R rently evolving towards greater resolution. In an

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1 1640 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 41, NO. 12, DECEMBER 1993 A Deterministic Approach to Predicting Microwave Diffraction by Buildings for Microcellular Svstems J Thomas A. Russell, Member, IEEE, Charles W. Bostian, Fellow, IEEE, and Theodore S. Rappaport, Senior Member, IEEE Abstract-Designers of low-power radio systems for use in urban areas would benefit from accurate computer-based predictions of signal loss due to shadowing. This paper presents a propagation prediction method that exploits a building database and considers the three-dimensional profile of the radio path. Models and algorithms are provided that allow the application of Fresnel-Kirchhoff diffraction theory to arbitrarily oriented buildings of simple shapes. Building location information used by the diffraction models is in a form compatible with a geographic information systems (GIs) database. Diffraction screens are constructed at all building edges, for both horizontal and vertical orientations, in order to consider all possible diffractions and to compute field contributions often ignored. Multiple buildings and edges of the same building that introduce multiple successive diffractions are considered with a rigorous, recursive application of the diffraction theory that requires sampling the field distribution in each aperture. Robust and computationally efficient numerical methods are applied to solve the diffraction integrals. Tests of the software implementation of these methods through example runs and comparisons with 914-MHz continuous-wave measurements taken on the Virginia Tech campus show promise for predicting radio coverage in the shadows of buildings. I. INTRODUCTION AD10 signal strength prediction methods are cur- R rently evolving towards greater resolution. In an urban area the principal obstructions to free-space radio propagation are buildings, and in a typical high-power radio cell they may number in the hundreds. The complexity of their interactive effects on field strength at any given point has traditionally forced the use of gross average models for path loss as a function of distance, which may include an empirical correction factor for the degree of urbanization [ll, or a theoretically derived diffraction loss based on an idealized layout of infinitely long buildings of uniform height [21. Recently, however, several researchers have shown the feasibility of using a specific, building-by-building description of the urban environment Manuscript received November 23, 1992; revised July 14, T. A. Russell is with Stanford Telecommunications, Inc., 1761 Business Center Drive, Reston, VA C. W. Bostian and T. S. Rappaport are with the Bradley Department of Electrical Engineering, Virginia Polytechnic Institute and State University, Blacksburg, VA IEEE Log Number and ray-tracing methods to determine signal strength [3], [41, E]. In the future, wireless phone communications will serve densely populated urban areas with plentiful mini-base stations or radio ports, each of which will cover areas of several blocks at most [6], [7]. Thus the resolution of the urban area is increased and the number of included buildings reduces to a small number, increasing the significance of individual buildings (10-20 db impact seen in downtown Ottawa [8]). An analogy may be made with terrain obstructions in a large-scale rural radio system. Methods currently employed for system design in such an area compute the diffraction introduced by each hill explicitly rather than statistically, including multiple diffraction either with geometrical construction methods [9], or with numerical sampling methods [lo] to more rigorously characterize the effects. We suggest taking a similarly careful approach to modeling the individual impacts of buildings, while recognizing the different form that these obstructions take. Of particular relevance is that hills usually fall off smoothly in height while buildings generally have sharp drop-offs. Indeed, the use of the infinite knife-edge model for terrain diffraction is based on the fact that smooth variation in the height of a hill transverse to the direction of propagation can be neglected with only small error [lll. This model, however, is inappropriate as a general approach for building diffraction, as diffraction around the corner edges is neglected. Other approaches consider corner-edge diffraction [ 121, but neglect rooftop diffraction paths, requiring that the transmitter and receiver are both low with respect to all of the buildings. This is appropriate for such systems but excludes hybrid or intermediate schemes, where the transmitter may be mounted on a building and may be higher than a neighboring building. It should be expected that a multitude of system designs will emerge with new personal communication systems, and a general modeling approach that allows the system designer flexibility in transmitter and receiver placement should be useful. In this paper we propose a unified, three-dimensiona1 approach to predicting the diffractive shadowing introduced by a collection of arbitrarily oriented buildings of simple shapes, given the locations and heights of the buildings, the transmitter, and the receiver. Finite knife X/93$ IEEE

