Propagation Path Loss Prediction Using Parabolic Equations for Narrow and Wide Angles

13 7th European Conference on Antennas and Propagation (EuCAP) Propagation Path Loss Prediction Using Parabolic Equations for Narrow and Wide Angles Rôulo A. N. Oliveira 1, João Furtado de Souza, Fátia Nazaré Baraúna Magno 3, Klaus Cozzolino 4, Gervásio Protásio dos Santos Cavalcante 5 1 Instituto Federal do Pará, IFPA, Belé, Brasil, roulo.oliveira@ifpa.edu.br Faculdade de Física, Universidade Federal do Pará, UFPA, Belé, Brasil, furtado@ufpa.br 3 Faculdade de Física, Universidade Federal do Pará, UFPA, Belé, Brasil, fnb@ufpa.br 4 Faculdade de Física, Universidade Federal do Pará, UFPA, Belé, Brasil, cozolino@ufpa.br 5 Faculdade de Engenharia da Coputação, Universidade Federal do Pará, UFPA, Belé, Brasil, gervasio@ufpa.br Abstract In this paper the foralis of parabolic equations for narrow and wide is presented as a ethod for calculating the propagation path loss of a obile radio signal propagating in an urban foliated sei-confined environent. The results were copared with those obtained in easureent capaigns and odels existing in the literature. Keywords Finite difference; ixed Fourier transfor; narrow angle; parabolic equations; path loss; wide angle. I.INTRODUCTION For the planning of any obile counications syste it is iportant to know the behavior of electroagnetic waves propagating in the studied region. This propagating is affected by a nuber of factors that ay be of natural origin such as lightning, rain, vegetation or provoked by an as the effects of buildings, the noise of engines, transission lines and other. For a ixed-path environent, the electroagnetic waves suffer a great influence of the effect of ultiscattering, where this effect causes significant attenuation in the obile radio signal level, which are due the variations of the electrical properties of vegetation (leaves, changes) and soil. Thus, for a perfect study of the coverage area of obile counications syste, there is need to develop a odel to predict the losses occurred in the of signal propagation within the studied environents. Parabolic equation solvers provide a powerful odeling capability for propagation of electroagnetic waves over long distances in coplex environents. They are used extensively for predicting radar coverage in ducting environents over rough surfaces. Two priary classes of solvers are used in conjunction with the parabolic wave equation. The first is a finite-difference approach that relies on fine discretization of the spatial doain for accurate representation of the field propagation. The second is the split-step approach that ipleents propagation in the Fourier doain and utilizes a series of phase-screens to account for refraction effects [1], []. It was natural therefore for Egli to produce a odel based on plane-earth propagation, after observer that there was a tendency for the edian signal strength in a sall area to follow an inverse fourth-power law with range fro the transitter. However he also observed firstly that there was an excess loss over and above predicted and secondly that this excess loss depended upon frequency and the nature of the terrain [3]. The Lee odel is a power law odel, which takes into account the antenna height of the base station and the variation in terrain where the effective base station antenna height is deterined by the projection of slope terrain in near vicinity of the obile to the base station location [4]. This paper presents a odel for calculating the propagation loss of electroagnetic waves based on the foralis of parabolic equations [1], which has the great advantage of reduced coputational effort to calculate and sall argins of error. To validate the proposed odel, this paper presents the results of easureent capaigns conducted in three cities in Pará State (Brazil), where were used the frequencies of 9 MHz and 1.8 GHz for the tests, valuesused in obile radio in Brazil. These values were copared with easureents obtained using the prediction odel of parabolic equations and with Egli Model and Lee Model, odels existing in the literature. Initially, the environent was odeled considering streets with buildings and vegetation, and then, we applied the ethod of parabolic equations for the calculation of electric fields considering the electrical paraeters involved in the siulated environent. To solve the resulting parabolic equation, the finite difference schee of Crank- Nicolson ethod was used for narrow, up to 15º. For wide, up to 9, the ixed Fourier transfor was used. This paper is organized as it follows: the propagation odel, the ethod of parabolic equation (PE) and the ixed Fourier transfor, is described in section II; section III describes the environents; in section IV are presented the path loss odels; on section V shows the results; and section VI, the conclusion. II. THE THEORETICAL METHOD The two-diensional scalar wave equation can written as [5] 978-88-9718-1-8/13/$31. 13 IEEE 944

