A new formulation expressed in terms of Uniform Theory of Diffraction (UTD) coefficients for the prediction of the multiple diffraction caused by a series of buildings modeled as wedges, considering spherical-wave incidence, is presented. The solution, which has a certain heuristic nature, is validated with numerical results from technical literature and the particular cases of diffraction by buildings modeled as absorbing knife edges, as well as the one in which the mentioned buildings are replaced by flat-roofed parallel rows of blocks (building rows in cross sections considered to be rectangular in shape) are also analyzed. The computing time is reduced over existing formulations, especially when the number of buildings is large, and the results can be applied in the development of theoretical models, in order to predict a more realistic path loss in urban environments when multiple-building diffraction has to be considered.
If you can't find a tool you're looking for, please click the link at the top of the page to "Go to old article view". Alternatively, view our Knowledge Base articles for additional help. Your feedback is important to us, so please let us know if you have comments or ideas for improvement.
 The analysis of the multiple forward diffraction of radio waves past an array of buildings has been widely carried out in order to predict the propagation of UHF signals in urban environments for cellular mobile radio and other personal communication networks, achieving a solid agreement with measurements [Maciel et al., 1993; Erricolo et al., 2002]. This study has been realized for both the vertical plane [Bertoni, 2000; COST 231, 1999] and the horizontal plane [Zhang, 2000] and, in order to predict the above-mentioned multiple-building diffraction, many formulations have been proposed assuming a plane-wave incidence either based on Physical Optics (PO) [Vogler, 1982; Walfisch and Bertoni, 1988; Saunders and Bonar, 1991] or the Uniform Theory of Diffraction (UTD) [Juan-Llácer and Cardona, 1997; Neve and Rowe, 1994]. However, for microcellular mobile radio systems, in which the transmitting antenna is located at a certain distance from the array of buildings, a cylindrical or spherical-wave incidence assumption would be more appropriate for obtaining precise multiple diffraction loss predictions. In this sense, Xia and Bertoni proposed a PO-based solution, in which the field impinging over an array of successive absorbing knife edges of equal height which are modeling the buildings is represented by a multidimensional Fresnel integral expanded into a series of Boerma's functions [Xia and Bertoni, 1992]. Andersen gave a UTD solution for this multiple knife-edge diffraction, which includes slope diffraction, in order to solve the invalidity of a ray description when buildings are placed in the transition region near the shadow boundary [Andersen, 1994]. This solution can be applied for the analysis of a multiple diffraction caused by buildings of different heights; however, if a great number of the latter are considered, the computation time significantly increases. Zhang achieved an attenuation function for the prediction of the over-rooftop multiple forward diffraction, also replacing buildings by equal-height knife edges and defining a hybrid function which takes advantage of both UTD and PO [Zhang et al., 1999].
 In the case of multiple diffraction caused by buildings modeled as wedges, Holm proposed a UTD-based formulation, deriving an expansion for higher order diffracted fields and removing some of the shortcomings of the original set from the UTD when the incident field is not ray-optical [Holm, 1996]. Tzaras and Saunders described a heuristic UTD approach for multiple-wedge diffraction modeling which incorporates slope diffraction terms, obtaining a balanced efficiency in terms of computation time and accuracy [Tzaras and Saunders, 2001].