2 RUSSELL et al.: PREDICTING MICROWAVE DIFFRACTION BY BUILDINGS FOR MICROCELLULAR SYSTEMS 1641 edge models for building edges are constructed and every possible diffraction path is simulated in order to determine the significant sources of field and to avoid missing a possibly dominant signal path. A terrain diffraction method that recursively computes the multiple diffraction introduced by successive diffracting edges through efficient numerical techniques [lll is adapted for use with finite knife edges, forming the basis of a computer program implementation. This is a deterministic site-specific approach that requires no statistical parameters or correction factors. We apply the program to a campus area and show through comparisons with measurements at 914 MHz that the transition regions are well predicted as a receiver moves into the shadow of a building, and that diffraction predictions give a worst-case prediction of signal strength. Buildings are approximated with simple rectangular volumes identified by planar coordinates (such as Universal Transverse Mercator) of the four corners plus the height above sea level of the rooftop. More complex buildings may be composed of adjacent rectangular sections. The use of a geographic information systems (GIS) package as a software platform for input and output of the data required by the model is described in [13]. This building representation provides the information necessary to consider diffraction paths around and over the building. 11. DIFFRACTION PREDICTION METHOD The three-dimensional model for single diffraction by buildings introduced in [13] forms the basis of the multiple-diffraction model presented in this paper. This approach applies the Fresnel-Kirchhoff solution for diffraction through an aperture in a two-dimensional screen [14] which is approximately normal to the direction of propagation (the third dimension.) When the screen is modeled with only one dimension, this solution reduces to the infinite knife-edge model traditionally used in radio engineering. In our case the aperture is the area surrounding a building. We must divide this complex area into simple rectangular areas in order to use the approximate form of the Kirchhoff diffraction integral consisting of the product of two one-dimensional integrals rather than a computationally intensive two-dimensional integral. The diffraction field contributions from each area are computed and summed at the observation point to give the total field, resulting in multipath diffraction. Bachynski and Kingsmill [15] verified this multipath approach to diffraction as applied to thin, knife-edge obstacles with varying profiles transverse to the direction of propagation. They conducted scale-model measurements and saw a strong dependence on the transverse profile and substantial agreement with predictions. A. Multiple Diffraction Multiple diffraction effects must be considered if the wave incident on the secondary Huygens sources in the area surrounding on obstacle has been affected by a preceding edge, as in Fig. 1. The method we will use requires first evaluating the complex field at many Huygens sources, Q, in the aperture plane in order to approximate the field distribution, and then integrating over the aperture. This is the same general approach used by Whitteker [ 111 and Walfisch and Bertoni 121; however, while the solutions in both [ll] and [2] integrate only over the vertical dimension, our solution includes the product of integrals over both of the dimensions of the aperture plane. When there is an intervening horizontal edge (parallel to the x axis), as in Fig. 1, the field in the aperture (at points Q) varies with y based on the degree of shadowing by the intervening edge, while the dependence on x is a function primarily of the increased path length, which can be described analytically. This important distinction allows us to characterize the field distribution with discrete function evaluations in only the y dimension. A requirement of two-dimensional sampling would lead to excessive computation time. The derivation provided in the Appendix yields the following solution: where 52 =x2 ~ This solution provides the component of the complex field at an observation point, P, resulting from diffraction through the rectangular aperture defined by (xl, x,) and (yl,m) in Fig. 1, given isotropic radiation of unity amplitude at source, S. The shaded rectangles in the figure represent building faces. Lengths s and p are the distances from the origin of the aperture plane to S and P, respectively. This result reduces to Eq. (9) of [ll], or Eq. (1) of [2] when the limits on x extend to +a. To compute the field at intermediate points Q, the points are treated as new observation points, P'. The computational speed of the multiple diffraction procedure depends on the number of field evaluations, EQ(y>, required to obtain an accurate result. This number can be minimized by performing integrations over interpolating polynomials according to the method of Whitteker, [lo], [ll], [16], and Stamnes et al. [17]. The coefficients of amplitude and phase interpolation functions are computed based on the field samples. The amplitude interpolator is always quadratic and the phase function is either linear or quadratic depending on the size of the computed quadratic coefficient. The aperture field is characterized with a set of up to 80 finely spaced samples appended by.