13 7th European Conference on Antennas and Propagation (EuCAP) ψ ψ + + knψ= where k is the wave nuber and n the refractive index. Following Levy [6], we choose x as the paraxial direction and replace the function ψ ( x, z) by ikx e E( x, z) yielding the scalar equation [5] E E E 1 (1) + + ik + k n E = () This equation can be forally written as + ik ( 1 Q) + ik ( 1+ Q) E = (3) = 1 +, = 1+ Z [5]. One ay here note that the operator Z would represent a quantity that is sall copared to one. Equation (3) represents both forward and backward propagating waves and the part that represents forward propagating waves is where Q n k ( x z) + ik Q E = + ik + Z E = ( 1 ) ( 1 1 ) The siplest approxiation of (4) is obtained by using first-order Taylor expansions of the square-root and exponential functions. This yields the standard parabolic equation (SPE) [6] E E ik k ( n ) E = + + 1 This is the parabolic equation used for narrow angle (NA) in this paper. The error in (5) is going fro 1-7 for an angle of 1, to 1-3 for an angle of 1 and over 1 - for an angle of [6]. In this paper was used the finite difference schee of Crank-Nicolson applied to the standard parabolic equation. The approach of the central finite differences was calculated for the derivatives of first and second order in x and z, where ξ = ( x 1 + x is the idpoint in the solution ) fro x -1 to x range. Using E E x, z, j (4) (5) ( ) = j b= 4ik z x and a = k n ξ, z 1) z, and applying in (5) we obtain [6] ( j j ( ) ( ) E + b+ a + E + E = E + b a E E (6) 1 1 1 j j j+ 1 j 1 j j j+ 1 j 1 and Q ~ 1+ A+ 1+ B 1 (7) is the new wide- (WA) split operator proposed by Felt and Fleck, where A and B are defined as: A = 1 k and B = n z 1. This approxiation is exact for unifor edia and the equation is only valid for couting operators [6]. Substitution of (7) into (4) leads to the result [6] E i k + E ik ( n ) E = Equation (8) is the wide-angle parabolic equation used in this paper. The ixed Fourier transfor, ipleenting ipedance boundary conditions, was developed to enable propagation siulation in the lower atosphere over finitely conducting surfaces. The algorith for the ipleentation of ixed Fourier transfor in a discrete doain is described in [6]. The procedure to solve (8) can be written as [7] ( ) E x+ x z = e (, ) where ik n 1 x ( α ) α i x k p k Fs e U( x, p) π + α + p p i x k p k Fc e U ( x, p) + π α + p i x k k α z + e e K( x) ik( n 1) x ik( n 1) x U( x, p) = α Fs{ e E( x, z) } pfc{ e E( x,z )} (1) F S and F C are sine and cosine transfor, respectively, K(x) is defined as [7]: α z α f ( z) e dz; Re( α) K x = > ( α ) ; Re (8) (9) (11) and the coefficient α represents the properties of surface ediu in ters of relative coplex perittivity η [7] ik α = η vertical pol. (1.a) α = ik η horizontal pol. (1.b) 945

13 7th European Conference on Antennas and Propagation (EuCAP) III. DESCRIPTION OF THE ENVIRONMENTS The easureents were taken in a covered by radio signal transitted by fixed station in cities 1, and 3, in Pará State. These cities are characterized by areas of dense vegetation, cut through streets paved with the presence of saller buildings (see Fig. 1, 3, 4). The transitted signal was at 9 MHz, for city 1 and, and 18 MHz for city 3. In the city 1, the transitter was installed on a building in ANATEL (Agência Nacional de Telecounicações) using a collinear antenna (see Fig. ) with a gain of.14 dbi. In the city, the transitter used the radio base station fro OI-Cellular Copany; and the antenna was an onidirectional, with a gain of dbi radiating a signal CW. In the city 3, the transitter used the radio base station fro TIM-Cellular Copany with a panel antenna, with a gain of 17.5 dbi. The obile receiver traveled at a speed of approxiately 3 k/h along road inside a forest, for the three cities. The easureent results were recorded for off line processing. Figure. 