Other Diffracting Obstacles: Geometrical Theory of Diffraction

In many situations, diffraction over obstructions, such as hills and buildings, may be treated as if those obstructions were absorbing knife-edges. There are other cases, however, when it is necessary to account for the structure of the obstruction in a more specific way, including its shape and the materials of which it is built. One approach is to extend geometrical optics to

rn

54 Antennas and Propagation for Wireless Communication Systems

include diffraction, producing a formulation similar to Section 3.4. This results in the geometrical theory of diffraction (GTD). GTD was first devised by Joseph Keller in the 1950s [Keller, 62]. In the past, GTD has been applied mainly to the analysis of small shapes, such as antennas, or for calculating the radar cross-sections of complex objects. More recently, it has been successfully applied to the modelling of terrain [Luebbers, 84] and of buildings.

The central idea of GTD is that an extended version of Fermat’s principle may be used to predict the existence of diffracted rays, which may then be treated with the ease of any

r1

Transmitter Receiver

n = 1

0.6 r1

‘Forbidden’ region First Fresnel zone

Figure 3.18: 0.6 times first Fresnel zone clearance defines significant obstructions

Figure 3.19: Single knife-edge attenuation: the transmitter is at (10,5), the top of the edge is at (0,10), the colour indicates the attenuation in decibels and the black line represents the shadow boundary

Propagation Mechanisms 55

other ray in geometrical optics. Figure 3.20 shows a ray obliquely incident upon the edge of an obstacle at an angle0to the edge. Fermat’s principle for edge diffraction predicts that a cone of diffracted rays will be produced, where the cone has semi-angle0. This contrasts with reflected and transmitted rays, where only a single ray is produced at each interaction. In the simpler case of normal incidence,0¼=2 and the cone reduces to a disc.

Once the diffracting point is determined, the diffracted field is given by

Ed ¼ DEiAd ð3:34Þ

where

Ed¼ E_djj Ed?



; D ¼ D_jj 0 0 D?



and Ei¼ E_ijj Ei?



ð3:35Þ

Here E_ijj and Ei? are measured parallel and perpendicular to the plane of incidence, which contains the incident ray and the diffracting edge, while E_djjand E_d?are measured parallel and perpendicular to the plane of diffraction, which contains the diffracting edge and the diffracted ray. By analogy with Eq. (3.24), Ad is a spreading factor which depends on the distances to the source and field points and on the curvature of the diffracting edge.

In place of the Fresnel reflection and transmission coefficients used earlier, we now have coefficients D_jjand D_?. These are the diffraction coefficients, which describe the character-istics of the diffracting obstacle at the point of diffraction, assuming the frequency is high enough that the diffraction characteristics can be determined with regard to local character-istics only, rather than by those of the entire obstacle. Terms such as Eq. (3.34) can now be

0

Cone of diffracted rays

Incident ray

Diffraction point

0

Figure 3.20: Generation of edge-diffracted rays from a wedge according to GTD

56 Antennas and Propagation for Wireless Communication Systems

added to the geometrical optics field of Eq. (3.24), yielding a field which has the proper behaviour within the shadow region. GTD also provides a simple explanation for the fluctuations in the single knife-edge diffraction (Figure 3.15) for negative v: it arises from constructive and destructive interferences between the direct ray and the diffracted ray.

The diffraction coefficients are determined from one of a number of canonical problems.

These are diffraction problems for simple scattering objects, such as a half-plane [Volakis, 86], a wedge or a cone, which have been solved using exact methods for solving Maxwell’s equations for plane waves incident on these objects. The resulting solutions are reduced, via asymptotic assumptions, to terms which correspond to a ray description of the field. As with geometrical optics, these assumptions are only valid if the obstacle dimensions are large compared to a wavelength and if the spatial variation of the scattered field is not too rapid.

Keller’s original formulation of GTD, though simple, had the disadvantage that it did not predict the field correctly for field points in a Fresnel-zone-like region close to the shadow boundary – the transition region. GTD was therefore extended to the uniform GTD (UTD) which applies at all points in space [Kouyoumjian, 74].

Consider the wedge diffraction situation as shown in Figure 3.21. The case shown is perpendicular polarisation (?). For parallel polarisation (||), replace Eiand E_din the figure with H_iand H_d, respectively, and H_i and H_d in the figure withEiandEd, respectively.

Assuming the wedge to be a perfect conductor, the UTD diffraction coefficient for the case when the incident wave is normal to the diffracting edge is then as follows [Kouyoumjian, 74].

Note that the UTD diffraction coefficient is polarisation dependent, in contrast to the simpler theory presented in Section 3.5.2.

D_?;jj¼ 1

Figure 3.21: Geometry for wedge diffraction coefficient

Propagation Mechanisms 57

The function FðxÞ is an alternative definition of the Fresnel integral, FðxÞ ¼ 2j ffiffiffi

px e^jx

ð

1

t¼^pffiffi_xe^jt2dt ð3:38Þ

where aare the integers which most nearly satisfy

2na ¼  ð3:39Þ

The spreading factor Ad in Eq. (3.34) is given by A_d ¼ 1= ffiffi

pr

for plane cylindrical wave incidence ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

r_i=rðr þ riÞ

p for spherical wave incidence

ð3:40Þ

Keller’s GTD diffraction coefficient takes exactly the same form, but with the Fresnel integrals replaced by 1, since this is a reasonable approximation outside of the transition regions. The total field at any point in space is thus given by