3 1642 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 41, NO. 12, DECEMBER 1993 where and Efs is the free-space field at P. The total field at P is found by computing E#') for x' each of the N rectangular areas that together make up / the total unobstructed portion of a diffraction screen. (The aperture area construction is described in Sections 1I.B and 1I.C.) The individual contributions are summed either in a complex field sense that displays the multipath interference between diffraction fields, X N E(P) = E,(P), (4) n=l or in an average power sense that provides an estimate of the local mean field strength {F. at P, Fig. 1. Construction for computing diffraction by multiple successive building faces. The field at a Huygens source Q is found by treating it as a new observation point P'. E(P) = (5) three more widely spaced samples. These final three samples yield a pair of amplitude and phase functions which are integrated to infinity. Spacing and number of samples are discussed in Section I1.D. The derivation of (1) is based on a diffraction field at Q caused by a horizontal edge. If, on the other hand, the edge producing the diffraction field in the aperture is a vertical edge, such as the corner edge of a building, then the opposite of the above reasoning holds: variation with x affects the degree of shadowing more than variation with y, which is assumed to be only a function of free-space expansion. We obtain for vertical edges, where (2b) The lower limit on the range of x is --CO for a field propagating around the left side of the building, and the upper limit is -CO for a field propagating around the right side. Note that the need for (2) instead of (1) is determined not by the nature of the diffracting edge at Q, but by the nature of the diffraction field at Q, which is a function of the previous edge. When the field at aperture points Q is not affected significantly by prior edges (the single-diffraction case), the aperture field variation with both x and y depends only on free-space expansion, and the above results (1) and (2) reduce to Both of these solutions may find application in propagation effects analysis, but the latter, (5), would be appropriate for signal strength prediction for system design. At intermediate points, P' in Fig. 1, neither of these solutions is used. Instead, the contributing fields, E,(P'), are compared and only the largest field source is used in the solution for the diffraction at the next point. This is done in order to avoid the possibly large interference effects that (4) would predict, and the consequent difficulties in integrating over the phase and amplitude variations. The alternative, the power sum method given by (5), cannot be used at intermediate points because it does not predict phase. B. Construction of Aperture Screens The area surrounding a building already must be separated into simple rectangular areas for evaluation reasons (see above); we take the additional step to assign an aperture area to each diffracting building edge, both vertical and horizontal, and construct a screen so as to best model that edge. The formulation of the Kirchhoff integral in terms of Fresnel integrals requires that the aperture screen be approximately normal to the line connecting the source and observation points [14]. The faces of the buildings, however, will not in general align with such a plane. For simplicity, we make all aperture screens vertical, which requires only that the vertical component of SP is smaller than the horizontal [14]. The aperture plane is oriented normal to the horizontal projection of SP: the y axis is vertical, the x axis is horizontal, and the origin is at the point of intersection with SP. The building face is then projected onto this plane, yielding the bounds on the aperture region in terms of x and y. These bounds transform into the limits of integration according to Ob), (IC), and (2b). 1) RoofEdge Aperture Screens: The rules for construction of aperture screens are illustrated with a simple example (Fig. 2) where the transmitter (source, S) is

4 RUSSELL et al.: PREDICTING MICROWAVE DIFFRACTION BY BUILDINGS FOR MICROCELLULAR SYSTEMS 1643 Fig. 2. Perspective view of diffraction by a rectangular building. The xy plane is constructed to model the diffraction by the roof edge of the left building side. higher than both the obstructing building and the receiver (observation point, P), and diffraction is introduced only by the left corner edge and the two roof edges nearest the receiver. In this example the right corner is too distant relative to the other edges to contribute relatively significant energy and the roof edges away from the receiver do not obstruct the path; therefore there is only single diffraction and no field sampling is required. Consider first the roof edge of the building face visible on the left of the figure, through which SP passes. The aperture plane constructed for this edge is superimposed on Fig. 2. The bounds on x are illustrated in Fig. 3 with a bold section of the Xb axis. The second diffracting building face visible in Fig. 2, the front of the building, does not intersect SP though it is near enough that the roof edge will contribute diffracted field to P. The xy plane in such a case is the vertical plane that intersects the face at the vertical edge nearest to SP. The x axis of this plane is presented as x, in Fig. 3. 2) Comer-Edge Aperture Screens: Finally, consider the diffraction field contributed by the left rear corner edge of the building. The aperture screens constructed for vertical edges always intersect the edge. The diffraction field is computed by integrating over the infinite half-plane abutting the edge. For the small geographical areas covered by microcellular systems, and fairly flat ground, there will be little obstruction by the ground. In these cases we can assume y, = --x to avoid computations necessary to access terrain information. As shown in Fig. 3 (with the x, axis), the aperture region also goes to infinity in the -x direction. 