SCATTERING OF ELECTROMAGNETIC WAVES FROM A RECTANGULAR PLATE USING AN EXTENDED STATIONARY PHASE METHOD BASED ON FRESNEL FUNCTIONS (SPM-F)

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1 Progress In Electromagnetics Research, Vol. 107, 63 99, 010 SCATTERING OF ELECTROMAGNETIC WAVES FROM A RECTANGULAR PLATE USING AN EXTENDED STATIONARY PHASE METHOD BASED ON FRESNEL FUNCTIONS (SPM-F) C. G. Moschovitis Division of Information Transmission Systems an Materials Technology School of Electrical an Computer Engineering National Technical University of Athens 9 Iroon Polytechniou Str., Zografou, Athens GR-15780, Greece H. T. Anastassiu Hellenic Aerospace Inustry P. O. Box 3 GR-3009, Schimatari, Tanagra/Viotia, Greece P. V. Frangos Division of Information Transmission Systems an Materials Technology School of Electrical an Computer Engineering National Technical University of Athens 9 Iroon Polytechniou Str., Zografou, Athens GR-15780, Greece Abstract This paper presents an extension over a novel, three imensional high frequency metho for the calculation of the scattere electromagnetic (EM) fiel from a Perfect Electric Conuctor (PEC) plate, which is base on the Physical Optics (PO) approximation an the Stationary Phase Metho (SPM). This extension efines a new analytical metho which is prove to be very efficient in computer execution time an enhances the accuracy of its preecessor aroun the area of the main scattering lobe. This new analytical metho accomplishes high accuracy through the use of higher orer approximation terms, which imply the use of Fresnel functions (SPM- F metho). By using higher orer Fresnel approximation terms, no impact on the time efficiency of the SPM metho appears to Receive 1 April 010, Accepte 0 July 010, Scheule 1 August 010 Corresponing author: C. G. Moschovitis, (harism@noc.ntua.gr).

2 64 Moschovitis, Anastassiu, an Frangos occur, since the extene SPM-F metho just removes the troublesome vanishing enominators when the stationary point coincies with the eges of the scatterer. The SPM-F results are compare to other straightforwar numerical an exact solution methos for the same problem in the far fiel, Fresnel zone an the near fiel area of the scatterer. 1. INTRODUCTION In a recent publication [1], we have evelope an presente a novel three-imensional (3-D) high frequency analytical metho for the calculation of the vector potential A, an eventually the scattere electromagnetic fiel (EM) from a Perfect Electric Conuctor (PEC) plate geometry, which is presente in Fig. 1. This metho is base on Physical Optics (PO) approximation an Stationary Phase Metho (SPM) three imensional calculations. Comparisons with numerical integration en exact solution results yiele excellent agreement, except for the main scattering lobe. This iscrepancy is ue to the fact that the enominator of the first orer analytical approximation vanishes when the stationary point coincies with the en point locate at the eges of the scatterer, i.e., the rectangular plate. In this paper, we present a higher orer approximation term for the ege contribution, which is base again on the Stationary Phase Metho, but also inclues Fresnel functions, hence the name (SPM-F). Using Fresnel functions, the accuracy is improve an the newly erive metho lacs any singularities/inaccuracies in the vicinity of the main scattering lobe encountere in earlier publishe SPM results [1]. Vector potential A an scattere electric fiel E expressions are erive in the sections below, accoring to the moifie version of the metho that was presente in [1], but now all calculations inclue the new Fresnel terms. Fig. 1, below, shows the geometry of scattering of an electromagnetic (EM) wave with incient wavevector i from a perfect electric conuctor (PEC) rectangular plate. The scattere fiel is calculate at an arbitrary observation point R(x, y, z). Possible applications of the extene 3D-SPM-F metho inclue common raio-propagation tools in urban outoor environment, where D [] an 3D (uner evelopment by our research group) scattering an iffraction calculations tae place, incluing both first an secon orer scattering phenomena. In such microcellular geometries in D [], both transmitter an receiver are consiere to be well below rooftop level an well above the groun. A -D approach leas to fiel calculation by calculating single integrals, while a 3-D approach more complicate formulation presente in next sections

3 Progress In Electromagnetics Research, Vol. 107, Figure 1. Scattering from a PEC rectangular plate of imensions a b an c. of this publication requires the calculation of ouble integrals. Both single an ouble integral calculations may be achieve by using the appropriate Stationary Phase Metho with Fresnel functions presente here. The calculations are accurate an fast enough to avoi long simulation times, which constitute one of the most significant rawbacs that occur in raio-propagation tools, which often pertain to moern high frequency communication wireless networs. Then the scattering of EM waves from builing walls in the actual 3D raio coverage problem in urban outoor environment can be formulate through the metho presente in this paper. Other numerical an efficient methos for the same scattering problem for a rectangular plate have also been introuce by our research group [3]. The inclusion of Fresnel functions in three imensional SPM metho with analytical results for the calculation of EM wave scattering from rectangular plate geometries has not been ocumente in literature [4 15] an appears to be very useful for calculating the path loss in urban outoor environments [16]. In this paper (Section 5), three imensional SPM-F simulation results are compare to corresponing results obtaine in the Near\Fresnel-zone\Far fiel areas with other methos. These results valiate the excellent accuracy of the propose methoology with stanar numerical integration,

4 66 Moschovitis, Anastassiu, an Frangos 3D-SPM [1] calculation an Bounary Integral (BI) Multilevel Fast Multipole Metho (MLFMM) full wave Exact Solution results (ES) [17 3]. 3D-SPM-F metho presente here proves to be more accurate an equally fast in computation times when compare to 3D-SPM [1]. Finally, the finite conuctivity of the rectangular plate is examine in etail in Appenix D since walls, roas an roofs in the actual 3D raiocoverage problem, which eventually are moele as segments an plates [], are highly non-pec surfaces.. DEFINITION OF THE ELECTROMAGNETIC SCATTERING PROBLEM/APPROXIMATIONS AND CALCULATIONS.1. Electromagnetic Layout An electromagnetic wave is consiere to be incient in the irection of wave vector i upon a PEC rectangular plate of negligible thicness, accoring to the layout in Fig. 1. Accoring to Fig. 1, the plate vertices lie at the intersections of the lines x = a, b an y = c,. A pair of angles, namely θ i (TÔz) an ϕ i efines the irection of incience. Assuming that transmitter T lies at a point z > 0 far from the scattering plate, our first goal is the calculation of the vector potential A at an arbitrary observation point R(x, y, z) with z > 0, an subsequently, the calculation of the electric fiel E at the same observation point, which lies in the near, intermeiate (Fresnel) or the far (Fraunhofer) fiel region of the scatterer. The time epenence of the electric fiel is proportional to the factor exp(+jωt). In the cases of GSM, UMTS, Wi-Fi or Wi-Max networ technologies, the operating frequency is about 1 GHz or higher. At that frequency, scatterers such as builings are consiere to be electrically large, an current ensity inuce on the scatterer may be calculate with goo accuracy using the Physical Optics (PO) approximation. Then, by using appropriate three-imensional Green s function, the vector potential calculation leas us to an integral expression, which is in the form of an amplitue function multiplie by a complex exponential, i.e., the phase function. In this equation, it is possible to apply our propose SPM-F metho in orer to calculate the vector potential, an eventually the electric fiel. This two imensional scatterer is a necessary prerequisite in orer to moel propagation in an urban outoor environment, which consists of three imensional walls an other scatterers.