3 Partial view of the ixed-path and city tower Figure. 4 Partial view of the ixed-path and city 3 tower For the odels applications, the following set of paraeters were considered and showed in the Table I. Figure. 1 Partial view of the ixed-path city 1 Figure. Tower view of the city 1 TABLE I. PARAMETERS USED IN THE MODEL Paraeters Syb Values City 1 City City 3 Frequency f 9 9 MHz MHz 1.8 GHz Average height forest h 1 14 14 Transitter height h T 1 7 6 Mobile receiver height h R 3 3 3 Receiver antenna gain G R.14 db.14 db.14 db Transitted power P T 3 db db 44.5 db Road paved, width W 1 11 1 Vehicle position w 7.75 6.5 7. Forest relative perittivity [8] ε F 1.1 1.1 1.1 Forest conductivity.1.1 σ [8] F S/ S/.1 S/ Road relative perittivity [8] ε R.7.7.7 Road conductivity [8] σ R 4 S/ 9 S/ 9 S/ Width of the right lateral forest d 1 15 3 Width of the left lateral forest d 5 4 5 Transitter distance 5 to 45 to 45 to d range 56 13 13 946

13 7th European Conference on Antennas and Propagation (EuCAP) IV. PATH LOSS MODELS Several theoretical and experiental odels exist for to copute the path loss propagation in wireless counications, and each characterizes the environent with different view. In ixed-path, the shadowing, the scattering and the absorption caused for the vegetation can cause a significant path loss, which increases with the frequency. For the validation of the proposed odel we copared the easured data and soe classical odels in the literature described below. A. Egli Model Based on a series of easureents perfored over irregular terrain, Egli [9] proposes an epirical odel where the attenuation of the transitted signal depends on the law of the inverse fourth power between the transitter and obile receiver. The expression for the edian path loss is [1]: LdB = 4 log( d) + log f log h log h G G 76.3 b b + (13) where G b and G are the gains of transitting and receiving antennas, h b and h are the heights of the transitter and receiver antennas, respectively, d is the distance between the and f is the frequency. B. Lee Model It is a point-to-point propagation odel, and to get it is required three steps. The first is the creation of so-called standard conditions. To ake the prediction area - point and then the prediction point to point. The general expression of the odel for the loss of the received signal is [11]: 9, it was used the ixed Fourier transfor. The reason for this choice was that, if the finite difference schee was used, it would be necessary to work with penta-diagonal atrices, instead of the tri-diagonal, causing an increase in coputational tie [5]. To calculate the propagation path loss through the ethod of parabolic equations and subsequent coparison with the odels of Egli and Lee was used the equation below [1]: LdB = 36.57 + log f+ log µ log µ GT GR (15) where µ is the electric field at a reference distance (d ), µ the electric field received, f is the frequency in GHz, and G R and G T are the gains of transitting and receiving antennas in db, respectively. The refractive index is given by the following expression [13]: n = ε + iσ 1 r π f ε (16) where ε r is the relative perittivity, σ is the conductivity (S/), f is the frequency (Hz) and ε is the perittivity in the vacuu (F/). Figures 5, 6 and 7 show the path loss in db, depending on the distance d to the transitter, in Kiloeters, to the cities 1, and 3, respectively. f LdB = 13.77 + 3.5log d+ 1nlog α (14) 9 where d is the distance between the transitter and obile receiver, in k, f is frequency in GHz, n is an experient value chosen to be 3 in our siulation and α is the correction factor to the standard condition in db, given by: α = α + α + α + α + α ; where α 1 = (h T / 3.48), α = (h R / 1 3 4 5 3 ) k, α 3 =( P T /1), α 4 = (G T / 4 ) and α 5 = G R ; and choose the paraeter k to be equal to [11]. Figure.5 - Path loss with distance for city 1 V. RESULTS The data analyzed in this paper were obtained fro easureent capaigns took place in the Cities 1, and 3 in the State of Pará. In the Cities 1 and was used the frequency of 9 MHZ and the City 3 was used the frequency of 18 MHz. For the solution of the parabolic equation for narrow, up to 15, it was used the finite difference ethod proposed by Crank and Nicolson, and for large, up to 947

