Diffraction

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 48, NO. 2, MARCH 1999

589

A Comprehensive Double Knife-Edge Diffraction Computation Method Based on the Complete Fresnel Theory and a Recursive Series Expansion Method Hatem Mokhtari

Abstract— This paper deals with the mathematical study of the surface Fresnel integral (SFI) which enables a rigorous computation of the double knife-edge diffraction. The proposed method, which is based upon an analytical series calculation for the SFI, becomes valid whatever the conditions of reception. Besides, the mathematical procedure that is proposed here involves the incident shadow boundary neighborhood for both knife-edge obstacles. Results are compared with some reference well-known values which correspond to asymptotic cases, the global system for mobile communications (GSM) and DCS1800 frequency bands, and a set of measurements. The suggested analytical method has been shown to be in very good agreement with these expected particular theoretical values and field-strength measurements, where different shadowing conditions, including frequency dependence such as mobile communications bands, have been considered for the sake of comparisons. Index Terms— Diffraction, Fresnel theory, surface Fresnel integral.

I. INTRODUCTION

T

HE RADIO-WAVE propagation which involves diffraction, reflection, and refraction is an essential phenomenon especially for field-strength prediction regardless of the concerned service or system, and many authors have been interested by the behavior of the interaction of waves with obstacles. Diffraction is one of these important physical phenomena which have been investigated using different philosophies. The most commonly studied methods are geometrical theory of diffraction (GTD), which is well known as an extension of the geometrical optics and involves polarizations and the physical properties of the obstacles [1], and the Huyghens’ principalbased scalar approach [2], which considers only the secondary fictive sources above each obstacle and, thus, by essence, does not consider dispersive lossy edges. Besides, there has been a great deal of simplifying methods whose formalism derives somehow from one of these two main approaches, such as the Bullington [3] and Deygout [4] methods. Moreover, as a reference accurate method, Millington et al. [5] proposed an analytical calculation method for the attenuation by diffraction due to two successive obstacles. In their paper, they have made use of the complete Fresnel theory which, thus, takes account of the Huyghens’ principal and leads to the use of

Fig. 1. Geometry of double knife-edge system.

the surface Fresnel integral (SFI) [6]. The authors considered the diffracting screens to be nonabsorbent which simplifies notably their formalism. The used assumptions will neither be modified nor extended to any specific type of frequencydispersive screen. Only secondary sources will be considered and screens assumed to be perfectly conductive, as the initial investigation of Millington’s work stated. The purpose of our investigation is to first give a quick overview of this analytical method, to study in which conditions it can be valid, and finally suggest a mathematical treatment for the SFI using a series expansion recursive calculations which allow a comprehensive range of validity of this analytical computation method. As it has been previously pointed out, our efforts have been focused on how can this SFI be generalized and applied to any receiving situation. Indeed, as a reminder, Millington et al. [6] have derived asymptotic expressions for the SFI, but only restricted cases have been studied, namely, the situations where the observation point is located “deep in the shadow” for either obstacles or receiver, or in the case where either obstacle lies on the line of sight of the other. The calculations of the SFI that are considered in this paper find their application especially in the case where optical rays lie in this vicinity of the direct incident shadow boundaries (i.e., for both obstacles and the receiver), since accurate results are given by Millington et al. in the asymptotic conditions (i.e., line of sight case and severe diffracting conditions). Moreover, since the problem is of a mathematical nature, we find it necessary to describe in detail our procedure through the next sections starting with an overview of its initial formulations. II. OVERVIEW

Manuscript received September 19, 1996; revised August 18, 1997. The author is with the Cellular Engineering Technical Group, MOBISTAR N.V., 1200 Brussels, Belgium. Publisher Item Identifier S 0018-9545(99)00780-X.

OF

MILLINGTON’S METHOD

A double knife-edge geometry is illustrated in Fig. 1, where one can easily see the different utilized parameters according to the work achieved by Millington et al. [5].