3) Ouerall Single DifSraction Model: The three diffraction screens discussed above and shown in Fig. 3 are treated as though they are portions of the same overall aperture plane; however, we segment the infinite aperture plane into semi-infinite regions with vertical dividing lines and translate these regions along SP (a translation that has both vertical and horizontal components). When viewed along a straight line from P to S in Fig. 3, these aperture screen segments appear to form a single aperture screen. Conversely, when the apertures introduced by different building edges overlap when viewed in this way, we treat them as multiple (successive) diffraction screens. Some sample algorithms are presented in Section 1I.C for I.P Fig. 3. Top view of the situation in Fig. 2. The x axes of three aperture planes are represented by dashed lines and the portion of each axis that defines the aperture is in bold. Plane a models the left rear corner diffraction, b models the left side rooftop diffraction, and c models the front rooftop diffraction. sorting the edges of rectangular buildings according to whether they diffract the wave multiple successive times or diffract different portions of the wave front a single time. This construction method is designed to minimize error in estimation of diffraction by any particular edge by defining as precisely as possible the distances involved in the diffraction integral; at the wavelengths of interest for microcellular applications (less than 0.4 m> and the close proximity to large buildings, accuracy in the location of the edge is critical to accuracy in the result. In the process, some error will be incurred through the assumption of single diffraction through apertures that are offset from each other. However, it should be noted that two separated aperture segments constructed by our rules meet and share the same plane in transition regions when SP passes directly through a comer edge. (In Fig. 3, as P moves to the left x, and xb will meet, and as P moves to the right xb and x, will meet.) 4) Example: The movement of a receiver past a building is simulated for the situation pictured in Fig. 4: the transmitter is 30 m high and 150 m from the building; the receiver is 0 m high and 50 m from the building; the building is 10 m high, 50 m wide, and very long. The receiver travels a 60-m course parallel to the long edge of the building, moving from a line-of-sight situation into the shadow of the building. At the point marked in Fig. 4 the situation is identical to the one shown in Fig. 3. Fig. 5 presents the results of applying the diffraction integral (3) to each edge, showing that there are regions where each component is dominant, and demonstrating the necessity of computing each component. While the error in the power estimation would usually be less than 3 db if only the most dominant diffraction component is computed, this component is difficult to reliably identify beforehand, particularly in the general case of multiple diffraction. The total diffraction field is represented as a power sum, or local mean field strength, computed by (5). C. Application to a Collection of Buildings At the receiver point the database is searched for the nearest diffracting building according to an algorithm

5 ~ 1644 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 41, NO. 12, DECEMBER 1993 Transmitter. 1 - j Building Obstacle r60m lyobile Receiver Fig. 4. Top view of the geometry used for an example receiver run. At the position indicated, this is the same situation depicted in Figs. 2 and 3. TS d Distance Travelled By MoDile Receiver. m Fig. 5. Computed diffraction field contributions from the three aperture regions defined in Fig. 4. The total field represents an estimate of the local mean diffraction field strength. Fig. 6. Top view showing aperture constructions for all four roof edges. some error in multiple diffraction, see Section 1II.A. If the following three conditions are true of the diffraction parameters computed for a particular aperture according to (lb), (IC), and (2b), c1 < -0.78, 52 > 0.78, ql > -0.78, then the aperture provides sufficient clearance. If neither of the level 2 edges offers a sufficiently clear propagation path, then both edges are considered potential diffractors. However, before any time-consuming diffraction evaluations are made, each aperture should be checked to ensure that the diffraction path is capable of contributing significant fields strength. For example, if 40 db is considered the maximum diffraction loss of interest, and any of the following are true for a particular aperture: described in [131, which includes a coordinate system transformation to simplify geometric manipulations. The procedure for considering each of the eight building edges as possible diffractors is demonstrated below. 1) Roof-Edge Difiaction: The first step is to construct aperture screens according to the rules in Section 1I.B. The results are illustrated in Figs. 6 (roof edges) and 7 (corner edges), where one of the two dimensions of each aperture is indicated with a bold line, and the other dimension is out of the page. The four roof edges of a single building are grouped into two diffraction levels, where edges that belong to the same level define apertures that derive from a single aperture plane which has been segmented and translated. These levels are numbered according to the order in which they affect the wave; for example, in Fig. 6 the two apertures at the top of the figure are part of the first aperture plane encountered by the wave radiating from S. The two corresponding roof edges are considered level 1, and the other two edges are level 2. In this model, the level 1 edges diffract the wave once and the level 2 edges diffract the wave a second time and the edges making up a single level diffract different portions of the wave front. The level 2 edges are considered first: both edges are checked for obstruction of more than 55% of the first Fresnel zone. This clearance threshold is traditionally used in microwave radio links [MI; however, it leads to then at least 40 db diffraction loss will be incurred on that path and the aperture should be considered blocked. The rooftop diffraction field at an observation point is computed by first considering the level 2 edges of the nearest diffracting building and then working towards the transmitter by proceeding on to the level 1 edges and then to prior buildings in the search for prior diffracting edges. If none are found, then the single diffraction (3) is evaluated at P; otherwise samples are taken and the multiple diffraction (1) is evaluated according to the recursive procedure described in Section 1I.A. An important note regarding multiple diffraction should be made. While an integration over the aperture field at a prior diffracting edge is performed based on the slightly altered geometry at each sample point in the current aperture, only at the first sample point (nearest the edge) is a search performed for prior diffractors, and samples of the field adjacent to the prior diffracting edge are only taken once. 2) Comer-Edge DifSraction: The diffraction fields contributed by the building corner edges are summed with the rooftop fields to find the total building diffraction field. Fig. 7 illustrates an example. The corners are grouped according to whether they are on the left of SP or the right, and then sorted according to the order in which they are encountered by the wave by applying a simple sorting algorithm.