 For the analysis of the multiple diffraction caused by buildings replaced by plateaus of rectangular cross-sections, Luebbers proposed a solution which, presenting a heuristic wedge-diffraction coefficient extended to include slope diffraction, is valid for the case of lossy plateaus [Luebbers, 1989]. Moreover, Whitteker achieved a PO formulation for a multiple-rectangular plateau diffraction by creating a simple extension to the Fresnel-Kirchhoff theory of double knife-edge diffraction [Whitteker, 1990]. Hasslet proposed a PO-based method to predict the diffracted field strength in the shadow of a rectangular building [Hasslet, 1994]. Furthermore, a method based on the parabolic wave equation was given by Janaswamy and Andersen to predict path loss in an urban environment where buildings are assumed to be flat and reflective [Janaswamy and Andersen, 2000]. Holm described a new heuristic UTD diffraction coefficient for non-perfectly conducting wedges, which allows the analysis of diffraction over rectangular plateaus that is, at the same time, valid deep in the shadow region, where the Luebbers coefficient fails [Holm, 2000]. Finally, Erricolo and Uslenghi have developed a two-dimensional, ray-tracing, polygonal line simulator to analyze multiple diffractions over a series of rectangular buildings, comparing the results with measurements and obtaining a solid agreement [Erricolo and Uslenghi, 2001]. It should be pointed out that, in all of the above-mentioned works obtained for analyzing multiple diffraction caused by rectangular plateaus, there are only results considering a small number of buildings presented.
 In this paper, a new formulation expressed in terms of UTD coefficients for the prediction of the multiple diffraction produced by an array of wedges, considering spherical-wave incidence, is presented. The major advantage of the proposed solution is that, as only single diffractions over wedges are involved in the calculations, both the computation time and mathematical complexity are reduced over existing formulations, permitting a quick analysis of the multiple diffraction produced by a great number of buildings. Furthermore, the presented solution is valid for both the vertical and horizontal plane.
 This paper is organized as follows. Section 2 presents the theoretical model; the considered propagation environment is described and the new formulation is expressed. In section 3, our solution is compared to other methods from technical literature, including an analysis of these two particular cases: multiple diffraction by buildings modeled as absorbing knife-edges and diffraction caused by a series of rectangular plateaus. Some additional results for the latter case are also shown. Finally, the entire work is summarized in section 4.
2. Theoretical Model
 By considering the final solution for the analysis of a multiple diffraction produced by an array of perfectly conducting wedges given in [Rodríguez et al., 2004], a new and more rigorous formulation expressed in terms of UTD coefficients for the prediction of multiple diffractions caused by a series of n wedges, considering spherical-wave incidence, is proposed.
2.1. Propagation Environment
 An idealized representation of the propagation environment considered in this work can be observed in Figure 1, where buildings have been replaced by n wedges of the same height, which is relative to the base station antenna height H, constant inter-wedge spacing w, and the same interior angle γ. The transmitting point is assumed to be placed at an arbitrary height (above, level with, or below the average rooftop height), and is located at a certain distance d from the array of buildings, so that a spherical wave impinges on the first one with an angle of incidence α.
 The theoretical model presented in this paper is a hybrid UTD-PO formulation for spherical-wave incidence based on the final UTD-PO solution for the multiple diffraction of plane waves, as expressed in Juan-Llácer and Rodríguez  (which, at the same time, is derived from the final PO solution achieved by Saunders and Bonar [1991, 1994]); therefore, it makes use of the advantages of both UTD and PO theories.
 First, an explanation about how the Saunders and Bonar solution for plane-wave incidence is derived is described as follows. Figure 2 shows a series of N parallel knife edges of the same height, which is relative to the source height H and a constant spacing w between knife edges (assumed to be large when compared to the wavelength λ).
 The source, having an arbitrary height (above, level with, or below the edge height), is located at a great distance from the edges (d0 ≫ Nw). For application to mobile radio wave propagation in urban environments, these assumptions are not restrictive. The angle of incidence is then
In this case, the assumption R ≫ H is considered, and this implies small angles of incidence over the array of knife edges.
 Considering such a propagation environment, the Saunders and Bonar solution for the function of attenuation with an arbitrary number of edges [Saunders and Bonar, 1991, 1994] is achieved by solving the multiple integral created by Vogler , using a method proposed by Boersma . This solution overcomes the slowness and inaccuracies of the Walfisch and Bertoni  model. If the number of edges is very large, both solutions are exactly the same, but when the number of edges is small, the Saunders and Bonar formulation produces a best fit with measurements. Moreover, unlike the Walfisch and Bertoni method, the Saunders and Bonar solution is also valid when the source is located below or level with the knife edge height.