0018–9545/99$10.00  1999 IEEE

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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 48, NO. 2, MARCH 1999

The method as described in Millington’s paper neglects the lateral profile effect and considers only vertical Huyghens’ sources which led the authors to the expression for the (see Fig. 1) divided diffracted electric field at the receiver by the free-space one. The ratio of these two fields reads

computation method for solving (9) is merely based upon the series expansion of [8] given by

(10) (1)

and substituting (10) into (9) gives

where the used parameters are given by (2) (3)

(11)

(4)

or in a different form, assuming the convergence of the series in (11), as follows:

(5) (6)

(12)

(7)

and for the sake of computing (12), one could define a complex variable

have been studied for a restricted The properties of set of conditions only as we have mentioned above. remains then incompletely studied especially for intermediate Our mathematical calculations allow an accurate values of in the case where lies within a determination of range of finite small values. Referring to the above-mentioned paper [6], which describes the whole method of computing the in the following form: SFI, we may write (8)

(13) Using a single integration by parts of (13), the following recursive equation is easily derived: (14) Equation (14) is valid whatever be condition given by

with the initial (15)

where which leads to the calculation of

in (12) as the form below

(9) (16) is sufficiently high, one can easily derive the When Epstein–Peterson model [7] which considers the case where and of the obstacles to the individual Fresnel parameters be greater than 1.5. Accordingly, one should compute the correction factor as it has been proved to be sin in Millington’s paper [5]. At this stage, no intermediary expression, regarding the diffraction conditions, is available which led us to consider this interesting situation especially when the diffracted ray lies within a reasonable range around the direct incident shadow boundary.

where (17) and (18) Finally, substituting (16) into (8) yields

III. PROPOSED GENERAL SOLUTION Since propagation depends upon several parameters such as frequency, distances, and heights, we find it interesting to examine the situations where shadowing conditions (i.e., for both obstacle or receiver) are not very severe so as to complete the range of validity of Millington’s approach. Our

(19) Indeed, the very fast convergence of the series in (19) allowed us to sum its terms up to an integer fixed value, say , which is sufficiently high to obtain the required

MOKHTARI: COMPREHENSIVE DOUBLE KNIFE-EDGE DIFFRACTION COMPUTATION METHOD

TABLE I ASYMPTOTIC CASES i), ii),

AND

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TABLE III INPUT PARAMETERS FOR THE PREDICTION

iii)

* Computed with the new approach using the series solution given by (19)

TABLE II ASYMPTOTIC CASE iv) FREQUENCY 900 AND 1800 MHz, RESPECTIVELY

=

* Computed with the new approach using the series solution given by (19) ** See (20)

accuracy. Beyond that value (i.e., ), the series remains constant. Concerning the computation time, no more than 3.62 s are spent on a PC 80486/ 66 MHz when which is quite acceptable. IV. EXAMPLES OF CALCULATION AND NUMERICAL RESULTS As it has been stated previously, different situations for the receiver are taken into account in view of testing our method. For the sake of comparisons, we find it quite helpful to verify the validity of our calculations [i.e., (19) especially] in some particular asymptotic cases. The considered situations are those and are of equal spacing and both where: 1) distances obstacles of negligible heights; 2) obstacles are sufficiently separated and both obstacles of negligible heights; 3) obstacles are very close to each other and both obstacles of negligible heights; and 4) finally one obstacle is in the line of sight of the remaining one. In the case where the diffracted ray lies within a reasonable range around the incident shadow boundary, some results are thus given in order to show that in such case our method yields very good agreement with measurements and can also be suitable for land mobile radio communications frequencies. Furthermore, frequency is assumed to be a fixed value since the purpose is to compare the validity of our computation method on the basis of one tested frequency value at a time. Results are reported in Tables I–III. Since the second obstacle lies in the line of sight of the first one, its height is accordingly taken as (see Table II)

TABLE IV PROPAGATION LOSS WHEN THE DIFFRACTED RAY IS IN THE VICINITY OF THE INCIDENT SHADOW BOUNDARY

* Computed with the new approach using the series solution given by (19)

which is, by evidence, a frequency-dependent parameter and can be identified as the Fresnel parameter related to the main obstacle as it is commonly used in single knife-edge diffraction calculation. That is the reason why in Table II the operating frequency has been taken into account. Furthermore, the relation in (21) is seemingly an appropriate asymptotic approximation for the attenuation by diffraction in the case where the Fresnel parameter exceeds 1.5. Also, in this case (see Table II), results are in very good agreement. Coming now to the case where the diffracting conditions for both obstacles are neither severe nor on line of sight asymptotic situations, we suggest to compare some experimental results in view of a realistic comparison of our method with the measurements. Hence, results are reported in Table IV, where individual Fresnel parameters have been added in order to recognize that propagation is really around the incident shadow boundaries. Table III summarizes the used parameters which have been used by Giovanelli et al. [9]. * Computed with the new approach using the series solution given by (19). It can be noticed that, from Table IV, also for that case the Millington et al. method remains in very good agreement with the experimental data. Furthermore, the prediction by the Epstein–Peterson method gives the worst value, which is quite expectable because of its limited range of validity. Besides, prediction by means of the Deygout procedure reveals that the propagation loss is slightly pessimistic in view of the obtained value.