5 Progress In Electromagnetics Research, Vol. 107, Current Density Calculation Using Physical Optics (PO) Approximation Vector Potential Calculation Electric Fiel Calculation The well-nown current ensity inuce on the surface of the PEC rectangular plate ue to an incient plane wave with wavevector i, accoring to the PO approximation is presente in [, 4]. Then, by referring to [1] an by applying the PO current ensity approximation J s on the rectangular plate mentione above, the vector potential A(x, y, z) at an arbitrary observation point R(x, y, z) with z > 0 results in the following equation: A(x, y, z) = µ 0 πη [ˆx(E 0θ cos φ i E 0φ cos θ i sin φ i ) +ŷ (E 0θ sin φ i + E 0φ cos θ i cos φ i )] c a y = x =b e j(x K+y L) e j (x x ) +(y y ) +z (x x ) + (y y ) + z x y (1) where µ o : magnetic permeability of the vacuum (µ o = 4π 10 7 H/m); r = (x, y, z): position vector of observation; r = (x, y ): position vector of source current; η = µ 0 /ε 0 = free space impeance, = π/λ is the wavenumber, an K, L are constants which epen on the angles of incience (θ i, ϕ i ) [4]. In [1], the vector potential A of Eq. (1) has alreay been compute with three ifferent ways, i.e., SPM, stanar numerical integration (NI) in two imensions an exact solution (ES) [17 3]. SPM-F metho an results follow in sections below. The calculation of the scattere electric fiel E follows the calculation of vector potential A accoring to formula (A1). 3. STATIONARY PHASE METHOD WITH FRESNEL FUNCTIONS CALCULATIONS (SPM-F) The main contribution terms in stationary phase metho approximations are presente in Eqs. () (7), below. The core algorithm [1] is now explaine in etail an the ege point contribution terms will be change in orer to inclue the newly introuce higher orer approximation terms with Fresnel functions. Initially we calculate asymptotically the efinite Double Integral (DI) of the form: I () = c a b F (x, y)e jf(x,y) xy ()

6 68 Moschovitis, Anastassiu, an Frangos After the asymptotic calculations [5] tae place, the following expression is obtaine for Eq. (): I () F (x s, y s ) e jf(x s,y s ) jπδ 4AB C where the en-points contribution terms are neglecte here, an: : real wavenumber, relatively high; (x s, y s ): Stationary point of function f(x, y), i.e., the point where the partial erivatives of the phase function f(x, y) with respect to x an y vanish. For frequency f = 1 GHz, = πf/c 1 m 1, a value which for the scatterer s maximum imension uner consieration in this paper can be consiere relatively high for Stationary Phase Calculations [5]. In other wors, for the numerical implementations of Section 5, the electrical length of the rectangular scatterer s maximum imension (iagonal of the plate) is L max = 40π 1. Phase function f(x, y): Slowly varying, real, non-singular function, inepenent of. Amplitue function F (x, y): non-singular function, may be complex [5], shoul also be inepenent of. In the present wor, F (x, y) is a real-value function. a, b, c, : limits of the ouble integral. Here it shoul be emphasize that the stationary point must be place within the surface bounaries, i.e., b < x s < a, < y s < c. A, B, C: Constants relate to the secon erivatives of the phase function f(x, y), accoring to the efinitions: (3) A = 1 f xx (x s, y s ) (4) B = 1 f yy (x s, y s ) (5) C = f xy (x s, y s ) (6) δ: the value of δ is etermine from the relative values of the constants A, B an C: +1, 4AB > C, A > 0 δ = 1, 4AB > C, A < 0 (7) j, 4AB < C To emphasize a previous statement, Eq. (3) is use only in the case when the stationary point lies between the finite limits of the plate, otherwise the SPM contribution vanishes, uner the present consierations. The SPM calculation term enote in Eq. (3) is not taing into account the contribution from the finite limits of the plate, (a, b,

7 Progress In Electromagnetics Research, Vol. 107, c, ). In [6], an for the case of single integral calculations (SI), the term over the interval (a, b) is extene to the inefinite interval (, + ) an correction terms relate to the ege contributions a, b are subtracte. The core algorithm of our propose metho is to generalize this metho [6] for the case of Double Integrals (DI) by aing ege calculation terms base on the Fresnel function (see Section 3., below). In Section 3.1, below, we summarize this metho [6] for the case of a Single Integral (SI). Furthermore, in Section 3., we apply initially the equations of the single integral to appropriate integran functions. Afterwars, by efining appropriate new integran functions, we apply the SPM metho regaring the secon integration of the ouble integral, always paying attention to inclue the secon orer Fresnel function ege approximation terms. The final result is nine integration terms, eight of which tae into account the Fresnel function ege terms, as explaine in Section 3. below, resulting in a new analytical calculation of the ouble integral in a novel close form (to our nowlege not ocumente to ate in the literature [4 15]). This novel form uses nine terms of stationary phase, which contain the total information about the finite limits of the plate, (a, b, c, ). The complexity of the corresponing mathematical expressions (elaborate analytical formulas), an the complicate calculations of Fresnel values lea us to implement analytically this metho in a stanar MATLAB computational pacage, both for the case of single an ouble integral. As it will be explaine in Section 6, below, Stationary Phase Metho analytical calculations, even with ege calculation terms base on Fresnel functions, are very fast in terms of actual computation time. All relative mathematical formulas are escribe below SPM-F Single Integral Calculations In the case of the Single Integral (SI), the proceure for the calculation of the limits contribution to the integral is presente in [6]. In this case, the SPM integral is efine by: I = a b F (x) exp (jf (x)) x (8)