6 ~ RUSSELL ef al.: PREDICTING MICROWAVE DIFFRACTION BY BUILDINGS FOR MICROCELLULAR SYSTEMS 1645 TS P- AP Fig. 7. Top view showing aperture constructions for the diffracting corners. Thus, for each building we consider two corner-edge diffraction fields, one around the left of the building and one around the right. Typically only one of these will survive the test for significant diffraction field, a test which is analogous to that for roof edges. Multiple corner edges in a single group are considered as possible multiple successive diffractors, following a procedure similar to that for roof edges. Considering only a single group at a time, first the nearest corner edge to the receiver point, then each prior comer moving towards S is checked first for blockage and then for clearance. At the corner nearest S another search is conducted for prior diffracting buildings. We note in Fig. 7 only one diffracting corner edge on the left, and two on the right (there is sufficient clearance at the corner edge on the right that is nearest PI. Also, we note that the aperture plane next to the top corner in the right is at a different angle. This arises through the procedure whereby we sample the field in the aperture of the diffracting, right-side corner nearest the receiver and consider sample points as new observation points P'. The aperture plane at the preceding diffracting corner is then defined relative to the new line SP'. 3) Connected Buildings: Connected buildings are defined as those sharing an edge. This is the representation used in our model for buildings with rooftop sections of different heights; it is evidenced in the database by two corner coordinates on each of two buildings matching to within some error tolerance. Connected buildings require some exceptions to the general rules described in this section. For example, the diffraction around a corner that is connected to a taller building (and does not really exist as a diffracting edge) is not computed. Rather, the true corner of the connecting building is considered a diffraction source in the same manner as the other corners of the original building. D. Field Sampling The integration range over y in (1) or x in (2) is divided into two subranges: a finite region that extends at least until the ray connecting S with the sample point clears the obstructing edge, and a semi-infinite region (if the full range itself is semi-infinite) wherein the field amplitude oscillates about the free-space level, converging to this level at infinity. The first region is characterized with a large set ( ) of finely spaced samples, and the second region with three relatively widely spaced samples. We set the sampling interval for the first integration region, A,, at about 2/3h. Walfisch and Bertoni [2], using linear representations of both amplitude and phase and some different assumptions in the integral formulalion, found that point spacing of less than a wavelength should result in an error in the integrand of less than 0.8%. Our error in the integrand, using quadratic polynomials, should be less. We tested the sensitivity to sample spacing by reducing it to 0.1 m and m and increasing the number of points in order to hold approximately constant the sampling range, and saw that the variation in the computed power was less than 0.2 db for all test cases, including up to four diffracting edges. The sampling interval for the second region, A,, is set to 3.0h. Ref. [ll] advises that the value of A, be comparable with the change in height for which the clearance of the obstructing edge changes by one Fresnel zone, corresponding to the half-period of the field oscillations, but this value varies depending on the portion of the curve sampled. Fortunately, tests that held A, to various values and varied NI over a wide range (thus shifting region 2) showed only small variations, except for some anomalies traced to a phase problem addressed below. These tests were performed on various building geometries with up to four successive diffracting edges. The best performance was found with A, = 3.0h, where variations in computed diffraction field were less than 1 db peak-to-peak. These tests demonstrate an insensitivity to both the size of the sampled region and the portion of the curve sampled to determine the interpolating polynomials for the extrapolation to infinity. Indeed, sampling two portions of the same amplitude curve, one yielding a parabola with positive curvature and one with negative curvature, did not yield significantly different results. This was not true, however, of the phase interpolator. The phase of the integrand is the sum of the phase of the aperture field with a.rry2/hp term that describes propagation from the obstacle to the observation point, P. Thus the phase at P due to individual Huygens sources increasing in distance from the diffracting edge shows a parabolic shape with a positive curvature and an oscillation (the aperture phase) superimposed. If the aperture phase oscillations are strong enough relative to the quadratic term, certain sets of samples will result in a phase function with a negative curvature, giving a wildly incorrect result which has been witnessed in many situations. Hence, while the result is not sensitive to small variations in the coefficients of the phase parabola, it is necessary that the function have a positive curvature so that in the extrapolation to infinity, the phase will go to positive infinity. This is ensured by requiring that the first of the three samples of region 2 is at a point of minimum phase slope. At such a point the phase curvature is zero and going positive. Fig. 8 shows the phase of the field in the aperture with the points of minimum phase slope marked. These points occur approximately where the ob-

