A Model for Radio Propagation Loss Prediction in Buildings using Parabolic Equations

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1 006 IEEE Ninth International Symposium on Spread Spectrum Techniques and Applications A Model for Radio Propagation Loss Prediction in Buildings using Parabolic Equations F. N. B. Magno, Z. A. Valente, J. F. Souza, J. C. Costa, G. P. S. Cavalcante. Universidade Federal do Pará Belém Pará - Brasil Abstract This paper presents a model for propagation loss prediction in indoor mobile communications using parabolic wave equation (PE). The implicit finite differences scheme of the Crank-Nicolson type is applied in order to get the solution of the parabolic equation. The propagation is considered in 5 0 in the direction paraxial. To validate this model, a measurement campaign was accomplished at a typical five floor building of a university using a frequency of 850 MHz. The loss predicted by this model had a mean error of 4.69 db and standard deviation of 3.85 db in relation to the measured data. Keywords Parabolic Wave Equation, Finite Difference, Indoor, Loss Propagation I. INTRODUCTION Wireless communication systems needs great details in its designing. This is due to the reduction of size cells in the mobile system and the increasing number of wireless networks, such as WLANs. Traditionally, propagation models based on empiric and semi-empiric models have been used in several environment. However, as the traffic has increased, the size cells has reduced, these models no more supply good predictions. As a consequence, deterministic methods have been used []. Some methods proposed for calculation of indoor wave propagation, such as the ray tracing and the numerical solution of Maxwell s equations, request quite a lot of computation time and great capacity of memory. For this reason, for demanding less time and amount of memory compared with the method totally elliptic, the parabolic approach of the wave equation has been used to solve scattering problems when the main interest is to determinate the characteristics of the channel fading []. The indoor environment is extremely complex and there is the impact of several propagation mechanisms, such as reflection, rough surface scattering, diffraction and transmission through walls and furniture. For this reason, it is necessary to know the materials used in the construction, and their electric properties [3]. The model presented in this paper is based on the method of the parabolic equation using the finite difference scheme of Crank-Nicolson to calculate the path loss in a building of five floors. A complex refractive index was considered to all the materials existing in the building and furniture located inside the studied environment. To validate the proposed model a measurement campaign was carried out in a 5 floors building of the Superior Studies of the Amazon Institute (IESAM - Belém-PA- Brazil). In this building, the floors are formed by classrooms, teaching laboratories and long corridors. A system transmitter was positioned on the ground floor and a receiver mobile system was moved along the corridors of the 5 floors. The frequency used was 850 MHz. This work is organized as it follows: the propagation model, the method of parabolic equation (PE), is described in section II; in section III is described the finite differences method used in the parabolic equations calculation; finally, on section IV the loss values predicted by the proposed model are compared with the results obtained during measurement campaign. II. PARABOLIC EQUATION METHOD The radio wave propagation modeling requires the solution of wave equation with proper boundary conditions [4] x z kn0 where the field,, is considered y independent, k and n are the propagation constant and refractive index, respectively. We define u, representing the electric field, as follows ikx, u x z e x, z () By substituting () in () and some approximations, a partial differential equation for u can be extracted. u u u x x z ik k n u This equation can be factored as [4] ik Q ik Qu 0 x x () 0 (3) (4) /06/$ IEEE 35