8 70 Moschovitis, Anastassiu, an Frangos an it can be written subsequently as: + I = F (x) exp (jf (x)) x + b + a + F ( x) exp (jf ( x)) x = F (x) exp (jf (x)) x a F (x) exp (jf (x)) x + b + F (x) exp (jf (x)) x F F (x) exp (jff (x)) x = I o I a I b (9) where functions FF an ff are efine from equations: F F (x) = F ( x) (10) ff (x) = f ( x) (11) I o term is calculate accoring to [6] or equivalently accoring to [5], as: I o = π f (x o ) F (x o) exp ( [ j f (x o ) + π 4 sgn { f (x o ) }]) (1) where x o enotes the stationary point for which f (x o ) = 0. For the other two terms, higher orer approximation terms using Fresnel functions are taen into account: I q = ε q F (q) exp { jf (q) ju } q f x F r ± [u q ] (13) x =q where q = a, b an ε q = sgn (q x o ) (14) u q = f f x x (15) x x =q =q F r ± (u q ) = ( exp ±jt ) t (16) u q where the combination (exp { jf (q) ju q}, F r+ [u q ]) is use ( ) in Eq. (13) when f x 0, an the combination x =q

9 Progress In Electromagnetics Research, Vol. 107, (exp { jf (q) + ju q}, F r [u q ]) in the same equation is use when ( ) f x < 0. Accoring to notation (see also Section 5 below), x =q secon orer approximation terms with Fresnel functions are vali when parameters u q 3.0 [6], otherwise simple SPM calculations [1] apply. Using the above formulas (9), (1), (13) (16) of Stationary Phase Metho with Fresnel functions (SPM-F), we have presente a complete approximate formula for the asymptotic calculation of the Single Integral (SI) using Fresnel functions (SPM-F). 3.. SPM-F Double Integral Calculations In this section, we will erive the close-form analytical formulas for the calculation of the Double Integral [DI, Eq. ()] accoring to the newly propose enhance analytical SPM-F metho. Our final results will be inclue in Eqs. (0), (A8), (A9), (4), (9) an (39) below. Integrals of the form of Eq. () are frequently encountere uring the proceure of calculating the vector potential A in common electromagnetic problems. Assuming an incient plane wave which is scattere by a perfect conucting plate, we calculate the vector potential A using SPM-F approximations, an subsequently, the electric fiel E at an arbitrary observation point R(x, y, z). For the calculation of the vector potential, we assume current ensity istribution accoring to the Physical Optics (PO) approximation, which is sufficiently accurate for electrically large scatterers (i.e., L max 1, where L max is the maximum imension of the scatterer). This result is important, because it provies a very fast metho of calculation, as compare to stanar numerical integration techniques. Furthermore, it provies an analytical (instea of numerical) metho for the calculation of the scattere electric fiel. Extening these results, we may obtain an expression for the preiction of path losses in an electromagnetic propagation scenario. We rewrite Eq. () with slightly moifie variables so that it complies with the form of Eq. (8) above, as follows: I (; x, y, z) = c a y = x =b F ( x, y, z, x, y ) e jf(x,y,z,x,y ) x y (17) An alternative, far more complicate way for calculating Eq. (17) is by using an appropriate combination of rotation an change of variables in the ouble integral, as escribe in brief in [6], a metho that we intentionally avoie here see Appenix B.

10 7 Moschovitis, Anastassiu, an Frangos Equation (17) refers to functions F an f, which are functions of two variables, so we have to calculate a ouble integral, an the algebraic expressions must be moifie properly. Using Eq. (9), Eq. (17) is calculate in two steps, in orer to tae into account the contribution of all four limits of the plate (a, b, c, ). The first step is relate to the calculation of the inner (single) integral of Eq. (17), an it is escribe by Eqs. (18), (19) an (A) (A4) below: Ĩ q = sgn (q x o ) F ( q, y ) exp { jf ( q, y ) j(u q (q, y )) } [ ( f x F r ± uq q, y )] (18) x =q,y =y ( u q q, y ) = f x x (19) x =q,y =y x =q,y =y f where q = a, b an ( an ±) combination varies accoring to Section 3.1 an x o = x o (y ) is the curve etermine by equating the first partial erivative of the phase function f(x, y ) with respect to x with zero, replacing the variables x = x o an y = y, an solving with respect to x o as explaine in Eq. (A5) Area Division Approach Equation (17) is extene properly, to inclue calculations involving Eqs. (18), (19), (A) (A4). Accoring to our propose first approach, the omain surrouning the rectangular patch is being ivie into four separate regions (regions 1 4, while region 0 correspons to the whole x-y plane) accoring to the pattern in Fig.. Accoring to Fig., Eq. (17) may be rewritten as in Eq. (A6) or, equivalently as in Eq. (A7) or (0). I (; x, y, z) = I 0 I 1 I I 3 I 4 (0) Term I[0] = I 0 is calculate accoring to Eq. (3), using the notation of Eq. (1) [where Eq. (1) has been written for the Single Integral case] see also Eq. (A8). Equations (3) an (A8) are equivalent expressions, in a slightly ifferent form, but leaing to the same result. Regaring the calculation of integral I[1] = I 1, this is calculate in the following two steps: First the inner integral (from x = a to x = + ) is calculate accoring to Eq. (18). Then by efining

11 Progress In Electromagnetics Research, Vol. 107, Figure. Eq. (17). Ranges of integration for the calculation of integral of new appropriate amplitue an phase functions (F 0 an f 0 ) by the following equations: F (0) [ ( f x F r ± ua y )] (1) x =a,y =y ( u a y ) = f f x x x =a,y =y x =a,y =y () ( ) f (0) ( y ) =f ( f x a, y ) x =a,y =y x x =a,y =y (3) f the outer integration (for < y < + ) is performe again using the SPM of Eq. (A4). The final result of this proceure is obtaine by Eq. (A9), where y 0 is a moifie stationary point efine by Eq. (A10). In the same way integral I[] = I is calculate. The final result