13 7th European Conference on Antennas and Propagation (EuCAP) Figure.6 - Path loss with distance for city Figure.7 - Path loss with distance for city 3 The average error, rs error and standard deviation are shown in Table II for cities 1, and 3, respectively, with the easured. TABLE II - AVERAGE ERROR, STANDARD DEVIATION AND RMS ERROR TO THE CITIES City 1 City City 3 Model Average Error Standard Deviation rs Error PE wide 1.95 1.54.49 PE narrow 3.78 3.7 5.3 Lee 3.98 3.38 5. Egli 3.35 3.14 4.59 PE wide 4.91 3.41 5.97 PE narrow 4.3 3. 5.4 Lee 5.3 4.7 6.47 Egli 5.56 4.35 7.6 PE wide.19 1.83.85 PE narrow 3.6.8 3.98 Lee 4..87 4.9 Egli 3.68.41 4.4 VI. CONCLUSION This paper presented a study of the behavior of the obile radio signal propagating into urban foliated sei-confined environent. To characterize this region, it was used the easureents obtained in the City 1, and 3, in the State of Pará, where a obile receiver went into across a road of a region of the Aazon forest. Thus, it was possible to perfor a coparative analysis of these easures with the theoretical results obtained by the odel proposed here, based on the foralis of parabolic equations for narrow and wide of propagation and of two classical odels: Lee and Egli. It was also noted that there was a fast processing the data using parabolic equation odel for narrow and wide. In this paper, the refractive index was considered coplex for the proposed odel. Thus, it was conclude that the behavior of obile radio signal propagating in the urban foliated sei-confined environent is best estiated by the proposed odel. Through the coparative analysis between these easures with the theoretical results obtained by the odel proposed, based on the foralis of parabolic equations for narrow and wide of propagation, and the two classical odels, Lee and Egli,, it was possible to observe that there were not significant differences in the values of the errors using the proposed odel for narrow or wide of propagation. But, it was possible to observer that a best results were obtained using the parabolic equation odel than the odels of Lee and Egli. REFERENCES [1] M. Levy, Parabolic Equation Methods for Electroagnetic Wave Propagation, 1st edn., London: Institution of Electrical Engineers,, pp. 1-36. [] C. R. Sprouse, and R. S. Awadallah, An Angle-Dependent Ipedance Boundary Condition for the Split-Step Parabolic Equation Method, IEEE Transactions on Antennas and Propagation, vol. 6, no., pp. 964-97, February 1. [3] J. D. Parsons, The Mobile Radio Propagation Channel, 1st edn., United States of Aerica: John Wiley & Sons, pp. 54,. [4] S. R. Saunders, Antennas and Propagation for Wireless Counication Systes, 1st edn., England: John Wiley & Sons Ltda., pp. 158, 1999. [5] P. D. Hol, Wide-Angle Shift-Map PE for a Piecewise Linear Terrain- A Finite-Difference Approach, IEEE Transactions on Antennas and Propagation, Vol. 55, No. 1, pp. 773-789, October 7. [6] M. Levy, Parabolic Equation Methods for Electroagnetic Wave Propagation, The Institution of Electrical Engineers, London,. [7] P. Valtr e P. Pechac, Doain Decoposition Algorith for Coplex Boundary Modeling using the Fourier Split-Step Parabolic Equation, IEEE Antennas and Wireless Propagation Letters, Vol. 6, pp. 15-155, 7. [8] G. P. S. Cavalcante, M. A. R. Sanches, and R. A. N. Oliveira, Mobile Radio Propagation along Mixed Paths in Forest Environent, Proceedings of IMC 99, SBMO/IEEE, 1999, pp. 3-34. [9] J. J. Egli, Radio Propagation Above 4 MC Over Irregular Terrain, Proceedings of the IRE, pp. 1383-1391, October 1957. [1] G. Y. Delisle, J.-P. Lefèvre, M. Lecours, and J.-Y. Chouinard, Propagation Loss Prediction: A Coparative Study with Application to the Mobile Radio Channel, IEEE Transactions on Vehicular Technology, Vol. VT-34, No., pp. 86-96, May 1985. [11] M. Alshai, T. Arslan, J. Thopson, and A. Erdogan, Frequency Analysis of Path Loss Models on WIMAX, 3 rd Coputer Science and Electronic Engineering Conference (CEEC), 11. [1] L. W. Li, T. S. Yeo, P. S. Kooi, M. S. Leong, and J. H. Koh, Analysis of Electroagnetic Wave Propagation in Forest Environent along Multiple Paths, Progress In Electroagnetics Research, PIER 3, pp. 137-164, 1999. [13] R. K. Wangsness, Electroagnetic Fields, John Wiley & Sons, United States of Aerica, pp. 34 appendices, 1979. 948