(20) V. CONCLUSION Referring once again to Millington’s work, the attenuation in this case is given by (21) where the dimensionless parameter

is given by (22)

The computation of the SFI is indeed analytically intractable unless one could use numerical two-dimensional (2-D) improper integration which is well known as a time-consuming procedure. However, Millington et al. [5] solved this problem in some conditions using series expansion and appropriate approximations for a limiting set of conditions. Besides, the method we have proposed in this investigation, which has been tested in several diffracting conditions, has been proved to be accurate enough to be taken as a comprehensive method for

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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 48, NO. 2, MARCH 1999

the double knife-edge diffraction calculation using the Fresnel scalar approach derived by Millington et al. analyses. However, since the novel proposed approach gives very good results with the asymptotic Epstein–Peterson method [7], where the diffracting conditions are very severe, for computation time reasons, one could use this approximation rather than the series approach as it has been studied in this paper. However, in intermediate diffracting conditions, our method becomes more suitable in view of its accuracy as mentioned by the comparisons. Furthermore, the implemented series expansion and iterations method can be extended to the multiple knife-edge problem which has been studied by Vogler [10] whose diffraction model is limited to up to ten obstacles. However, our series expansion approach requires, for the multiple diffracting edges, a rigorous study of convergence since the integration domain becomes a hypervolume rather than a 2-D surface as it has been previously demonstrated.

REFERENCES [1] R. J. Luebbers, “Finite conductivity uniform GTD versus knife-edge diffraction in prediction of propagation path loss,” IEEE Trans. Antennas Propagat., vol. AP-32, pp. 70–75, Jan. 1984. [2] C. A. Balanis, Advanced Engineering Electromagnetics. New York: Wiley, 1989. [3] K. Bullington, “Radio propagation at frequencies about 30 Mc,” Proc. IRE, vol. 35, no. 10, pp. 1122–1136, 1947. [4] J. Deygout, “Multiple knife-edge diffraction of microwaves,” IEEE Trans. Antennas Propagat., vol. AP-14, no. 4, pp. 480–489, 1966.

[5] G. Millington, R. Hewitt, and F. S. Immirzi, “Double knife-edge diffraction in field-strength prediction,” in IEE Monograph 507E, Mar. 1962, pp. 419–429. [6] G. Millington, R. Hewitt, and F. S. Immirzi, “The Fresnel surface integral,” in IEE Monograph 508E, Mar. 1962, pp. 430–437. [7] J. Esptein and D. W. Peterson, “An experimental study of wave propagation at 830 Mc,” Proc. IRE, vol 41, no. 5, pp. 595–611, 1953. [8] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products. New York: Academic, 1980. [9] C. L. Giovanelli, “An analysis of simplified solutions for multiple knifeedge diffraction,” IEEE Trans. Antennas Propagat., vol. AP-32, pp. 297–301, Mar. 1984. [10] L. E. Vogler, “The attenuation of electromagnetic waves by multiple knife-edge diffraction,” NTIA Rep. 20, Boulder, CO, Oct. 1981.

Hattem Mokhtari was born on September 18, 1964 in Constantine, Algeria. He received the B.S. degree in physics electronics from the University of Constantine, Constantine, in 1986, the M.S. degree in electronics from the University of Nancy, Nancy, France, in 1987, and the Ph.D. degree in electronics from the University of Metz, France, in 1992, where he worked on guided multiwire propagation phenomena such as in-circular lossy tunnels. In September 1992, he joined TDF-C2R, the research center of T´el´eDiffusion de France, where he was involved in several projects dealing with frequency planning in radio communications and broadcasting networks, radio-wave propagation modeling, planning tools design, and antenna pattern synthesis and theoretical modeling. Since September 1997, he has been with Mobistar, Brussels, Belgium, the second Belgian GSM operator, where he is in charge of radio parameters optimization and implementation of modern features such as microcellular concepts and slow-frequency hopping in view of reducing the effect of interferences in dense urban environments and, hence, improving the overall speech quality and geographical coverage. He is the author of several technical papers, most of them dedicated to theoretical upstream research topics. He was designated as a potential evaluator within the European Commission for the behalf of “Telematics Applications Programme” projects from 1997 to 1998.