12 74 Moschovitis, Anastassiu, an Frangos is: + x =b I = F ( x, y ) e jf(x,y ) x y exp j f (03) (y 03) + π 4 sgn f (03) y π y F (03) (y 03) y =y 03 (4) f (03) y =y 03 where functions F 03 an f 03 are efine by equations: F (03) [ ( f x F r ± ub y )] (5) x =b,y =y ( u b y ) = f f x x x =b,y =y x =b,y =y (6) ( ) f (03) ( y ) =f ( f x b, y ) x =b,y =y x x =b,y =y (7) f an y 03 is a moifie stationary point efine by: f (03) (y ) y = 0 (8) y =y 03 Regaring the calculation of integral I[3] = I 3 in Eqs. (A7), (0), this is also calculate in two steps as follows: First the inner integral (for b x a) is calculate using the SPM-F formulation through Eqs. (18), (19), (A) (A4), an subsequently the outer integration (for < y ) is calculate through Eq. (18). Then the final result is given below Eqs. (9) (35): I 3 = y = a x =b F ( x, y ) e jf(x,y ) x y = y = ( I 0 I a I b) y = I 3.1 I 3. I 3.3 (9)

13 Progress In Electromagnetics Research, Vol. 107, where integrals I[3.1], I[3.] an I[3.3] are calculate as: y = { I 3.1 = I oy = sgn ( y 01 )F(01) () exp jf (01) () j(u(1) ())} [ f (01) y F r ± y = f () = (01) f (01) y y y = u (1) I 3. = y = u (1) y = ] () (30) (31) { I ay = sgn ( y 0 )F(0) () exp jf (0) () j(u() ())} [ f (0) y F r ± y = f () = (0) f (0) y y y = u () I 3.3 = y = u () y = ] () I b y = sgn ( y 03 )F (03) () exp { jf (03) [ f (03) y F r ± y = f () = (03) f (03) y y y = u (3) u (3) y = ] () () j(u(3) (3) (33) ())} (34) (35)

14 76 Moschovitis, Anastassiu, an Frangos where functions F 01 an f 01 are efine by the following equations: F (01) ( y ) π ( = f x F xo, y ) x =x o,y =y { exp (j π4 sgn }) f (36) x x =x o,y =y f (01) ( y ) = f ( x o, y ) (37) y 01 is a moifie stationary point efine by: f (01) (y ) y = 0 (38) y =y 01 an functions F 0, f 0, F 03 an f 03 have been efine previously in Eqs. (1), (3) an (5), (7). Finally, integral I[4] = I 4 in Eqs. (A7), (0) is calculate in the same way with integral I[3] an the final result is given by Eqs. (39) (45) below: I 4 = + a y =c x =b F ( x, y ) e jf(x,y ) x y = + y =c ( I o I a I b) y = I 4.1 I 4. I 4.3 (39) + { I 4.1 = I oy = sgn (c y 01 )F(01) (c) exp jf (01) (c) j(u(1) c (c)) } y =c [ f (01) y F r ± y =c f c (c) = (01) f (01) y y y =c u (1) u (1) c y =c ] (c) (40) (41)

15 Progress In Electromagnetics Research, Vol. 107, I 4. = + y =c { I ay = sgn (c y 0 )F(0) (c) exp jf (0) (c) j(u() c (c)) } [ f (0) y F r ± y =c f c (c) = (0) f (0) y y y =c + I 4.3 = u () y =c u () c y =c ] (c) (4) (43) { I b y = sgn (c y 03 ) F (03) (c) exp jf (03) (c) j(u(3) c (c)) } [ f (03) y F r ± y =c f c (c) = (03) f (03) y y y =c u (3) u (3) c y =c ] (c) (44) (45) Regaring the above calculations of integral I(; x, y, z) in Eq. (A7), we note here that the moifie stationary points y 01, y 0, y 03 an x 0 will be inclue in the final result, only if y 01, y 0, y 03 (, c) an x 0 (b, a). In any other case, the contribution regaring these integral terms is zero. Also, note here that the stationary point (x s, y s ) of Eq. (3) coincies with the stationary point (x o, y 01 ) of Eqs. (A5), (38). Before proceeing to analytical calculations, we note that for large angles of scattering parameters u a, u b, u (1) c, u () c, u (3) c, u (1), u(), u(3) in Eqs. (19), (), (6), (31), (33), (35), (41), (43) an (45) are larger than 3.0 (u a, u b, u (1) c, u () c, u (3) c, u (1), u(), u(3) > 3.0) thus the replacement formulas in Eqs. (18) are not uniform accoring to [6], an equations without Fresnel functions [1] must be use for best accuracy (i.e., combination formulas). Combination formulas are easy to accomplish with an aitional control variable, being able to switch simulation results to appropriate formulation between SPM-F an SPM for large

16 78 Moschovitis, Anastassiu, an Frangos scattering angles. A secon metho of calculation of the ouble integral of Eq. (6), which iffers slightly from the metho escribe above (an which finally is not use in the numerical simulations provie in Section 5 below), is presente in Appenix B. A comparison of these two methos is provie in Appenix C. 4. VECTOR POTENTIAL CALCULATIONS USING SPM In orer to calculate the vector potential A from Eq. (1) by using the SPM-F metho escribe in Section 3 above, we first efine the amplitue F (x, y) an phase functions f(x, y) by: F (x, y ) = 1 (x x ) + (y y ) + z (46) f(x, y ) = x K + y L (x x ) + (y y ) + z (47) Note here that the functions F, f satisfy the appropriate conitions for SPM properties, given above (below Eq. (3)). Subsequently, accoring to the propose SPM-F metho, in Sections 3. an 3.3, the application requires various proceures, such as calculation of the partial erivatives, the solution of complicate systems of equations, computation of aitional coefficients an implementation of various computational proceures. All the above are calculate by using the symbolic toolbox of MATLAB computational pacage. First, we calculate the stationary point for the function f(x, y ) by: f x f x=xs x (x s, y s ) = 0 (48) y=y s f y x=xs y=y s f y (x s, y s ) = 0 (49) By substituting function f(x, y ) from Eq. (47), we obtain the following system of equations: x x K + s = 0 (x xs ) +(y y s ) +z y y L + s = 0 (50) (x xs ) +(y y s ) +z The solution of the above system yiels the stationary point (x s, y s ) of functionf(x, y ). Note that for the EM scattering problem uner consieration, one stationary point can exist at maximum,

