## 1. Introduction

[2] 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].

[3] 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].

[4] 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.

[5] 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.

[6] 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.