2 where these two terms correspond to progressive and regressive waves respectively and Q is the distinguishing pseudo-differential operator defined by [4] Q k z n Using the progressive wave equation u ik Qu x xz,. (5) (6) measurements. That address has two different buildings, one of them has old architecture, is tumbled for the Historic Heritage Department, and the other, named block, is a five floors modern architecture building located behind the first one mentioned. In block building, the one used to be the test environment, exists corridors, computer science laboratories, laboratories for experimental works, classrooms, bathrooms, etc. There are different materials involved, such as brick walls, doors and teacher and student s wooden tables, group of laboratories benches made of formic, glass and aluminum in the windows, steel in the elevators, etc. we have the formal solution for the electric field u as: ikxq, u x x z e u x, z. (7) Defining Q q 0,5q, where q n, that k z characterizes the approach for small angles, up to 5 0 [5], in (7), using a first-order Taylor expansions of the exponential term and also of the square-root, a standard parabolic equation is found (SPE) [4] u u xz, ik xz, k n xz, u xz, z x III. FINITE DIFFERENCE METHOD 0 (8) In this paper was used the finite difference scheme 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, respectively, and used in (8) obtaining m, j m, j m, j m, j m, j u z u z u z u x z u x z ik z xm m, j m, j k n z u z 0 where m xm xm is the midpoint in the solution from m x m- to x m range. Using uj u xm, z, b4ik z x and m aj k n m, zj z j in (9), we obtain [4] m m m m m m m m uj baj uj uj uj baj uj u (0) j IV. DESCRIPTION OF THE ENVIRONMENT The model was developed to study the electromagnetic waves propagation in indoor environments. Different geometric structures and material using in the building were considered. The Instituto de Estudos Superiores da Amazônia (IESAM) was the building used for campaign of (9) Fig.. Sight of the building of the Instituto de Estudos Superiores da Amazônia V. RESULTS A plane wave vertically polarized in 850 MHz was transmitted for simulation using the parabolic equation method. The paraxial direction was chosen, with approach for small angles, up to 5 0. Also was considered only the dimensions of length and width of studied environments. A FORTRAN program was developed to simulate (0). The measurement setup consisted of a transmitter system, composed by a sweeping generator (model HP 8375A), an amplifier (ZHL - 4W) and an antenna (3 dbi gain monopole). During the measurement campaign, the transmitter set was located in the end of the corridor in the ground The receiver system was composed by a receiving antenna (.5 dbi gain monopole), a spectrum analyzer (HP 8593E), a LNA amplifier, an acquisition board AD-DA (LAB JACK U), a computer for storage data, and a prototype of a vehicle with a 5th wheel, that allows to measure the distance traveled by the receiver system [6]. The mobile receiver system traveled the corridors of all the 5 floors of the building (see Fig. ), measuring and storing the signal received and the distance traveled by the system. The receiver antenna was linked to the spectrum analyzer, measuring the level of the signal received. The prototype of a vehicle with a 5th wheel measured the distance traveled by the system. The intensity of the signal measured for the spectrum analyzer and the signal measured for the prototype were sent to the computer through a converter board. The data stored during the measurement campaign were treated and processed for subsequent use. 35

3 4th floor 3rd floor nd floor st floor Ground-floor Fig.. Transmitter located in the ground floor and the receiver covering the diverse floors of the building After the collection and treatment of the data the path loss was founded. Due to the differences in the electric characteristics between all the existing materials, it is necessary to use the values of the electromagnetic constants regard to the different kind of materials applied. Table I shows the values of relative permittivity and conductivity of the materials used. Fig. 3. Measured and Predicted Propagation loss versus distance in the ground TABLE I RELATIVE PERMITTIVITY AND CONDUCTIVITY OF THE MATERIALS Materials Relative permittivity Conductivity (S/m) Brick wall [7] Wood [6] Formic [8] The refractive index is given by the following expression [9] n i r f 0 () where r is the relative permittivity, is the conductivity (S/m), f is the frequency (Hz) and 0 is the permittivity in the vacuum (F/m). The path loss is calculated by Fig. 4. Measured and Predicted Propagation loss versus distance in the first LdB f u ug G R () ( ) 36,57 0log0 0log0 0 0log0 T where u 0 is the field in the distance of reference (d 0 ), u is the received field, f is the frequency in GHz, and G T and G R are the transmitter and receiving antennas gain in db, respectively. The Fig. 3, 4, 5, 6 and 7 show the path loss, in db, versus distance, in meter, for the calculation using parabolic equation and the experimentally results. Fig. 5. Measured and Predicted Propagation loss versus distance in the second 353