17 Progress In Electromagnetics Research, Vol. 107, as expecte from physical intuition, which correspons to specular scattering. The solution is foun by using the MATLAB computational pacage, thus yieling the following result: ( ) K z (x s, y s ) = x + 1 K L, y + L z (51) 1 K L Having efine the stationary point (x s, y s ), we chec whether this point is locate within the imensions of the plate. This means both x s an y s must lie within (b, a) an (, c) respectively. We continue the application of the metho by calculating the A, B, C parameters, by using Eqs. (4), (5) an (6): y yy s + ys + z A = (x xx s + x s + y yy s + ys + z ) 3 x xx s + x s + z B = (x xx s + x s + y yy s + ys + z ) 3 C = (x x s ) (y y s ) (x xx s + x s + y yy s + y s + z ) 3 (5) (53) (54) Substituting the values of the stationary point (x s, y s ) from Eq. (51) to Eqs. (46) (47) we obtain: 1 F (x s, y s ) = ( ) ( ) (55) K z 1 K L + L z 1 K L + z ( ) ( ) K z L z f(x s, y s ) = x + K + y + L 1 K L 1 K L ( ) K z ( ) L z + + z (56) 1 K L 1 K L Subsequently, through the efinition of Eqs. (46) (47), the DI SPM metho escribe in Sections 3. an 3.3 above is use to calculate the vector potential A, Eq. (1), where the contribution from the plate eges (bounaries a, b, c, ) are also taen into account. The DI of the vector potential A, Eq. (1), is therefore calculate accoring to the efinition of F, f by Eqs. (46) (47), formula (3), an Eqs. (0) (45) an (A8) (A10). We may also note here that the results of Eqs. (3) an (A8) are completely ientical. Also note that Eqs. (A9), (4), (30), (3), (34), (40), (4), (44) refer to the contribution of the vertices (a, b, c, ) of the rectangular plate.

18 80 Moschovitis, Anastassiu, an Frangos Finally, the two components A x an A y of the vector potential A are obtaine accoring to Eq. (1) by: A x = µ 0 πη [E 0θ cos φ i E 0φ cos θ i sin φ i ] F (x s, y s ) e jf(xs,ys) jπδ 4AB C A y = µ 0 πη [E 0θ sin φ i +E 0φ cos θ i cos φ i ] F (x s, y s ) e jf(x s,y s ) jπδ 4AB C (57) (58) Having calculate the vector potential A, we procee to calculate the scattere electric fiel E as in [1]. 5. NUMERICAL RESULTS USING MATLAB Regaring the numerical implementation, finally yieling the vector potential an scattere electric fiel for the geometry of Fig. 1, the proceure is as follows: the vector potential A is calculate by Eq. (1), where the corresponing integration is calculate through the methos of Sections 3 an 4. Subsequently, regaring the calculation of the three components of gra (iva) in the formulae of the scattere electric fiel [1], this can be performe in MATLAB environment in two ifferent ways. In the first way, function A(x, y, z) is given in analytical form an the appropriate ifferentiations are calculate analytically in MATLAB environment. In the secon way, the appropriate ifferentiations for calculating gra (iva) are performe in MATLAB environment using stanar numerical ifferentiation techniques. We have implemente both these techniques in our simulations an excellent agreement was observe. Furthermore, in our numerical simulations we compare results base on our propose 3D analytical SPM-F metho with 3D analytical SPM metho [1], numerical integration technique using Gaussian quarature [] an Finite Element Bounary Integral (FEBI) fullwave exact solution (see Ref s. [17 3]). In the case of the numerical integration technique, a numerical integration tolerance error level of 10 6 is specifie, which is of satisfactory accuracy an leas to acceptable run-time in our simulations. With woring frequency set equal to 1 GHz, the simulation parameters, accoring to the geometry of Fig. 1 are provie in Table 1:

19 Progress In Electromagnetics Research, Vol. 107, Table 1. Double integral simulation parameters. Symbol Quantity Value λ Wavelength 0.3 m r Observation istance Near Fiel area 1, Fresnel area 1, Far Fiel area 1 ϕ i Angle of incience 5 θ i Angle of incience 45 ϕ s Observation angle 45 θ s Observation angle 0 90 E 0θ Constant relate to emitte fiel (θ i ) 1 V/m E 0ϕ Constant relate to emitte fiel (ϕ i ) 1 V/m a b x axis plate imension 0λ, 40λ, 60λ, 80λ c y axis plate imension 0λ, 40λ, 60λ, 80λ 1 As calculate from the corresponing Fresnel an Far Fiel area istances, given by well nown expressions as in Ref. []. In Figs. 3, 4 an 5, numerical results are isplaye for a plate of imensions 0λ 0λ for the far fiel, Fresnel zone an near fiel areas respectively. As expecte for all these cases, the maximum value for the main scattering lobe is observe for angle of observation θ s equal to angle of incience θ i. Also we note that in the far fiel area the number of the sie lobes is proportional to the electrical size (L) of the plate (where is the wavenumber = π/λ, an L = a b = c ) Far Fiel Area Results Far Fiel Approximation For the Far Fiel Area, calculations may be further simplifie accoring to the following approximations: r r = (x x ) + (y y ) + z = x + y + z = r (59) for the amplitue factor an r r = (x x ) + (y y ) + z = r xx + yy r for the phase factor. (60)

20 8 Moschovitis, Anastassiu, an Frangos (a) Vector Potential, A ( ) θs (b) Total Electric Fiel, Etot ( ) θ s Figure 3. 1 GHz simulation results for plate of imensions 0λ 0λ an observation istance 600 m (Far fiel area). (a) Magnitue of vector potential A as a function of observation angle θ s. (b) Magnitue of total electric fiel E as a function of observation angle θ s.

21 Progress In Electromagnetics Research, Vol. 107, (a) Vector Potential, A ( ) θs (b) Total Electric Fiel, Etot ( ) θ s Figure 4. 1 GHz simulation results for plate of imensions 0λ 0λ an observation istance 100 m (Fresnel area). (a) Magnitue of vector potential A as a function of observation angle θ s. (b) Magnitue of total electric fiel E as a function of observation angle θ s.

22 84 Moschovitis, Anastassiu, an Frangos (a) Vector Potential, A ( ) θs (b) Total Electric Fiel, Etot ( ) θ s Figure 5. 1 GHz simulation results for plate of imensions 0λ 0λ an observation istance 5 m (Near fiel area). (a) Magnitue of vector potential A as a function of observation angle θ s. (b) Magnitue of total electric fiel E as a function of observation angle θ s.