 The field at the receiver, shown in Figure 2, which is relative to the free-space field, is given by
and Sn(t)—considering the starting value S0(t) = 1—is given by
where the complex function FS(X) is proportional to a Fresnel integral.
 Careful analysis of this solution reveals some useful properties. As can be seen in (4), the total field at the receiver is the average of the N contributions, due to the fields being diffracted several times in a special way. The case for N = 2 is illustrated in Figure 3b, where S1 is the solution for the case N = 1 (Figure 3a).
 Without taking account of the phase term [exp(jt2)], the total field at the receiver is
and may be seen as the arithmetic mean of two contributions. One of them corresponds to the field diffracted by a knife edge, evaluated at a distance 2w from such edge. The other takes account of the field diffracted once by an edge
which is diffracted again by another edge and evaluated at a distance w from the latter.
 The interesting point is that the total diffracted field is expressed in terms of single-edge diffractions. This property is essential for reaching solutions in terms of UTD-diffraction coefficients that will be presented next.
 In order to obtain the UTD field diffracted by a single edge, for harmonic time dependence [exp(jωt)], the phasor field just before the edge in Figure 4, for hard (vertical) polarization, can be assumed as Eo = 1 (N = 0).
which is the sum of the direct field and the diffracted field.
 For α < 0,
which is the diffracted field.
 If α = 0, then
which is independent of frequency and distance.
 The angle of incidence α is given by (1), k is the wave number, and the diffraction coefficient D(ϕ, ϕ′, L), for hard (vertical) polarization, is defined in Kouyoumjian and Pathak , along with the notation in Figure 4, as
is called the transition function and is defined in terms of a Fresnel integral. F(X) may be expressed, after a number of transformations, in terms of the Fresnel sine and cosine integrals, in order to make it suitable for computer implementation, as can be seen in the following:
 As can be seen in Figure 4, the incident shadow boundary (ISB) occurs when = + α and the reflection shadow boundary (RSB) occurs when = − α. Where there are small angles of incidence, the fields are always calculated in the transition region, very close to the incident shadow boundary (ISB), and it becomes necessary to use the UTD (not the Geometrical Theory of Diffraction—GTD), in order to remove the Geometrical Optics (GO) discontinuity along this boundary and also to give an appropriate solution for the field in this transition region. Furthermore, the fields are evaluated at a wide angular separation from the reflection shadow boundary (RSB). This implies cos ≈ 1, and if L ≫ λ, the transition function is
 Therefore, it is possible to make an approximation for the diffraction coefficient
 The fact that the diffraction coefficient is proportional to a Fresnel integral—as the function FS(X) is, in (4)—allows the consideration of a hybrid UTD-PO solution for calculating the multiply diffracted field.
 Another interesting consideration is that the argument of the transition function
is approximately equal to the square of the parameter t defined in (3), when the angle of incidence α is small and L = w.
 Therefore, considering the previously explained procedures, the total field at the receiver in Figure 2, for plane-wave incidence over an array of multiple co-linear knife edges, can be expressed by the following hybrid UTD-PO formulation [Juan-Llácer and Cardona, 1997; Juan-Llácer and Rodríguez, 2002]. Assuming that the phasor field at the reference point, indicated in Figure 2, is E0 = 1, for N ≥ 1, three cases can be distinguished:
 1. Source above the edge height (α > 0)
where k is the wave number, α is the angle of incidence defined in (1), D(ϕ, ϕ′,L) is the diffraction coefficient for a knife edge, given by (16), and
is a distance parameter. Note that direct fields have been included in (18).
 2. Source below the edge height (α < 0)
 It should be noted that, for a single-edge diffraction (N = 1), the expressions (18) and (20) are the same as (7) and (8), respectively, when L = w.
