Abstract
Estimation of the scattering of electromagnetic waves by buildings and other obstacles is very important for wireless communications. In our previous work, we have examined the diffracted fields by two horizontal edges which make an arbitrary angle. In this paper, we compare the diffracted fields between parallel edges and edges with arbitrary angle. The numerical calculation results show that when the angle between two edges increases, relatively strong orthogonal polarized wave components appear, but the principal components are almost the same as the incident wave.
1. Introduction
Electromagnetic waves can be scattered by edges of the buildings in urban areas or edges of the mountain (ridges) for long communication. Thus, the communication quality deteriorates. Therefore, it is important to estimate propagation path from the viewpoint of the effective utilization of the radio wave resources. However, the prediction of the propagation path is not easy.
Until now, there are many research works dealing with estimation of the radio wave propagation path in the urban areas [1, 2]. These papers mainly analyzed and examined the propagation path of radio waves when in the urban area there are many buildings. For the case when the waves exceed building roof, Zhang analyzed propagating radio wave over a large number of buildings [3].
In our previous works, we have examined the diffracted fields by two horizontal edges which make an arbitrary angle [4, 5]. But it is almost impossible to find diffraction points on these edges analytically [6], so we used a method which can find the diffraction points by repetition process [5]. However, it should be noted that edges are not always horizontal. Thus, it is necessary to examine the diffracted field by edges which change in both horizontal and vertical directions [7]. We also discussed the fields diffracted by wedges and showed that wedge shapes, materials, and incident polarization do not have significant effect on diffracted fields except near the wedge surfaces [8].
In this paper, first we survey the double diffraction by two edges which make an arbitrary angle. Then, we refer to coordinate transformation between a total coordinate system and an edge-fixed one and explain the method for finding the diffraction points on two edges with an arbitrary angle which change in both horizontal and vertical directions. For the analysis of diffracted fields, we used GTD (geometrical theory of diffraction) [9, 10]. We make the analysis by including the slope diffraction considering Holm’s method [11, 12]. The numerical calculations are carried out on perfect conductor. The results show that when the angle between two edges increases, relatively strong orthogonal polarized wave components appear, but the principal components are the same as the incident wave.
This paper is organized as follows. In Section 2, we present diffraction by two edges containing coordinate transformation and the method of finding the diffraction points and introduce the diffraction by two edges. In Section 3, we present the numerical examples. Finally, some conclusions are given in Section 4.
2. Diffraction by Two Edges
2.1. Aspect of Double Diffraction
In Figure 1(d), the aspect of double diffraction by edges 1 and 2 is shown. We express edge () by , which makes an arbitrary angle in horizontal and vertical planes. The wave from a source is diffracted twice, at point on and at point on , and it reaches an observation point . We use three coordinate systems , and , and the origins of and are, respectively, at and of coordinate system . The edges and agree with -axsis and -axsis of coordinate systems and . It should be noted that the subscript in means the coordinate system , but the subscript 0 of is omitted. Figure 1(a) shows the front view of two wages. Figures 1(b) and 1(c) show the top view and side view of Figure 1(d). The wedges are supposed to rotated by angle () counterclockwise in a horizontal plane as shown in Figure 1(b) and inclined by angle in a vertical plane in Figure 1(c) (in Figure 1, and are drawn as and ).
(a) Front view of two wedges
(b) Top view of two edges and
(c) Side view of two edges and
(d) Aspect of double diffraction by two wedges
2.2. Coordinate Transformation
It is difficult to analyze the diffracted field by edges with an arbitrary angle. But the analysis becomes easier by using coordinate transformation as described below.
Let us consider Figure 1 to explain the coordinate transformation. The origin of coordinate system () is put at of coordinate system , and is supposed to be rotated by counterclockwise around -axis (-axis) as shown in Figure 1(b). Furthermore, the -axis is inclined by around -axis (Figure 1(c)). A point on is obtained from a point on by the following equation: On the other hand, a point on is obtained from a point on as follows: In addition, the relationship between a vector on and a vector on is shown as follows:
2.3. Diffracted Field by Edge
Firstly, we would consider a diffraction by an edge (, ) of a wedge . The aspect of diffraction at on an edge on the coordinate system is shown in Figure 2. The wedge which constitutes has the external angle and it consists of 0-plane and -plane which are in the angle of , from the axis. The source point , the observation point , and a unit vector of the current at are expressed as , , and of coordinate system . The electromagnetic wave emitted from the source is diffracted at point on and reaches the observation point . According to GTD, as the angle between the incident path and ( axis) is equal to the angle between the path after the diffraction and [9], the point is easily obtained as follows: where and are the distances from the to the points and , respectively. Thus, the distances, the angles, the unit vectors and , and so on in Figure 2 can be calculated.
