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 An approximate approach is derived to enable a computer time-efficient prediction of multiple screen diffraction for grazing incidence. Comprehensive computations of multiple screen diffraction at up to ten randomly arranged screens were carried out using the Vogler method. On the basis of these numerical results, a suitable polynomial approximation is derived. In order to verify the approximation, the computations are repeated with different random screen distribution, and the deviation between the Vogler method and polynomial approximation is statistically evaluated.
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 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.
 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].
 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].
 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. , 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].
 On the basis of a series representation pertaining to propagation over irregular terrain given in the work of Furutsu , 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.
 Vogler's integral representation forms the basis of a geometrical optic solution derived by Holm . 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].
 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.
 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.
2. Vogler Method
 For glancing incidence Vogler derived an integral representation for multiple screen diffraction. For an arrangement of screens as outlined in Figure 1 the reduction of field strength A due to multiple diffraction loss is given by
with the geometrical parameter βi
 Here k represents the wave number, r the distance between the screens and Θ the diffraction angle. The corresponding geometry is shown in Figure 1. Θ is assumed to be positive if the top of the subsequent screen (or the reception point, respectively) is in the shadowed region. The underlined quantities are complex values. Only βi is of interest for the following sections and, thus, explained in detail above. The other quantities in equation (1) are defined in the work of Vogler .
 Vogler found a solution of the multiple integral (1) in terms of sums over repeated integrals of the error function. The overall reduction of field strength A is calculated by the sum
I(ni, βi) represents the repeated integral of the error function, N is the number of screens and M0 is the upper limit of the sum. The other parameters are defined in the work of Vogler . With respect to computational effort, the sum (equation (3)) represents a more suitable solution than numerical integration of equation (1) or comparable integral representations. However, the computation time of the Vogler method increases rapidly with the number of screens.
 The computation time primarily depends on the upper limit M0 of the sum in equation (3), which can be determined by convergence investigations. For the following discussion all Θi are assumed to be positive. In order to guarantee a remaining error below one percent at worst (that means all βi ≅ 0 for a number of 10 screens which is the maximum number regarded in this paper) M0 should be set to 18. Furthermore, M0 is proportional to the number of screens and inversely proportional to the average value of the geometrical parameter βI, according to equation (5). Applying both, M0 can be set considerably lower than 18 in many cases and the computation time can be reduced. Nevertheless, the average computation time seems to be too high for practical purposes like radio network planning.
3. Approximation of the Vogler Method
 As discussed in the previous section the numerical effort of the Vogler method is probably too high for large area field strength prediction. Thus with respect to this purpose, a more time-efficient approach to predict multiple screen diffraction loss would be desirable which, in addition, should not significantly reduce the accuracy of the prediction.
 A time-efficient way is given by a polynomial approximation of the Vogler method. In order to determine such an approximation, comprehensive numerical investigations are carried out. The multiple diffraction loss is calculated for scenarios consisting of arbitrarily arranged screens with βi randomly distributed within the interval 0 ≤ βi ≤ 1.3. According to equation (5), the real quantity βi is given by the complex quantity βi as defined in equation (2) divided by the square root of j. Owing to the lower limit 0 ≤ βi only positive diffraction angles Θi are regarded. That means, screens, which do not intercept the path between the neighboring screens, are not taken into account. The upper limit βi = 1.3 corresponds with a magnitude of the UTD transition function F close to one (F ≅ 1). Thus for βi > 1.3 no incident transition region field is given and βi = 1.3 can be regarded as the upper limit for grazing incident. As one result of these numerical investigations it is found that the average value βav of the geometrical parameter βi
can be regarded as a suitable variable of the polynomial approximation.
Figure 2 shows the relationship between the multiple screen diffraction loss Lmsd (given in logarithmic units) and βav for a number of 5 screens. The prediction is done for 500 screen arrangements. In order to get a wide range of βav and particularly small values of βav the geometrical parameter βi is stepwise uniformly distributed as given in Table 1. According to Figure 2 the multiple screen diffraction loss primarily depends on βav. The loss increases with increasing βav and the values are distributed in a fairly small range around a local mean. Therefore the results shown in Figure 2 verify that βav represents a suitable variable for the polynomial approximation.
Table 1. Overview of the Intervals Within βi Is Uniformly Distributeda
Range of βi
Number of Arrangements
For each interval the multiple screen diffraction loss is calculated for the number of screen arrangements given in the second column. The (real) geometric quantity β is given in equation (5).
0 ≤ βi ≤ 0.1
0 ≤ βi ≤ 0.3
0 ≤ βi ≤ 1.3
 In addition to the Vogler method, Figure 2 contains the results of the Epstein-Peterson method, which is commonly used for radio network planning. In order to compare both methods the relationship between the corresponding geometrical parameter β and ν
is used, which is valid for the range of β considered in this paper. ν represents the well-known Fresnel parameter [Born and Wolf, 1975]. Two effects can be observed by comparing the results of both methods. For βav > 0.7 the Epstein-Peterson method tends toward over-optimistic results, probably for reasons generally discussed in literature [Beyer, 2004; Causebrook and Davis, 1971; Deygout, 1966; Pogorzelski, 1982]. However, it is of more interest with respect to the topic of this paper that the deviation between the Vogler method and the Epstein-Peterson method increases rapidly for βav → 0. Finally, for βav ≅ 0.05 the diffraction loss predicted by the Epstein-Peterson method is 10 dB too high. A similar result could also be expected for other methods applying Fresnel diffraction for reasons mentioned in the introduction. Thus the results in Figure 2 confirm the need for an alternative method to predict multiple screen diffraction loss in the case of grazing incidence.