 This solution is in agreement with that which was derived by Lee . Therefore, regarding the above, the total field at the reference point indicated in Figure 1 can be calculated by following the same methodology expressed in Juan-Llácer and Rodríguez  for plane waves, that is, “the observed field at the reference point may be seen as the average of n possible fields from n different phase reference planes of n wedges”, but extended to the case of spherical-wave incidence. In this sense, the final solution is obtained using the summations of finite terms and satisfying a recursion relation, so that the successive (more than once) use of the UTD in the transition zone is avoided, that is, only single diffractions over wedges are involved in the calculations, achieving an easier and faster solution while demonstrating that it is not necessary, in some situations, to introduce slope diffraction (suggested in Andersen  and Tzaras and Saunders ) for the prediction of multiple building diffraction losses. A clear explanation about how this new formulation is constructed is described as follows.
 Firstly, considering the base station antenna above the average rooftop height (H ≥ 0), when n = 0 (absence of wedges), the observed field at the reference point in Figure 5 (the field that impinges on the first wedge) can be expressed as
being that Ei is the relative amplitude of a spherical source and k is the wave number.When n = 1, the field arriving at the reference point in Figure 6 can be written as
where D(ϕ, ϕ′, L) is the diffraction coefficient for either a perfectly conducting wedge (given in Kouyoumjian and Pathak ) or a finitely conducting wedge (given in Luebbers ), depending on the type of buildings we are modeling. It should be noted that the addends have been grouped in order to express the total field as a function of E(0) (total field calculated in the previous iteration) and, moreover, the spreading factor for a spherical wave as well as the proper L distance parameter have been taken into account in the diffraction term.
 When n = 2, the multiple diffraction appears, and E(2) can be obtained—as expressed before and by following the same way as the final solution presented in Juan-Llácer and Rodríguez  is formed—as the average of two field contributions coming from two spherical wave fronts impinging on every wedge, as can be observed in Figure 7. Therefore,
 In this case, since the emitting wave is spherical, and so that the average of the contributions can be performed following the same methodology as that which was used for the plane-wave formulation in Juan-Llácer and Rodríguez , that is, in order to calculate an arithmetic mean of a number of contributions coming from identical spherical wave fronts and impinging over every wedge with the same angle of incidence α (as it occurs for plane-wave incidence), the following consideration has been taken. In order to calculate E″(2), a “virtual spherical source” must be located at a distance Ro from the second wedge (the same distance which separates the source from the first wedge), bearing both a relative amplitude [E(1)·Ro] (which ensures a field E(1) over the second wedge—which was calculated in the previous iteration—due to the spherical decay with the distance 1/R0) and an elevation angle α relative to the height of the wedges. In this way, it can be ensured that an equal spherical wave front as the one considered for the first wedge will impinge into the second wedge with the same angle of incidence (as in the plane-wave incidence case).
 Hence, the final expression for E(2) can be written as
 It should be noted that, in the direct term of E″(2), the phase is related to the “real” emitting point [exp(−jk(R2 − R1))], and not to the “virtual” one, like the relative amplitude is. Furthermore, with the proposed method, both the spherical spreading factors and the distance parameters L are only dependent on the distance from the wedge to the receiving point (s) (see Figure 4) when the number of wedges varies, since the distance from the source to the wedges (s′) is always equal to R0, and such is desired in order to follow the same methodology as that which was used for the plane-wave solution, in which the spreading factor is 1/.
 Therefore, if we generalize the previous process in the case of n wedges, always placing these “virtual sources” at a distance Ro from all of the wedges except the first one in order to generate equal spherical wave fronts impinging on every wedge with the same angle of incidence, we obtain a straightforward, fast, and recursive final solution for spherical-wave incidence, which perfectly converges with the one given in Juan-Llácer and Rodríguez  for the analysis of the multiple diffraction of plane waves (when the source is located at a great distance from the array of buildings), and which avoids the use of multiple integrals (as occurs in Xia and Bertoni ) as well as the successive (more than once) use of UTD diffraction in the transition zone (only single diffractions over wedges are involved in the calculations), thus demonstrating, as mentioned before, that it is not necessary, in some situations, to introduce higher order diffraction terms (suggested in Andersen  and Tzaras and Saunders ) when predicting multiple building diffraction losses.