23 Progress In Electromagnetics Research, Vol. 107, Since variables x an y are calculate from x r = sin ϑ s cos ϕ s (61) y r = sin ϑ s sin ϕ s (6) the integral of Eq. (17) is transforme to I = e jr r c y = a x =b e jx (sin ϑ i cos ϕ i +sin ϑ s cos ϕ s ) x e jy (sin ϑ i sin ϕ i +sin ϑ s sin ϕ s ) y (63) [( ) I = e jr (a b) (a b) (c ) sinc r [( (c ) sinc ) (sin ϑ i sin ϕ i + sin ϑ s sin ϕ s ) ] (sin ϑ i cos ϕ i +sin ϑ s cos ϕ s ) ] (64) Furthermore, the vector potential A, an electric fiel E will be simplifie to the following expressions: e jr A (x, y, z) = µ 0 (a b) (c ) {ˆx (E 0ϑ cos ϕ i E 0ϕ cos ϑ i sin ϕ i ) πη r +ŷ (E 0ϑ sin ϕ i + E 0ϕ cos ϑ i cos ϕ i )} [( ) ] (a b) sinc (sin ϑ i cos ϕ i + sin ϑ s cos ϕ s ) [( ) ] (c ) sinc (sin ϑ i sin ϕ i + sin ϑ s sin ϕ s ) (65) E (x, y, z) = jωa (x, y, z) (66) Far Fiel Numerical Results an Comments From the above results (Fig. 3), it can be easily conclue that the propose 3D analytical SPM-F metho yiels results of excellent accuracy for all angles of observation. Also the errors of respective results in SPM metho [1] are significantly reuce from 4 B [1] to a maximum of B here. Also we note that any errors that exist in SPM-F metho are reuce in the case of a larger scatterer, i.e, 80λ 80λ scatterer (not shown here), as expecte once SPM-F is a high frequency asymptotic metho. Finally, an error of about 30 B, which appears at several observation angles between the stanar numerical

24 86 Moschovitis, Anastassiu, an Frangos integration metho an the analytical sinc function result, is explaine ue to the highly oscillatory behavior of the inuce physical optics currents on the surface of the rectangular plate. 5.. Fresnel Area Results an Comments In this case (Fig. 4), we observe results of very satisfactory accuracy. Errors of about 4 B [1] were reuce significantly to about 0.1 B. Any errors will be reuce in the case when the scatterer is electrically larger (not shown here/see comment at Section 5.1. above) Near Fiel Area Results an Comments In the case of near fiel observations, the results (Fig. 5) are foun to be rather accurate, since errors of 15 B in [1] were reuce to 4 7 B at maximum here. By physical intuition, we expect a larger numerical error at the near fiel area, since it appears that for this case the fiel contribution from only one stationary point an the iffracting eges of the scattering plate is not enough for very accurate calculations. Once again, these errors can be correcte when the scatterer is electrically larger (not shown here/see comment at Section 5.1. above). Comparison graphs in Figs. 3 5 above inicate the improve behavior aroun the area of the main scattering lobe between the asymptotic results of the SPM-F an SPM [1] methos. 6. RUN-TIME CONSIDERATIONS OF THE PROPOSED ANALYTICAL METHOD Table below compares the run-time neee for all four methos (SPM-F, SPM [1], Numerical Integration NI [] an Exact Solution ES [17 3]) for the cases of 0λ an 80λ plate scatterers on a stanar estop personal computer with Pentium M GHz processor an 1 GB of RAM memory. ES results of a full-wave exact metho [Bounary Integral (BI) Multilevel Fast Multipole Metho (MLFMM)/Exact Solution (ES)] [17 3] simulate on an AMD Athlon XP 800+ (faster processor) are provie only for the case of 0λ scatterer, since it is meaningless to perform such a tas for the case of a 80λ scatterer. As shown in Table, Stationary Phase Metho with Fresnel functions is much faster than numerical integration an almost equal in computation time to 3D SPM [1]. Moreover, as it is clearly seen below in Table, we note that the run-time for the propose 3D analytical SPM-F metho is almost inepenent of the electrical length of the scatterer, unlie the numerical integration (NI) metho

25 Progress In Electromagnetics Research, Vol. 107, where the run-time increases rastically with the electrical length of the scatterer. Furthermore, for the propose SPM-F the run-time is Table. Total simulation time for calculations at 90 ifferent scattering angles on a stanar estop PC 1. far fiel Fresnel area near fiel far fiel Fresnel area near fiel Metho Run-Time (sec) DI SPM-F 0λ.98 DI SPM 0λ.91 DI NI 0λ 30.6 DI ES 0λ 645 DI SPM-F 0λ 3.34 DI SPM 0λ 3.5 DI NI 0λ DI ES 0λ 645 DI SPM-F 0λ 4.37 DI SPM 0λ 4.31 DI NI 0λ DI ES 0λ 645 DI SPM-F 80λ.96 DI SPM 80λ.9 DI NI 80λ DI SPM-F 80λ 3.31 DI SPM 80λ 3.07 DI NI 80λ DI SPM-F 80λ 4.1 DI SPM 80λ 4.11 DI NI 80λ Numerical Integration to SPM-F calculations run-time ratio 1 Personal computer with Pentium M GHz processor an 1GB of RAM memory for the cases of 0λ an 80λ scatterers

26 88 Moschovitis, Anastassiu, an Frangos almost inepenent of the region of observation (far fiel or Fresnel area or near fiel), while for the numerical integration metho the corresponing run-time increases rastically as the observation point moves from the far fiel to the near fiel area. Finally, it is ascertaine in Table that full-wave exact methos are much slower in actual computer computation time than PO base methos for the electrically large scatterers consiere in this paper. Using the above comparison results, we can easily realize the usefulness of the propose SPM-F metho for the urban outoor raio-coverage problem mentione in Section 1 []. A first D approach [] to a common raio propagation scenario consists of two rectangular builings. In this case, the following scattering mechanisms [] must be calculate: 8 scattering phenomena of first orer (3 scattering mechanisms an 5 iffraction mechanisms), an a total of 93 phenomena of secon orer (31 scattering-iffraction mechanisms, 9 ouble scattering mechanisms an 33 iffractionscattering mechanisms). A simple aition leas us to the conclusion that in a typical an simple real worl scenario of two builings, a total of 96 calculations of integrals of Eq. (1) relate to scattering phenomena are require. Therefore, for this simple scenario an for one scattering angle of observation the acceleration in our raio coverage simulation tool will be 93 times the acceleration corresponing to that provie above for one single plate. Aitionally, after realizing Figure 6. Total scattere electric fiel comparison results between PEC an non-pec [groun an wall] rectangular plates of imensions 0λ 0λ in the far fiel area (600 m), operating frequency 1 GHz.