 In order to determine a suitable polynomial approximation, the multiple diffraction loss is computed by the Vogler method for 2 ≤ N ≤ 10 screens. For each number of screens the diffraction loss is calculated for 500 randomly arranged screen arrays with βi distributed according to Table 1. Afterward, polynomials with a maximum order of 4 have been fitted to the computed diffraction loss. These investigations show that the multiple screen diffraction loss in logarithmic units related to free space propagation can be sufficiently approximated by a polynomial of order 4.
 One disadvantage of a polynomial of order 4 is that its coefficients are very sensitive to even slight variations of the input data when fitting the polynomial within the complete range 0 ≤ βav ≤ 1.3. This can be avoided by applying the polynomial of order 4 only for small values of βav (e.g., βav < 0.1) and using a polynomial of order 2 for the remaining range of βav. However, this approach is not covered further in this paper.
 The constant coefficient of the polynomial can be determined without fitting but based on an analytical solution for a special case. Lee  found an analytical solution for multiple screen diffraction by an array of parallel equally spaced screens of the same height lying in a plane together with the source and the receiving point. The multiple diffraction loss related to free space propagation is given by
 Such an arrangement corresponds with βav = 0 and, thus, for βav → 0 the multiple diffraction loss predicted by the approximation should converge to Lee's solution. Therefore the constant term of the approximation is set to LLee and the polynomial approximation Lappr of the multiple diffraction loss in logarithmic units is finally given by
 The remaining parameters a1 to a4 must be determined by separate polynomial fitting for each number of screens. As mentioned above, the polynomial is particularly fitted to multiple screen diffraction related to free space propagation. Thus free space propagation loss L0 must be added in equation (8) to get an absolute value for multiple diffraction loss.
4. Numerical Verification of the Polynomial Approximation
 In order to verify the approximation, comprehensive computations are carried out. The sets of randomly distributed βi used for polynomial fitting differ from those used for the numerical verification. Furthermore, the seed of the random number generator is changed for each number of screens. Thus the results discussed below can be regarded as widely representative. Figure 3 shows the results of the Vogler method together with the approximation for 5 screens. We observe good agreement between the visually estimated local mean of the Vogler method and the approximation. This is confirmed by the corresponding statistical evaluation given in Figure 4. The figure shows the distribution of the deviation ΔL between the diffraction loss of the approximation Lappr and the Vogler method Lvogl
All values are given in dB again. From the cumulative distribution in Figure 4 we obtain that 80% of the values for ΔL are in between ±1 dB.
 In order to estimate the error of the approximation for a number of screens between 2 and 10, the mean and the standard deviation of ΔL is calculated. As dB values resulted in polynomial approximation, the mean and the standard deviation of ΔL are also calculated on the basis of dB values. According to Figure 5 the mean of ΔL is close to 0 dB regardless of the number of screens. The standard deviation of ΔL increases with the number of screens from 0.1 dB to 1.3 dB for 10 screens.
 The results discussed previously verify that the multiple screen diffraction loss in logarithmic units is sufficiently approximated by a polynomial of order 4. Compared with the Vogler method the computation time is significantly reduced by applying the approximation. On the other hand, a remaining error of the approximation exists. However, this error seems to be less compared with absolute values of the multiple diffraction loss and, furthermore, when assessing the quality of the approximation we should consider the accuracy usually given for field strength prediction in radio network planning.
 The approximation can also be applied to the case when one or more βi are greater than 1.3 (β = 1.3 corresponds with the boundary of the transition region). Then, the radio path can be divided into subpaths consisting of subsequent screens with βi lower than 1.3. The multiple diffraction loss of each subpath is separately calculated with the approach described above. The individual diffraction loss of each obstacle with βi > 1.3 is calculated with an appropriated method (e.g., Geometrical Theory of Diffraction, Fresnel diffraction). The overall path loss is the sum over the losses of the subpaths and the individual diffraction losses.
 A deterministic error of the approximation occurs when βi is constant for successive screens. For values of βi greater than zero the probability of such cases occurring is very low. More probable is the case that βi is zero for a successive number of screens which, for example, occurs in areas where the obstacles are of the same height. For this special case a correction of the previously described approach was found which is not further covered in this paper. The correction consists of a simple dual slope model with parameters similar to those which are introduced above. Please note that Lee's solution according to equation (7) can be applied if βi = 0 for all screens.
 For the approach introduced in this paper all screens are assumed to have a positive diffraction angle. That means in particular, that obstacles, which are very close to, but do not intercept the path between the neighboring obstacles are not taken into account. Therefore for a more general application, the approach should be further extended in order to include this case.
 In order to reduce the prediction time of multiple screen diffraction loss in the case of grazing incidence a polynomial approximation of the Vogler method is suggested and investigated in this paper. Comprehensive computations of multiple screen diffraction loss at up to 10 screens are carried out. For each number of screens 500 different sets of randomly distributed geometrical parameter β are assigned to the array of screens. The magnitude of β is distributed within 0 ≤ β ≤ 1.3 and all diffraction angles are assumed to be positive. These computations verify that the multiple diffraction loss related to free space propagation in logarithmic units can be sufficiently approximated by a polynomial of order 4. In order to verify the method the computations previously described are repeated with different random parameters. The standard deviation of the difference between the approximation and the Vogler method increases with the number of screens but, finally, is kept below 1.1 dB for 10 screens. The remaining deviations between the approximation and the Vogler method are low compared with absolute values of multiple diffraction loss. Therefore by applying the approximation the prediction time of multiple screen diffraction loss can be significantly reduced without considerable loss of accuracy.
 The author would like to thank to the reviewers for their helpful comments and valuable suggestions for further investigations.