 In this way, our final theoretical model (considering that the base station antenna is above or level with the average rooftop height (H ≥ 0)) can be finally expressed as:
 To evaluate the case in which the base station antenna is located below the average rooftop height (H < 0), (30) must be used without the term exp(− jk · (Rn − Rm)).
 In this section, in order to validate the proposed formulation, several comparisons with other methods from technical literature are presented, analyzing the multiple diffraction caused by two perfectly conducting wedges, as well as the particular cases of diffraction by buildings modeled as absorbing knife edges and the diffraction caused by buildings modeled as flat-roofed, parallel rows of blocks (building rows in cross sections, considered to be in rectangular shapes).
3.1. Multiple Diffraction Over Two Perfectly Conducting Wedges
 A comparison between our solution, normalized to the free space field (in this case –20log10∣Enormalized∣), and the one given by Holm (up to order 20) [Holm, 1996], considering that n = 2, H varies from −4 m to 4 m, w = 75λ, d = 75λ, f = 900 MHz, and three different interior wedge angles can be observed in Figures 8 and 9, for soft and hard polarizations, respectively.
 A solid agreement can be noted, in both figures, between the two simulated solutions. The maximum deviation comparing our method and Holm's appears for γ = 150°, where differences of 0.63 dB and 0.83 dB can be found for soft polarization (H = 1.5 m) and hard polarization (H = 4 m), respectively.
3.2. Multiple Diffraction Over an Array of Absorbing Knife Edges
 In this case, the considered propagation environment can be observed in Figure 10, where buildings have been modeled as n parallel, absorbing half-screens (knife edges).
 By considering the above-mentioned scheme, the proposed formulation can now be written (if n ≥ 1) as
 Base station antenna above the average rooftop height (H ≥ 0):
where now D(ϕ, ϕ′, L) is the diffraction coefficient for a knife edge, as given by (16).
 Base station antenna below the average rooftop height (H < 0): (32) must be used without the term exp(− jk · (Rn − Rm)).
 The results for the electric field intensity at the reference point relative to the free-space field are shown in Figures 11 and 12 when w = 50 m, d = 50 m, and there are n knife edges, different values of H varying in steps of 1.25 m, and frequencies of 900 MHz and 1800 MHz, respectively.
 Although the calculations use only integer values of n, a continuous curve has been drawn to assist in visualization. In order to establish a comparison, the PO-based solution defined in Xia and Bertoni  is also shown in the plots.
 A solid agreement can be found between the two formulations in both figures. The maximum deviation, when comparing the two methods, is placed at H = 3.75 m when n = 100, so that a difference of 1.94 dB for 900 MHz can be observed, and at H = 2.5 m when n = 100, for the case in which the frequency is 1800 MHz, where a difference of 1.76 dB can be observed.
 For comparison purposes, it should be noted that, for example, our depicted solution in Figure 11, when n = 20, required 0.062 seconds to compute, whereas the PO result took 7.203 seconds, which points to a significantly larger computational effort.
 Furthermore, it should be pointed out that, when H = 0, the results obtained with the proposed solution emit a behavior almost equal to the function 1/(n + 1), as expected [Lee, 1978].
3.3. Multiple Diffraction Over a Series of Flat-Roofed Buildings
 For this particular case, an idealized representation of the considered propagation environment can be observed in Figure 13, where n buildings of the same height, which is relative to the base station antenna height H, having the same thickness v, have been taken into account, assuming that their cross sections are rectangular and that there is a constant inter-building spacing w.