7 1646 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 41, NO. 12, DECEMBER 1093 sorted the diffracting building edges and computed the diffraction field for the building configurations tested. The agreement of the predictions with measurements is discussed in this section. The multiple diffractions were computed in 5-10 s per successive edge on a 386SX personal computer with a math co-processor. The computation time for characterizing the field at 300 points along a one-dimensional track is on the order of a half hour, with one or two buildings obstructing the radio path at any given time. 0 3 n.fi DiFrcncc nhme 4qe. m Fig. 8. Field variation in aperture with height above edge. Points of minimum phase slope and minimum amplitude are marked according to the degree of clearance over a prior diffracting edge. Results for two transmitter heights are presented. The height of both edges is 10 m. structing edge is cleared by n = 2k Fresnel zones, where k = 0,1,2,..-. These are also the points of minimum amplitude. We impose two requirements on the size of the first integration region: some minimum number of samples (we use 25) must be taken, and the sampling must continue at least until clearance is obtained. The former ensures that even where there is only some obstruction of the first Fresnel zone and the line of sight is clear, there is still sufficient characterization of the diffraction field; the latter ensures that obstructed portions of the amplitude and phase curves are fully characterized and that the samples in the second region indeed fall within the oscillatory region. These requirements are most efficiently met for a variety of shadowing depths through employing an adaptive procedure for choosing NI. The phase of the field in the aperture has positive curvature at the very beginning of the unobstructed region. Thus in the case of relatively deep shadowing, where many samples are taken in region 1, the minimum number of samples is exceeded by the first sample point to have clearance, and this point constitutes the boundary point. However, in the cases where the second edge is in a shallow shadow, or obstructed by a subpath obstacle, and the minimum range for region 1 extends into the oscillatory region, we must ensure positive phase curvature on region 2 by ending region 1 not at the minimum range, but extending it to the next point of minimum phase slope. The algorithm can be stated as follows: If the clearance at sample 25 is less than 0, continue sampling in region 1 until clearance is achieved; If the clearance at sample 25 is greater than or equal to 0, continue sampling in region 1 until the next integer k such that Fresnel zone clearance n = 2k RESULTS The C-language computer program implementation of the algorithms proposed here successfully located and A. SensitiLiiy Analyses As a test of the multiple-diffraction procedures, consider two buildings with heights of 15 m and 10 m for buildings 1 and 2, numbered from the transmitter towards the receiver. The buildings are wide enough that the corner-edge diffraction fields are negligible; the successive edges of the buildings are at distances of 100 m, 120 m, 150 m, and 200 m from the transmitter; the receiver is at 250 m. With a receiver height of 0 m, we lower the transmitter from a height of 30 m to only 5 m and simulate the diffraction introduced at 900 MHz. The solid line in Fig. 9 plots the results of applying the algorithms described here, while the other lines represent the forced exclusion of certain edges, demonstrating the impact of neglecting certain diffractions. For example, if only the last edge before the receiver (the trailing edge of building 2) is considered, a technique often used in predicting approximate field strength, the error is on the order of 20 db at deep levels of diffraction. The discontinuities seen in some of the curves of Fig. 9 are the result of our assumption that edges providing clearance of at least 55% of the first Fresnel zone can be neglected. This is approximately where the single-diffraction amplitude curve first crosses 0 db (free-space loss); as clearance increases past this point the power continues to increase before returning to oscillate about 0 db. This clearance threshold is traditionally used in microwave radio links [ 181, but we found that in the multiple-diffraction situation, the field in the shadow of the second edge encountered by the wave can be near a maximum of the oscillations at the point where the first edge has a 0.55R, clearance (where R, is the radius of the first Fresnel zone). The sudden consideration of the first edge then causes a sharp increase in power. B. Comparisons with Measurements The software was tested against measurements taken near two buildings on the Virginia Tech campus: Patton Hall and Davidson Hall. Only the building coordinates, the locations of the transmitter and receiver, and the frequency, 914 MHz, were input to the program; as there were no adjustments of parameters, these are blind tests of the models. Both of the measurement areas were chosen to isolate a single building while avoiding obvious sources of strong specular reflections. The signal radiated by a continuous-wave transmitter was received with a 6-ft-high antenna on a mobile cart and stored in the

