## 1. Introduction

[2] Field strength prediction in macro cells needs to take the propagation behavior of paths obstructed by a number of transverse obstacles into consideration. In general, these obstacles are modeled as perfectly absorbing half planes (also referred to as knife edges or screens) and the problem to be solved is multiple knife edge diffraction.

[3] In literature, several solutions were suggested for this problem. Many of them apply Fresnel diffraction to calculate diffraction loss of each individual obstacle and consider the overall path loss as a composite of these individual diffraction losses. One well-known method applying this procedure is the Epstein-Peterson method [*Epstein and Peterson*, 1953]. In this method the individual diffraction loss of an obstacle is calculated by regarding the previous one as a source and the following one as a receiving point. The Epstein-Peterson method was modified and extended several times and alternative approaches applying similar principles were introduced [*Causebrook and Davis*, 1971; *Deygout*, 1966; *Giovanelli*, 1984].

[4] The methods mentioned above give reliable results as long as the diffraction angle Θ as defined in Figure 1 is quite large. However, they failed where Θ is small, which occurs with grazing incidence. Now, the diffraction point on top of a screen is close to the shadow boundary of the previous screen and, thus, the incident field has rapid spatial variations in the vicinity of the diffraction point. This is called incident transition region field in the Uniform Geometrical Theory of Diffraction (UTD) [*Kouyoumjian and Pathak*, 1974]. Such an incident field contradicts the conditions of Fresnel diffraction, which requires an incident homogeneous cylindrical or spherical wave [*Born and Wolf*, 1975], at least locally in the vicinity of the diffraction point. Thus methods involving Fresnel diffraction are not valid in the case of glancing incidence. The same applies for approaches calculating multiple screen diffraction with help of the UTD [*Xia and Bertoni*, 1992].

[5] From the discussion above we found that for glancing incidence more rigorous solutions for multiple screen diffraction must be applied which give accurate results even for incident transition region fields. An exact solution for multiple screen diffraction is given by Kirchhoff-Huygens, which implies a multiple integral. Thus to calculate the diffraction by *N* screens a multiple integral of order *N *requires solving [*Born and Wolf*, 1975]. An analytic solution of this multiple diffraction integral was introduced by *Millington et al.* [1962], at least for two screens. However, Millington's solution becomes complicated for three screens and impractical for more than three screens [*Pogorzelski*, 1982]. Thus for more screens the Kirchhoff-Huygens integral must be solved numerically which, for example, was done to derive the Walfisch-Bertoni model [*Walfisch and Bertoni*, 1988]. An integral representation similar to Kirchhoff-Huygens can be derived by using repeated physical optic solutions [*Balanis*, 1989] for each screen [*Xia and Bertoni*, 1992]. In any case, numerical integration becomes complicated and time-consuming for a higher number of screens although the computation time can be significantly reduced using the Fast Fourier Transformation (FFT) instead of numerical integration [*Berg and Holmquist*, 1994].

[6] On the basis of a series representation pertaining to propagation over irregular terrain given in the work of *Furutsu* [1963], Vogler derived an integral representation for multiple screen diffraction in the case of glancing incidence [*Vogler*, 1982]. Regarding computation, Vogler found a more suitable solution of this multiple integral in terms of sums over repeated integrals of the error function. The validity of this approach is not limited by the number of screens but the computation time increases rapidly with the number of screens.

[7] Vogler's integral representation forms the basis of a geometrical optic solution derived by *Holm* [1996]. Holm transforms the multiple integral of the Vogler solution into a sum containing products of higher-order derivatives of the well-known Fresnel integral. He shows that this formula can be expressed as a sum over products of higher-order derivative of the UTD diffraction coefficients. For nonglancing incidence the higher-order derivatives can be neglected and the UTD representation of Holm turns to the formulation of the GTD usually used to calculate multiple screen diffraction. Thus Holm's approach represents a general solution for multiple screen diffraction. However, for the case of glancing incidence, the computation time increases rapidly with the number of screens [*Beyer*, 1999].

[8] To summarize, for glancing incidence several solutions are available to calculate multiple screen diffraction. However, with respect to practical purposes like radio network planning, these solutions are not sufficient because their numerical effort is too high.

[9] In this paper an approach is suggested for computer time-efficient prediction of multiple diffraction loss for an array of arbitrarily arranged screens in the case of glancing incidence. The approach represents a polynomial approximation of the multiple diffraction loss yielded by the Vogler method. The approximation is formulated as a function of only one average geometrical parameter of the screen arrangements and of the number of screens. The approach is numerically verified for up to 10 screens.