27 Progress In Electromagnetics Research, Vol. 107, (a) (b) Figure 7. RCS of a 5λ 5λ square plate calculate with SPM-F, IBCM [4] methos, operating frequency 10 GHz. Simulation results for observation istance r = 600 m (Far Fiel area): (a) Raar Cross Section (RCS) σ θθ /λ (B) (vertical, TM polarization) as a function of monostatic scattering angle θ s, (ϕ s = 0 ). (b) Raar Cross Section (RCS) σ ϕϕ /λ (B) (horizontal, TE polarization) as a function of monostatic scattering angle θ s, (ϕ s = 0 ).

28 90 Moschovitis, Anastassiu, an Frangos that the above acceleration rates in computation time refer only to one observation point (position an angle), these figures have to be multiplie by the number of resolution cells in the urban scene in orer to calculate the total acceleration in computation time for the actual raio coverage problem. Finally, an aruous trial to compare the above methos with full-wave exact solution results (BI-MLFMM FEBI [17 3]), emonstrates faster rates up to 1000 times. In the case of an exact solution with Eventual BIM (pure MoM), a ifficult tas not performe by our research group, the estimate run-time for the solution of the problem (10,074 unnowns) is approximately 60*3600 sec, whereas the estimate memory require woul approximately be equal to 130 GB of RAM. 7. CONCLUSION FUTURE RESEARCH In this paper, we presente a novel full 3D analytical metho base on the Stationary Phase Metho (SPM) an Fresnel functions for the scattering of electromagnetic waves from electrically large conucting rectangular plates. The inuce currents on the rectangular plate are calculate through the Physical Optics metho an subsequently Stationary Phase Metho is use to calculate the contributions from the stationary point within the rectangular plate, as well as the contributions from the iffracting eges of the plate. This metho was foun to be much more accurate aroun the main scattering lobe compare to [1] an very fast, compare to stanar numerical integration methos [] an full-wave exact solutions [17 3]. Furthermore, the metho propose here can be use, for example, for calculating the path loss in urban outoor environments [, 16]. Finally, the results of [1] were significantly enhance in all aspects, an SPM-F metho as valiity an an asset to the use of Stationary Phase Metho techniques for the solution of electromagnetic problems [5, 6]. ACKNOWLEDGMENT The authors woul lie to than Dr. A. Tzoulis for proviing the numerical results of Sections 5, 6 an Appenix D base on the exact BI/MLFMM metho.

29 Progress In Electromagnetics Research, Vol. 107, APPENDIX A. BASIC FORMULATION OF THE SPM DOUBLE INTEGRAL CALCULATION E(x, y, z) = jωa j ω gra (iv (A)) (A1) Ĩ = a x =b F ( x, y ) e jf(x,y ) x Ĩ = Ĩ 0 Ĩ a Ĩ b Ĩ 0 π ( = f x F xo, y ) x =x o,y =y ( exp j [f ( { x o, y ) + π4 sgn f f (x, y ) x I (; x, y, z) = = 0 x =x o,y =y = c y = a x =b x x =x o,y =y }]) F ( x, y ) e jf(x,y ) x y F ( x, y ) e jf(x,y ) x y + x =a x =b y = a + y =c x =b a x =b F ( x, y ) e jf(x,y ) x y F ( x, y ) e jf(x,y ) x y F ( x, y ) e jf(x,y ) x y F ( x, y ) e jf(x,y ) x y (A) (A3) (A4) (A5) (A6)

30 9 Moschovitis, Anastassiu, an Frangos I (; x, y, z) = = x =b + a y =c x =b + + I 0 = ( exp ( j π 4 c a y = x =b F ( x, y ) e jf(x,y ) x y F ( x, y ) e jf(x,y ) x y F ( x, y ) e jf(x,y ) x y F ( x, y ) e jf(x,y ) x y { sgn + + x =a y = a x =b F ( x, y ) e jf(x,y ) x y F ( x, y ) e jf(x,y ) x y F ( x, y ) e jf(x,y ) x y = πf (x s, y s ) ( ) ( )}) f x +sgn f x x =x s,y =y y s =x s,y =y s ) f x x =x s,y =y s ( f y x =x s,y =y s ) ( ) f x x y =x s,y =y s (A7) exp (jf (x s, y s )) (A8) I 1 = + + F ( x, y ) e jf(x,y ) x y π f (0) y F (0) (y 0) y =y 0 exp j f (0) (y 0)+ π 4 sgn f (0) y (A9) y =y 0 f (0) (y ) y =0 (A10) y =y 0 x =a

31 Progress In Electromagnetics Research, Vol. 107, APPENDIX B. ALTERNATIVE APPROACH FOR THE CALCULATION OF THE SPM DOUBLE INTEGRAL Here, we present an approach for the above calculation which is a variant to that propose in Section 3.3. In this alternative approach the ouble integral uner evaluation, Eq. (A7), namely the integral I (; x, y, z) = c a y = x =b F ( x, y ) e jf(x,y ) x y (B1) is calculate using SPM in a straightforwar way, that is first by integrating over variable x (b x a) an then by integrating over variable y ( y c). The inner integration is performe accoring to Eqs. (18), (19), (A) (A4). Subsequently, the outer integration over y is performe in a way very similar to that escribe in the previous Section 3.3. Then, the final result of this secon approach is presente by the following set of equations, Eqs. (B) (B6): I () I 1 I I 3 I 1 = I 01 I c (1) I (1) I = I 0 I c () I () I 3 = I 03 I c (3) I (3) The terms of Eq. (B3a) are given by: I 01 π = f (01) y F (01) (y 01) y =y 01 exp j f (01) (y 01) + π 4 sgn f (01) y y =y 01 { I c (1) = sgn (c y 01 ) F (01) (c) exp jf (01) (c) j(u (1) c (c)) } [ ] f (01) y F r ± u (1) c (c) y =c (B) (B3a) (B3b) (B3c) (B4a) (B4b)

32 94 Moschovitis, Anastassiu, an Frangos I (1) { = sgn ( y 01 ) F (01) () exp [ f (01) y F r ± y = jf (01) u (1) ] () () j(u(1) ())} (B4c) The terms of Eq. (B3b) are given by the following equations: I 0 π = f (0) y F (0) (y 0) y =y 0 exp j f (0) (y 0) + π 4 sgn f (0) y (B5a) I () { c = sgn (c y 0 ) F (0) (c) exp [ f (0) y F r ± c y =c { = sgn ( y 0 ) F (0) () exp [ f (0) y F r ± y = I () jf (0) u () ] (c) jf (0) u () ] () y =y 0 (c) j(u() c (c)) } () j(u() ())} (B5b) (B5c) Finally, the terms of Eq. (B3c) are given by the following equations: I 03 π = f (03) y F (03) (y 03) y =y 03 exp j f (03) (y 03) + π 4 sgn f (03) y (B6a) y =y 03

