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FIGURE 5.3. The transmitter and receiver locations near the building at the plane ground surface. Here (x0 ; y0 ; z0 ) and (x1 ; y1 ; z1 ) are the coordinates of the transmitting and receiving points, respectively.
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In the following section, we examine these effects in the case of a vertical dipole. 5.5.1. The Electric Field of the Vertical Dipole Let us consider a rectangular building illuminated by a vertical electric dipole placed at the point fx0 ; y0 ; z0 g above a plane ground surface. A coordinate system is placed at an arbitrary point {x, y, z}, and the point of observation is placed at point fx1 ; y1 ; z1 g (see Fig. 5.3). The dipole eld is represented by the electric Hertz s vector [7,14] i p expfikrg r 5:13
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where p is the moment of the electric dipole. Using the relations between electric and magnetic eld components and the Hertz s vector components for the eld of the vertical electric dipole, we get [7] x x0 y y0 eikr r r2 5:14
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x x0 2 z z0 2 eikr r2 r y y0 z z0 eikr Ez E0 r r2 z z0 eikr Hx H0 r r Hy 0 x x0 eikr : Hz H0 r r
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In a spherical coordinate system, the source and observation coordinates can be written, respectively, as: x0 r0 sin y0 sin j0 ; x1 r1 sin y1 sin j1 ; y0 r0 cos y0 ; y1 r1 cos y1 ; z0 r0 sin y0 cos j0 ; z1 r1 sin y1 cos j1 : 5:16
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We also introduce the coef cient of re ection R from the plane ground surface, the formulation of which can be found in 4. The coordinate system is shown in Figure 5.3. We will analyze the eld diffracted at the edge and roof of the building and the loss of the diffracted eld compared to the dipole eld in free space, as well as changes of eld polarization for both waves re ected from the walls and ground and those diffracted from the building edges. 5.5.2. Diffraction from the Edge of a Perfectly Conductive Building To de ne the eld diffracted from an ideal conductive edge of a building, we use Keller s method of GTD described in References [5,6,9,10]. The relationship between the eld diffracted from a wedge and the incident eld is [5,6] eikr Ed p Eo P DE r0
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where E0 P is an incident wave eld at point P x; y; z on the edge of a building and the diffraction matrix DE has the form:
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3 07 6 6 7 ! ! 6 7 7 cos b sin c c0 sin c c0 cos c 1 1 2eip=4 6 6 07 DE p 6 cos c0 7 A1 A2 cos c0 A1 A2 6pk sin b 6 7 ! ! 6 7 4 cos b cos c c0 cos c c0 5 sin c 1 1 0 cos c0 A1 A2 cos c0 A1 A2 0 2
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The main parameters in expressions (5.17) (5.18) are as follows: cos b y1 y0 = y1 y0 2 b a 2 1=2 ; cos c0 h y0 =a; cos c y h =b; 2 2 A1 1 2 sin c c0 ; A2 1 2 sin c c0 ; 3 3 y by0 ay1 = a b ; a l x0 d z0
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Here, 2l is the building length, and 2d is its width. Let us take a cross-section of a vertical wedge at the plane y y0 y1 0 and get the coordinates x and z to zero at the vertical wedge. We do this to determine the diffracted eld at an arbitrary point fx1 ; y1 ; z1 g as a function of the distance from the source fx0 ; y0 ; z0 g. The derivations for the polar eld components Edy and Edj are Edy Edx sin j Edz cos j; Edj Edx cos j Edz sin j 5:20