Et¼ EiðA0þ UrRArþ DAdÞ ð3:41Þ where A0, Arand Adare appropriate spreading factors and Ur¼ 1 when a reflection exists and U_r¼ 0 otherwise. The total field around a 0 wedge (i.e. a conducting half-plane) is illustrated in Figure 3.22. The influence of the three regions marked in Figure 3.21 is clearly

0 50 100 150 200 250 300 350

−20

−15

−10

−5 0 5 10

Angle of Incidence, φ₀ , [deg]

Field [dB]

Region (1) Region (2) Region (3)

Figure 3.22: UTD solution for total field around a conducting half-plane. Here0¼ 90 ; r ¼ 10l and the polarisation is parallel

58 Antennas and Propagation for Wireless Communication Systems

visible. In region 1, the visible region, the field is the sum of the direct ray and the diffracted ray, plus a ray reflected from the surface. Since the wedge is perfectly conducting, the reflected wave has the same amplitude as the incident ray and complete cancellation of the two fields occurs at intervals, since the diffracted ray has negligibly small amplitude by comparison. In region 2, there is no reflection point on the wedge which can obey Snell’s law, so no reflection exists, and the diffraction is still very small, so the field is nearly at its free space value. In region 3, the shadow region, only the diffracted ray is present, and it diminishes in amplitude in a similar way to the simple knife-edge approximation (compared with Figure 3.15).

UTD will be applied to practical propagation problems in Chapters 12 and 13.

3.6 CONCLUSION

All of the major propagation mechanisms have been introduced in this chapter, namely, reflection, refraction or transmission, scattering and diffraction. These complement the basic effects of attenuation and phase velocity which were introduced in Chapter 2 and provide a basis for practical prediction of wireless communication system performance in the rest of this book. Combinations of these mechanisms will be seen later to account for all of the observed effects in the wireless channel.

REFERENCES

[Balanis, 89] C. A. Balanis, Advanced engineering electromagnetics, John Wiley & Sons, Inc., New York, ISBN 0-471-62194-3, 1989.

[Beckmann, 63] P. Beckmann and A. Spizzichino, The scattering of electromagnetic waves from rough surfaces, Macmillan, New York, ISBN 0-63010-108, 1963.

[Ca´tedra, 99] M. F. Ca´tedra and J. Pe´rez-Arriaga, Cell planning for wireless communications, Artech House, ISBN 0-89006-601-9, 1999.

[ITU, 527] International Telecommunication Union, ITU-R Recommendation 527-3: Elec-trical characteristics of the surface of the earth, Geneva, 1992.

[James, 80] G. L. James, geometrical theory of diffraction for electromagnetic waves, Peter Peregrinus, London, ISBN 0-906048-34-6, 1980.

[Keller, 62] J. B. Keller, Geometrical theory of diffraction, Journal of the Optical Society of America, 52 (2), 116–130, 1962.

[Kouyoumjian, 74] R. G. Kouyoumjian and P. H. Pathak, A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface, Proceedings of IEEE, 62, 1448–1461, 1974.

[Luebbers, 84] R. J. Luebbers, Finite conductivity uniform GTD versus knife edge diffraction in prediction of propagation path loss, IEEE Transactions on Antennas and Propagation, 32 (1), 70–76, 1984.

[Volakis, 86] J. L. Volakis, A uniform geometrical theory of diffraction for an imperfectly conducting half-plane, IEEE Transactions on Antennas and Propagation, 34 (2), 172–180, 1986.

PROBLEMS

3.1 Prove Eqs. (3.13) and (3.15).

Propagation Mechanisms 59

3.2 Repeat Example 3.1 for the case of perpendicular polarisation and for the case where the polarisation is at 45 to the scattering plane.

3.3 Prove the expression for the Brewster angle (3.19).

3.4 When the angle of incidence of a plane wave onto a plane boundary between media is equal to the Brewster angle, all of the energy in the incident wave becomes transmitted into the second medium. At what incidence angle (the critical angle) does all of the energy in the incident wave become reflected?

3.5 Prove expressions (3.21) and (3.22) for the Rayleigh criterion.

3.6 Polaroid sunglasses reduce glare from road surfaces by permitting only one polarisation to be transmitted. Using the Fresnel reflection coefficients, explain whether vertical or horizontal polarisation should be transmitted.

3.7 A plane wave at 900 MHz is incident from free space onto a material with relative dielectric constant of 4. What are the phase velocity and wavelength of the refracted wave?

3.8 A transmitter and a receiver separated by 10 km operate at 400 MHz and are at the same height above the Earth. Relative to the transmitter, how much lower must an absorbing diffracting obstacle situated at the centre of the path be for negligible diffraction loss?

Calculate the diffraction loss produced when the obstacle is increased to 10 m above the transmitter height.

3.9 Use Keller’s GTD to repeat Problem 3.7, for both horizontal and vertical polarisations, assuming a conducting knife-edge in place of the absorbing knife-edge.

3.10 A microwave link is to be deployed in an urban area at 17 GHz. The transmitter antenna is to be located on a rooftop at 15 m. The receiver antenna is to be installed at 5 m above ground level. Determine the maximum height of a building at the centre of the path if transmitter and receiver antennas are separated by 5 km.

3.11 Determine the surface height variation required to consider a surface rough if a 900 MHz wave is incident on the surface at 30 to the normal.

3.12 Repeat problem 3.10 for the case when the radiowave is at a frequency of 60 GHz.

Explain your answer.

60 Antennas and Propagation for Wireless Communication Systems