4 Fig. 6. Measured and Predicted Propagation loss versus distance in the third Fig. 8. Propagation loss predicted versus distance with signal passing through corridor and laboratory. The Table II shows the average error, standard deviation and rms error for some floors studied in the test environment. TABLE II CALCULATION OF AVERAGE ERROR, STANDARD DEVIATION AND RMS ERROR Floor Average Error Standard Deviation Rms Error Ground-floor First floor Second floor Third floor Fourth floor The Table III shows the path loss exponent for the ground floor, first, second, third and fourth floors. It is noticed that this factor depends on the environment and the scenarios [0]. Fig. 7. Measured and Predicted Propagation loss versus distance in the fourth The Fig. 8 shows the simulation of the path loss versus the distance with the signal while transposing many obstacles in the corridor and the existing laboratory placed in the end of the floor, at the second, third and fourth floor, that had been considered previously. It is noticed, in this case, that the path loss increased when passes through the laboratory, where there are a group of laboratories formic benches to used by the students and wooden table destined to the teacher. This alteration that can be seen in the Fig. 8 is due refraction and reflection on the brick wall and diffraction occurred in the corners of benches and tables. TABLE III PATH LOSS EXPOENENT Path Loss Exponent Floor PE Experimental Ground-floor.5.5 First floor.6.8 Second floor.5.8 Third floor.3.3 Fourth floor Average

5 VI. CONCLUSION It is observed that the use of this model increased the velocity of data processing and does not require great capacity of computer memory. To process a grid of 000 x 400 was used a time of processing of approximately 4.30 seconds in a computer Pentium III, 700 MHz, 56 MB. This suggests that this model can be applied for large environments. This is possible considering that the implicit finite difference scheme of the Crank-Nicolson type a tridiagonal matrix is solved instead of a matrix with all the elements different of zero. In the applied model the complex refractive index was considered what increases your precision. It was observed for all the floors that the path loss increases with the distance of the transmitter, what corresponds to the increase of the uncertainty caused by the diversity of the architectural configuration, geometry, variety of material used in the building, furniture, etc. Considering all the analyzed floors, the biggest error found between the simulated path loss and the values measured was of approximately 5 db and 4 db for the standard deviation. Observing the low time of processing and the values calculated for the error we can to consider that the presented model is a good alternative to indoor environments. Considering that, in this paper only two dimensions were applied, the next simulation to be studied will take in consideration tree-dimensional indoor environment. Also simulations for other environments and frequencies of.8;.4 and 5.0 GHz will be carried on. [8] C. A. Balanis, Advanced Engineering Electromagnetics, John Wiley & Sons, United States of America, pp. 50-6, 989. [9] R. K. Wangsness, Electromagnetic Fields, John Wiley & Sons, United States of America, pp. 34 appendices, 979. [0] C. C. Chong, Y. Kim and S. Lee, Statistical Characterization of the UWB Propagation Channel in Various Types of High-Rise Apartments, IEEE Communications Society, pp , 005. ACKNOWLEDGEMENTS The authors would like to thanks Prof. Dr. José Maria Filardo Bassalo and Prof. Dr. Klaus Cozzolino for the outstanding contribution in our learning. REFERENCES [] S. Grubisic, W. P. Carpes, C. B. Lima, P. Kuo-Peng, Ray-Tracing Propagation Model Using Image Theory With a New Accurate Approximation for Transmitted Rays Through Walls, IEEE Transactions on Magnetics, Vol. 4, No. 4, April 006. [] N. Noori, H. Oraizi, A Parabolic Wave Equation Approach for Modeling Propagation Through Windows, IEEE, 3 rd International Conference on Computational Electromagnetics and Its Applications Proceedings, pp. -4, 004. [3] A. Aragón-Zavala, B. Belloul, V. Nikolopoulos, S. R. Saunders, Accuracy evaluation analysis for indoor measurement based radio-wavepropagation predictions, IEE Proceedings.-Microwave Antennas and Propagation, Vol. 53, No., February 006. [4] M. Levy, Parabolic Equation Methods for Electromagnetics Wave Propagation, The Institution of Electrical Engineers, London, pp. 4-40, 000. [5] E. Premat, Prise en Compte d effets Météorologiques dans une Méthode d Eléments finis de Frontière, Thèse de docteur, France, pp. 59-6, 000. [6] R. A. Lima, R. N. S. Barbosa, J. C. Rodrigues, A. A Neves, S. G. C. Fraiha, H. S. Gomes, G. P. S. Cavalcante, Path Loss Semi-Empirical Model for Indoor Mobile Communication at 800 MHz Band (in Portuguese), presented at the th Telecommunication Brazilian Symposium, Belém, Brazil, 004. [7] D. I. Axiotis, M. E. Theologou, GHz Outdoor to Indoor Propagation at high Elevation Angles, IEEE, PIMRC