8 RUSSELL et al.: PREDICTING MICROWAVE DIFFRACTION BY BUILDINGS FOR MICROCELLULAR SYSTEMS tr-20 U L $-25 a.-?-30 U 0 - a! w-35 Edges considered: ' I \ k Free Space Field -_ ---_- Measurements --_ $ 1 Predictions - ' " " " " ' I 5 Transmitter height, m Fig. 9. Computed multiple diffraction by two buildings of heights 15 and 10 m, as a function of transmitter height. In all but the solid line, various diffracting edges are purposefully neglected to test the impact on predicted field strength. -80' ' I " " " " Distonce along path, m Fig. 10. Comparisons with 914-MHz measurements taken of the diffraction introduced by Patton Hall at a receiver passing in front of it. The transmitter was placed on the roof of another building. on-board computer while the cart was propelled along a sidewalk. For the Patton Hall measurements the transmitter was placed on top of another building, providing the case of propagation from above the obstacle. The receiver began with a line-of-sight to the transmitter and passed into the shadow of Patton Hall. The measured data was filtered with a 20h Hamming window to provide an estimate of the local mean field strength [19]. The predicted mean diffracted field strength is compared with that measured in Fig. 10. The onset of signal loss as the receiver passes behind the building is predicted, but the predictions become pessimistic when the receiver passes into a deep shadow. Note that the predicted diffraction field strength increases towards the end of the measurement track. This is due to the increasing strength of the diffraction contribution from around the far side of the building as the receiver nears that end of the building. Davidson Hall has sections with three different heights, and provided a successful test of the algorithms for modeling a more complex building. The measurements simulate a typical situation in the proposed microcellular system architecture: the transmitter was placed 15 ft high on the sidewalk, similar to the proposed lamp-post placement of low-power base stations, and the 6-ft-high receiving antenna simulates a pedestrian with a personal communication device. The two stations were on perpendicular sidewalks that ran along two sides of the building. The transmitter was placed 50 ft from the junction of the sidewalks; the receiver began measuring at the junction, where a line-of-sight existed, and then passed into the shadow of the building. At the corner the building is 31 ft high. The comparisons with predictions, plotted in Fig. 11, show a slow drop in the signal level at m on the track that matches well in shape but exhibits large prediction error that may be due to a 2-3-m error in locating the various elements. Again, though, as the predicted Free Space Field Measurements 8-60 LY.-. W ~ Predlctlons - - ~ - -_ ' j0 I -80' 3 " " " I ao ' I 100 Distance along track, m Fig. 11. Comparisons with 914-MHz measurements taken adjacent to Davidson Hall. The receiver passed down the sidewalk along one side of the building while the transmitter radiated from the top of a 15-ft-high pole around the corner of the building. diffraction field falls below a certain threshold, the measured signal power levels off at a higher power than the prediction. A similar result was reported by Rappaport and McGillem [20] on tests in the indoor factory environment. The explanation in that case appears reasonable here too: that signal is arriving by alternate paths through scattering or reflections by other elements of the environment. In both of our measurement runs, the mean of the measured signal never fell more than approximately 20 db below the free-space level, whereas the predicted diffraction field fell up to 35 db below free space. Experiments on land mobile satellite propagation [21] indicate that foliage and telephone poles typically scatter energy with about 12 to 20 db loss, with a nearly uniform distribution with azimuth. Our measurements were taken during summer, and in the vicinity there were both leafy foliage and tall tree trunks resembling telephone poles, so the evident floor on the attenuation of the received signal is readily

9 1648 IEEE TRANSACTIONS ON ANTEWAS AND PROPAGATION, VOL. 41, NO. 12. DECEMBER 1993 explained. Low-level reflections from distant buildings also are likely. IV. CONCLUSION An overall approach to prediction of diffraction by a specified assortment of buildings has been proposed as a prediction tool for a small number of buildings. Using simplified geometries for buildings, the three-dimensional profile of the radio path is surprisingly easy to access and to use as a basis of diffractive shadowing prediction, although there are a number of rules and algorithms required. The time requirement, dominated by the numerical multiple-diffraction procedures, should not be preclusive for system design requirements given a small number of shadowing buildings. In the prediction of local mean field strengths, samples can be spaced on the order of 5-10 wavelengths. A two-dimensional prediction grid for small areas thus appears feasible, particularly if a highperformance workstation is used as the computing platform. A graphics workstation would also aid in the input/output of the building information required for this site-specific approach. Unfortunately, the availability of building heights and locations may be a significant constraint at this time. The computational precision in sensitivity analyses was about k1 db. Up to 5 db of additional error may arise from the assumption that an edge can be neglected if it presents less than 55% Fresnel zone obstruction; a different criterion is thus desirable. It is clear that diffraction is only one process by which the signal can change direction to penetrate the shadow zone of an obstacle; reflections and other scattering mechanisms must be included in a comprehensive approach to field strength prediction. Absent these, the diffraction field is a pessimistic or worst-case estimate of the received field strength that approximates the received field best when the diffractive shadowing is not deep and strong specular reflections are not present. The measurement comparisons, though limited, appear to confirm this. The diffraction prediction models, however, are not conclusively validated in these comparisons. This is an early step in an ambitious new approach: further refinement and validation of both the approach and the models is warranted. APPENDIX The multiple-diffraction integral for perfectly absorbing finite knife edges given by (1) is derived from the following form of the Kirchhoff diffraction integral [14], expansion (no prior diffraction). and can be separated out of the integrand, leaving The first term of the integrand, E(Q), describes the propagation from S to Q, and the remaining terms, eikrp