Therefore, the single diffracted field at point on the is obtained by the following equations [10–12]: where is a constant and is the wave number. The and are the distances between and and between and , respectively, and . The and are unit vectors of and the current at the source . The is the dyadic diffraction coefficient of the diffracted wave from to until it reaches . The diffracted field in (5) described by is easily transformed to on using (3).
2.4. Diffraction Points and on Two Edges
When edges make arbitrary angle in a plane or they are parallel in different heights, the diffraction points can be found easily [5]. But, when they make arbitrary angle in the space, it is impossible to find the diffraction points analytically. So, we use the following method to find the diffraction points [7] as shown in Figure 3.(1)Let us assume , which is the diffraction point on as given in preceding subsection, to be a first diffraction point (subscript 1 means diffraction point on and superscript (1) means 1st step for finding the last diffraction point).(2)The temporary diffraction point on the edge is given as follows (see Figure 3(a)).(a)A plane can be defined containing the point and .(b) is rotated around with a constant distance of as shown in Figure 2; then reaches a point on the plane .(c)Then using the distance from to () and the distance from to (), the first temporary diffraction point on is given by (3)The 2nd temporary diffraction point on the edge is given as follows.(a)A plane can be constructed using the point and .(b) is rotated around with the length shown in Figure 3(a), and then reaches a point on the plane as shown in Figure 3(b).(c)Using and , the 2nd temporary diffraction point on is given by (4)After that, we repeat the processes (2) and (3) until becomes small enough. Then, we can find the last diffraction points .
(a) Plane
(b) Plane
2.5. Double Diffracted Fields
The double diffracted field by the point on the edge and the point on the edge is calculated by the following equation considering the slope diffraction [10–12]: where and is the direction angle at the point toward the point measured from the 0-plane of the wedge . The first term in the parenthesis of (8) is the first-order diffraction and the second term is the second-order diffraction (: the so-called slope diffraction). When we execute (8), the dyadic diffraction coefficients and are calculated on and , respectively. Then, they are transformed to by (3), and the double diffracted fields by and are obtained by performing scalar product in (8).
2.6. Total Field
The direct field from a source to the observation point is obtained as follows: where is the distance between and , and the and are unit vectors of and the current at for the coordinate system .
The single diffracted field by an edge and double diffracted field by and can be obtained by (5) and (8). Considering direct field from a source and the field reflected by surfaces of the wedges, the total electric field at an observation point is calculated as the sum of these fields:
3. Numerical Examples
In this section, we present the numerical calculation. The calculations were carried out for the case where the origin of coordinate system is located at of coordinate system and for at , where and are the heights of each origin for the coordinate system as shown in Figure 1. The angles of edges with -axes are , , , and as shown in Figures 1(b) and 1(c), but we do not consider the field diffracted by an edge formed at an intersection of two wedges and the field reflected by the wedge surfaces and the ground. The source is at , the observation point is at , and in (5), (8), and (9).
When the difference or becomes large, two edges cross each other near to -axis or -axis as shown in Figure 1, and the diffraction points and/or go out the calculable range. So, we firstly checked the limitation of , and values. We found that we can nearly calculate diffracted fields when or .
In Table 1 the aspect of converging the locations of the diffracted points on two edges and is shown. Let the heights of the source and the edges be m, m, and m and let the angles and be and , respectively. Table 1 shows the effect of converging locations of and the diffraction angles at m and 300 m. In this table, is the number of repetitions, , and . According to GTD, the angle between the incident ray and is equal to the angle between and the diffracted ray as shown in Figure 2. Table 1(a) is for horizontal edges () and Table 1(b) is for and . The number of repetitions has a tendency to increase when , , and/or become large. Although the calculations were carried out for many cases of various angles , and various positions of the # origins, their results show that the number of repetitions is relatively small.
