20 log0:5 expÀ0:95 v 0 < v < 1 À20 log0:4 Àf0:1184 À0:38 À0:1 v 2 g 1=2  1 < v < 2:4 À20 log0:225= v v > 2:4 8 > < > : 3:11 Propagation over Irregular Terrain 39 Figure 3.7 Plots of the Fresnel integral in terms of the diraction parameter v (Cornu's spiral). The approximation used for v > 2:4 arises from the fact that as v becomes large and positive then eqn. (3.8) can be written as     E E 0     3 2 1=2 2pv an asymptotic result which holds with an accuracy better than 1 dB for v > 1, but breaks down rapidly as v approaches zero. Ground re¯ections The previous analysis has ignored the possibility of ground re¯ections either side of the terrain obstacle. To cope with this situation (Figure 3.10), four paths have to be taken into account in computing the ®eld at the receiving point [5]. The four rays depicted in Figure 3.10 have travelled dierent distances and will therefore have dierent phases at the receiver. In addition the Fresnel parameter v is dierent in each case, so the ®eld at the receiver must be computed from E  E 0 X 4 k1 Lv k  expjf k 3:12 In any particular situation a ground re¯ection may exist only on the transmitter or receiver side of the obstacle, in which case only three rays exist. 40 The Mobile Radio Propagation Channel Figure 3.8 Diraction loss over a single knife-edge as a function of the parameter v. 3.4 DIFFRACTION OVER REAL OBSTACLES We have seen earlier that geometrical optics is incapable of predicting the ®eld in the shadow regions, indeed it produces substantial inaccuracies near the shadow boundaries. Huygens' principle explains why the ®eld in the shadow regions is non- zero, but the assumption that an obstacle can be represented by an ideal, straight, perfectly absorbing screen is in most cases a very rough approximation. Having said that, and despite the fact that the knife-edge approach ignores several Propagation over Irregular Terrain 41 Figure 3.9 Nomograph for calculating the diraction loss due to an isolated obstacle (after Bullington). [...]... dimensions which are large compared with the wavelength of transmission Neither hills nor buildings can be truly represented by a knife-edge (assumed in®nitely thin) and alternative approaches have been developed 3.4.1 The uniform theory of di raction The original geometric theory of di raction (GTD) was developed by Keller and his seminal paper on this subject [6] was published in 1962 By adding di racted... produced a widely used heuristic di raction coecient More rigorous work on wedges with ®nite conductivity had been undertaken earlier by Maliuzhinets [9] To illustrate the theory very brie¯y, we consider a two-dimensional diagram of a wedge with straight edges (Figure 3.11) It is conventional to label the faces of the wedge the o-face and the n-face We measure angles from the o-face The interior angle of... pessimistic 3.6 PATH LOSS PREDICTION MODELS The prediction of path loss is a very important step in planning a mobile radio system, and accurate prediction methods are needed to determine the parameters of a radio system which will provide ecient and reliable coverage of a speci®ed service area Earlier in this chapter we showed that in order to make predictions we need a proper understanding of the factors... prediction technique operates in a `point-to-point' mode However, if the terrain pro®le is not available, the report gives techniques for Figure 3.22 Geometry of a transhorizon radio path 58 The Mobile Radio Propagation Channel Table 3.1 Estimated values of Dh Type of terrain Dh Water or very smooth plains Plains Hills Mountains Rugged mountains 0±5 $ 30 80±150 150±300 300±700 estimating these path-related... horizon distances The angular distance for a transhorizon path is always positive and is given by Propagation over Irregular Terrain 59 y ˆ ye ‡ di 8495 …3:41† where di is the length of the transmission path in kilometres In computing di raction loss using this technique it is necessary to express the distances d1 and d2 to two ideal (knife-edge) obstacles in terms of the horizon distances The expressions... kilometres as: H d 1 ˆ dL ‡ 0:5…72 165 000= fc †1=3 and d2 ˆ d1 ‡ …72 165 000=fc †1=3 …3:43† The v-parameters appropriate to obstacles at distances d1 and d2 are then computed from vb,i ˆ 1:2915yebi ‰ fc dLb di À dL †= di À dLm †Š1=2 vm,i ˆ 1:2915yemi ‰ fc dLm di À dL †= di À dLb †Š1=2 …3:44† with i ˆ 1 and 2 The di raction losses A1 and A2 are then estimated using A1 …dB† ˆ A…vb,1 † ‡ A…vm,1 † A2 …dB†... the di raction angles used in the Deygout method and reasons as follows In Figure 3.18 the di raction angle used in calculating the Propagation over Irregular Terrain 51 Figure 3.18 The Deygout di raction construction loss due to 02 (the main edge) alone is larger than the angle through which a ray from 01 must actually be di racted in order to reach the top of 03 The di erence increases when the individual... Having said that, and despite the fact that the knife-edge approach ignores several 42 The Mobile Radio Propagation Channel Figure 3.10 Knife-edge di raction with ground re¯ections important e€ects such as the wave polarisation, local roughness e€ects and the electrical properties and lateral pro®le of the obstacle, it must be conceded that the losses predicted using this assumption are suciently close... fourth-power law relating path loss to range from the transmitter, and the lognormal variation in median path loss (or signal strength) over a small area 3.6.2 The JRC method A method that has been in widespread use for many years, particularly in the UK, is the terrain-based technique originally adopted by the Joint Radio Committee of the Frequency (MHz) Figure 3.20 The terrain factor for base-to-mobile... program: Antenna heights above local ground Surface refractivity (250 to 400 N-units) E€ective Earth radius Ground constants Climate In addition it is necessary to provide a number of path-speci®c factors: E€ective antenna heights Horizon distances of the antennas, dLb and dLm Horizon elevation angles, yeb and yem Angular distance for a transhorizon path, ye Terrain irregularity parameter, Dh The . terrain, The Mobile Radio Propagation Channel. Second Edition. J. D. Parsons Copyright & 2000 John Wiley & Sons Ltd Print ISBN 0-4 7 1-9 8857-X Online ISBN 0-4 7 0-8 415 2-4 others in built-up areas, etc The Mobile Radio Propagation Channel Figure 3.10 Knife-edge di raction with ground re¯ections. where  D represents the dyadic di raction coecient of the wedge, s H and s are the distances along. consider a two-dimensional diagram of a wedge with straight edges (Figure 3.11). It is conventional to label the faces of the wedge the o-face and the n-face. We measure angles from the o-face. The