and l/rp, describe the phase shift and amplitude change, respectively, introduced between Q and P. The expression for the field at Q can be broken down as follows. We first make the assumption that the normal to the aperture plane makes a small angle with SP; this allows us to make the Fresnel phase approximation [141 and expand (AZ), eik(stx2 /2sty2 /2s) E(&) = r, (A41 When there is an intervening horizontal edge (parallel to the x axis), the character of the y dependence of the field at Q differs from that of the x dependence (Section 1I.A.). We separate them as E(Q> E ( )eik(x2/25) Q Y where, given no obstacle between S and Q, (A5 1 eik(s+y2 12s) EQ(y) E? (A6) rs but in the multiple-diffraction case, we substitute explicit evaluations of the field at Q as a function of y (at x = 0) for the relation (A6). The terms of the integrand in (A31 that represent the amplitude and phase change incurred between Q to P can be simplified using the standard amplitude and phase approximations [ 141, eikrp -e rp eik(p+x2 /2p+y2 /Zp) (A7) With the substitutions of (A51 and (A7) in (M), and rearrangement of terms, the solution for the field at P can be written P (AS) The double integral can now be separated into the product of two single integrals, and the integral over x may be written in the form of a Fresnel integral through a change of variable, yielding (11, where R is the aperture region and rs and rp are the distances from S and P, respectively, to a Huygens source point Q in the aperture. Assume for now that the field at a Huygens source Q is a function of free-space wave

10 RUSSELL et al.: PREDICTING MICROWAVE DIFFRACTION BY BUILDINGS FOR MICROCELLULAR SYSTEMS 1649 where REFERENCES Y. Okumura et al., Field strength and its variability in VHF and UHF land-mobile radio service, Reu. Elec. Commun. Lab., vol. 16, nos. 9 and 10, p , Sept.-Oct J. Walfisch and H. L. Bertoni, A theoretical model of UHF propagation in urban environments, IEEE Trans. Antennas Propagat., vol. 36, no. 12, p , Dec F. Ikegami et al., Theoretical prediction of mean field strength for urban mobile radio, IEEE Truns. Antennus Propugat., vol. 39, no. 3, pp , Mar A. Ranade, Local access radio interference due to building reflections, IEEE Trans. Commun., vol. 37, no. 1, pp , Jan K. R. Schaubach, N. J. Davis, and T. S. Rappaport, A ray tracing method for predicting path loss and delay spread in microcellular environments, in Proc. 42nd IEEE Veh. Technol. Conf., Denver, CO, May 1992, pp D. C. Cox, A radio system proposal for widesprcad low-power tetherless communications, IEEE Trans. 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L. Stutzman, Modeling and simulation of mobile satellite promgation, IEEE Trans. Antennas f ropanat.,.- Thomas A. Russell (S 83-M 85-S 90-M9 1) was horn in Camden, NJ, in He received the B.S.E.E. degree from the University of Virginia, Charlottesville, in 1986, and the M.S.E.E. degree from the Virginia Polytechnic Institute and State University, Blacksburg, in Since 1986 he has been employed with Stanford Telecommunications, Inc. in Reston, VA, where he is involved in propagation modeling, link simulation, and systems architecture studies for microwave satellite communications. Between August 1990 and December 1991 he was a Research Assistant with the Mobile and Portable Radio Research Group where his area of research was the prediction of diffraction by buildings. Charles W. Bostian (S 67-M 67-SM 77-F 92) was born in Chambersburg, PA, on December 30, He received the B.S., M.S., and Ph.D. degrees in Electrical engineering from North Carolina State University, Raleigh, in 1963, 1964, and 1967, respectively. After a short period as a Research Engineer with Corning Glassworks and a tour of duty in the U.S. Army, he joined the Faculty of Virginia Polytechnic Institute and State University. Blacksburp. in 1969 and is currently Clayton Ayre Professor of Electrical Engkeering. His primary interests are in propagation and satellite communications. Dr. Bostian was a 1989 IEEE Congressional Fellow with Representative Don Ritter. He is chair of the IEEE-USA Engineering R & D Policy Committee and serves on the IEEE-USA Technology Policy Council and the Congressional Fellows Committee. He is Associate Editor for Propagation of IEEE Transactions on Antennas and Propagation. In his off-duty hours, he is a performing folk musician, playing hammered dulcmer with the group Simple Gifts. Theodore S. Rappaport (S 83-M 84- S 85-M787-SM 90) was born in Brooklyn, NY on November 26,1960. He received the B.S.E.E., M.S.E.E., and Ph.D. degrees from Purdue University in 1982, 1984, and 1987, respectively. In 1988, he joined the Electrical Engineering faculty of Virginia Tech, Blacksburg, where he is an Associate Professor and Director of the Mobile and Portable Radio Research Group. He conducts research in mobile radio communica- tion system design and RF proparation -._ prediction through measurements -and modeling. He guides a number of graduate and undergraduate students in mobile radio communications, and has authored or co-authored more than 70 technical papers in the arcas of mobile radio communications and propagation, vehicular navigation, ionospheric propagation, and wideband communications. He holds a US. patent for a wide-band antenna and is co-inventor of SIRCIM, an indoor radio channel simulator that has been adopted by over SO companies and universities. In 1990, Dr. Rappaport received the Marconi Young Scientist Award for his contributions in indoor radio communications, and was named a National Science Foundation Presidential Faculty Fellow in He is an active member of the IEEE, and serves as senior editor of the IEEE Joumal on Selected Areas in Communications. He is a Registered Professional Engineer in the State of Virginia and is a Fellow of the Radio Club of America. He is also Dresident of TSR Technologies, a cellular vol. 40, no. 4, pp: , April radio and paging test equipment manufacturer.