In [8], we mentioned that the diffracted fields by wedges are not affected too much by their shapes, material, and incident polarization except near the surfaces of wedges. So, we show the following calculation for the perfectly conducting wedges whose inner angles are zero (). Figures 4–6 show the change of the total electric field and its incident and diffracted components with a distance from a surface of the wedge 2; when the wave is incident to the perfectly conducting wedges whose inner angles are , the heights of source and two edges are m, m, and m.
(a)
(b) , ,
(c) , ,
(a)
(b) , , and
(c) , , and
Figure 4 shows the frequency characteristics of the fields when the horizontal polarized wave () is incident and . In this figure, the fields are the total fields (), single diffracted fields () by the , double diffracted components (), and the slope diffracted components (), where the subscript means component. is the direct field from to without wedges. At m the direct field appears, so we called the point m (shadow boundary for an incident wave) and the region near to m is called a transition region. When the frequency is doubled, outside the transition region, , , and are deceased at 3 dB, 6 dB, and 12 dB, respectively [13]. For example, at m is dB, dB, and dB at 800 MHz, 1600 MHz, and 3200 MHz, and and are attenuated at 6 dB and 12 dB when frequency is doubled.
Figure 5 shows the fields when the horizontal polarized wave () is incident on the wedges. Figure 5(a) is the case when two edges and are parallel (). Only horizontal wave is diffracted and the total field increases according to distance because in this case the direct field is added in the range m.
Figure 5(b) is the case when , , and . As two edges are rotated in horizontal plane, the orthogonal components (, ) are generated, and when the angle between two edges increases, relatively strong orthogonal polarized wave components appear to the incident wave. But the horizontal components (, , , and ) change scarcely from Figure 5(a).
Figure 5(c) shows the case that two edges are not rotated in horizontal plane but in vertical plane (, , and ). Comparing Figure 5(c) with Figure 5(b), the orthogonal (, ) components are different from each other, but the principal components (, , and ) are almost the same as in Figures 5(a) and 5(b).
Figure 6 shows fields for vertically polarized incidence (). There are fields diffracted in and components. So, the total field is composed by and components. In Figure 6(a) the case when the edges and are parallel () is shown. Comparing Figure 6(a) with Figure 5(a), in Figure 6(a), the components of the diffracted fields , , and at are not zero because of the boundary condition for conductor. Then, the curve of starts with finite value at , but in Figure 5(a) the is zero. But both curves in Figures 5(a) and 6(a) almost agree in the region where is large. Figure 6(b) is the fields when , , and . The orthogonal components , , and are generated, but the principal components (, , , and ) change a little bit. Figure 6(c) shows the case for , , and . The component of single diffracted wave is zero, and the double diffracted components (, , etc.) are different from those of Figure 6(b), but the principal components are almost the same as in Figures 6(a) and 6(b).
4. Conclusions
In this paper, we considered the diffracted fields by two edges, parallel edges and edges with arbitrary angle. We used three coordinate systems , , and and considered coordinate transformation. Next, we explained the method to find the diffraction points on two edges. We analyzed the diffracted fields by two edges using the GTD. The numerical calculations were carried out for two wedges of perfect conductor for horizontal and vertical polarized incidence.
From the numerical results, we conclude as follows.(i)The diffraction points on two edges with arbitrary angle in horizontal and vertical planes can be confirmed by few repetitions of our method.(ii)The number of repetitions is increased when the angles and/or distance to the observation point are increased.(iii)When the frequency is doubled, outside the transition region, the single-, double-, and slope-diffracted fields are deceased 3 dB, 6 dB, and 12 dB, respectively.(iv)When the wave is incident perpendicularly to parallel edges, the diffracted fields have the same component as incident polarization, but the orthogonal component wave to incident wave is generated by edges with arbitrary angle.(v)When the angle between two edges increases horizontally and/or vertically, relatively strong orthogonal polarized wave components appear, but the principal components are the same as the incident wave.
In the future work, we would like to confirm numerical results by experiments.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.