 For multiple diffraction analysis purposes, this configuration can be seen as a series of wedges made of interior angle π/2 radians, joined two by two, forming the flat-roofed buildings. Therefore, assuming this geometry, when n = 1, and being
the field which reaches the left-placed wedge forming the rooftop of the first building, as indicated in Figure 14, where Ei is the relative amplitude of the spherical source, E(1) can be expressed as
is the field impinging over the left corner of the second building, which has been calculated following the method based on virtual spherical sources. D(ϕ, ϕ′, L) is the diffraction coefficient for a finitely conducting wedge, as given in Luebbers .
 Considering that n = 2, the field reaching the reference point of Figure 15, E(2), can be obtained as
 Therefore, the proposed formulation can now be written, for n ≥ 1 and considering that α > 0, as
and Em is the field reaching the left-placed corners of the rooftops, as observed in Figure 13. Thus, for m ≥ 1,
 It should be noted that, with the proposed theoretical formulation for this particular case, the analysis of the diffraction of spherical waves impinging on the array of buildings, with an angle of incidence smaller than zero (α < 0), could not be carried out, since, in this case, the above-mentioned spherical waves would hit the right-placed wedges which form the roof cross-sections from inside the building, which is meaningless for this purpose. Furthermore, it should be pointed out that, as expected, the proposed solution for this particular case perfectly converges to the one given in Rodríguez et al.  for the analysis of the multiple diffraction of plane waves caused by an array of rectangular plateaus when the source is located at a great distance from the array of buildings.
 In order to validate this formulation, a comparison between the total field normalized to the free space field obtained with its application and the field calculated with the method based on the use of the parabolic equation (PEM) that appears in Janaswamy and Andersen  is presented in Figure 16, considering two perfectly conducting buildings, the geometry depicted in the same figure (distance values in meters), and a frequency of f = 900 MHz, for both soft and hard polarizations.
 It should be pointed out that, for calculation with the solution presented in this work, the transmitter and the receiver have been turned around, so that an angle of incidence greater than zero can be assumed, firstly obtaining E(2), and then evaluating Etotal with the final diffraction down to the receiver, adding the contribution of the direct field for the Line of Sight (LOS) cases (H > 21 m). Solid agreement can be noticed between the two simulated solutions, except when values of H are near that of the grazing incidence case for soft polarization, in which our formulation gives an erroneously null field.
 A previously unreported analysis of the multiple diffraction caused by an array of perfectly conducting, flat-roofed buildings, assuming a spherical-wave incidence and considering up to n = 100, d = 30 m, v = 28λ, w = 22λ, f = 900 MHz and different values of H, can be observed in Figure 17, for both soft and hard polarizations, respectively. Again, although the calculations use only integer values of n, a continuous curve has been drawn to assist in visualization.
 Furthermore, results for the electric field intensity at the reference point marked in Figure 13, normalized to the free space field versus the number of buildings n, and considering that ɛr = 5.5 and σ = 0.023 S/m (values close to the buildings' actual electrical properties [Zhang et al., 1998]), and that d = 30 m, v = 28λ, w = 22λ, and f = 900 MHz, are shown in Figure 18, for different values of H and both soft and hard polarizations.
 It should be noted how, in this case, the solution is practically independent of the polarization considered, as it occurs in Rodríguez et al.  for the same scenario.
 A new formulation expressed in terms of the Uniform Theory of Diffraction (UTD) coefficients for the prediction of the multiple diffraction caused by an array of buildings modeled as wedges, considering spherical-wave incidence, has been presented. The solution has been validated through comparison with several methods from technical literature, including an analysis of particular cases of diffraction by buildings modeled as absorbing knife edges and the diffraction caused by buildings modeled as flat-roofed, parallel rows of blocks (building rows as cross-sections, considered to be in rectangular shapes). The major advantage of the proposed formulation is the fact that only single diffractions over wedges are involved in the calculations, achieving a faster and easier final solution and significantly reducing the computational time over existing formulations.