33 Progress In Electromagnetics Research, Vol. 107, I (3) { c = sgn (c y 03 ) F (03) (c) exp [ f (03) y F r ± c y =c { = sgn ( y 03 ) F (03) () exp [ f (03) y F r ± y = I (3) jf (03) u (3) ] (c) jf (03) u (3) ] () (c) j(u(3) c (c)) } () j(u(3) ())} (B6b) (B6c) APPENDIX C. COMPARISON OF THE AREA DIVISION APPROACH (SECTION 3.3) WITH THE DIRECT DOUBLE INTEGRAL CALCULATION APPROACH (APPENDIX B) Comparing the two methos of calculation mentione above, we note that Eqs. (B3b) (B6c) of Appenix B have alreay appeare uring our propose first approach in Section 3.3 [namely they are Eqs. (A9), (4), (30), (3), (34), (40), (4) an (44)]. Then, the only ifference between the two approaches, Section 3.3 an Appenix B respectively, is in the first term of the corresponing representations, namely between Eqs. (A8) an (B4a). Subsequently, after careful consieration of Eqs. (A8) an ((B4a), these are foun to be exactly the same except ) for the term: f x x y appearing at the enominator of =x s,y =y s Eq. (A8) [an not appearing at all at the enominator of Eq. (B4a)]. For this reason the first approach (Section 3.3) is consiere of higher accuracy than the secon approach (Appenix B), ue to the fact that it oes not ignore the mixe secon erivative of the phase function f(x, y ) mentione just above, when taing the appropriate Taylor expansion accoring to the SPM [5]. Then, in our numerical simulations of Section 5 above, we woul rather implement numerically the first approach (Section 3.3).

34 96 Moschovitis, Anastassiu, an Frangos APPENDIX D. SCATTERING FROM RECTANGULAR PLATE WITH FINITE CONDUCTIVITY For non Perfect Electric Conuctor (non PEC) rectangular plates, as is the case, for example, for the raio coverage problem in urban outoor environment examine in [], the scattere electric fiel as given in [1] shoul be multiplie, as a first approximation, with the appropriate Fresnel reflection coefficient. This coefficient epens on the electric characteristics of the surface an the operating frequency, as shown below: an R (θ i ) = sin(θ i) ε c cos (θ i ) (D1) sin(θ i )+ ε c cos (θ i ) (Fresnel reflection coefficient for perpenicular polarization) R (θ i ) = ε c sin(θ i ) ε c cos (θ i ) ε c sin(θ i )+ ε c cos (θ i ) (Fresnel reflection coefficient for parallel polarization) (D) where θ i is the angle of incience, ε c is the complex relative ielectric constant of the rectangular plate ε c = ε r j60σλ, ε r is its relative permittivity, an σ is its conuctivity. The values ε r, σ are chosen accoring to the literature [16], namely ε r = 15, σ = 7 S/m for the groun an ε r = 7, σ = S/m for the builing walls (roofs are not consiere at this point, once both transmitter an receiver in our preiction moel [] are consiere to be place well below rooftop level). Accoring to these values a comparison iagram for the scattere electric fiel results is presente in Fig. 6. As clearly seen from the results, in the far fiel area both SPM- F an BI-MLFMM [1] methos agree not only for the case of a PEC scatterer (as seen in Fig. 3 above), but also for the case of a non-pec scatterer (Fig. 6). Some minor errors (less than 0.1 B) are encountere again in the area of the main scattering lobe, as seen clearly from the zoom winow provie at the left top corner of the graph. The minor errors are justifie, since SPM-F is only a high frequency approximation. Also, minor ifferences observe for scattering angles θ s greater than 60 egrees are explaine ue to the inaccuracies of the PO currents calculate before applying our SPM-F metho in those large scattering angles. Another approximate approach for the scattering from non-pec rectangular plates might be implemente through the use of the Impeance Bounary Conition Metho (IBCM, [4]), which uses the normalize surface impeance Z s (efine an chosen accoring to

35 Progress In Electromagnetics Research, Vol. 107, Ref. [4]) an yiels the results of Fig. 7 for the Raar Cross Section (RCS) of a plate of imensions 5λ 5λ at 10 GHz at the Far Fiel area. In Fig. 7, some minor errors (less than 0.05 B as compare to the SPM-F/BI-MLFMM comparison in Fig. 6) are encountere mainly in the area of the main scattering lobe. These errors are significantly smaller than those shown in Figs. 3 6, ue to the fact that the operating frequency of 10 GHz is significantly higher than that of Figs. 3 6 (1 GHz), an SPM-F is by efinition a high frequency approximation metho. REFERENCES 1. Moschovitis, C. G., K. T. Karaatselos, E. G. Papelis, H. T. Anastassiu, I. C. Ouranos, A. Tzoulis, an P. V. Frangos, High frequency analytical moel for scattering of electromagnetic waves from a perfect electric conuctor plate using an enhance stationary phase metho approximation, IEEE Trans. Antennas an Propagat., Vol. 58, No. 1, 33 38, Jan Papelis, E. G., I. Psarros, I. C. Ouranos, C. G. Moschovitis, K. T. Karaatselos, E. Vagenas, H. T. Anastassiu, an P. V. Frangos, A raio coverage preiction moel in wireless communication systems base on physical optics an the physical theory of iffraction, IEEE Antennas an Propagation Magazine, Vol. 49, No., , Apr Papelis, E., H. Anastassiu, an P. Frangos, A time efficient near fiel scattering metho applie to raio-coverage simulation in urban microcellular environments, IEEE Trans. Antennas an Propagat., Vol. 56, No. 10, , Oct Jenn, D. C., Raar an Laser Cross Section Engineering, 9 33, American Institute of Aeronautics an Astronautics, Inc. Washington, Balanis, C. A., Antenna Theory: Analysis an Design, 9 96, John Wiley & Sons, New Yor, Graeme, L. J., Geometrical Theory of Diffraction for Electromagnetic Waves, 30 4, 61, 90 an , UK, IEE, Knott, E. F., J. F. Shaeffer, an M. T. Tuley, Raar Cross-section, n Eition, Artech House, Sirovich, L., Techniques of Asymptotic Analysis, , New Yor, Springer, Boroviov, V. A., Uniform Stationary Phase Metho, Lonon, UK, IEE, Jones, D. S. an M. Kline, Asymptotic expansion of multiple

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