Rather than use the moving window concept to save memory, one can accomplished that by adopting an efficient iterative method to solve the linear system. The result is a FDFD formulation with only one field component to evaluate and high wave absorption at the computation domain boundaries.

## Path Loss Analysis of WSN Wave Propagation in Vegetation

In Section III preliminary simulation results are generated. First the technique is validated through the canonical problem of an infinite magnetic line above a plane electric conductor in Section III. Then, Section III. B exhibits radiowave prediction data for a cellular system measurement campaign performed by the authors and described in [ 9 ].

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Section III. C shows the two-dimensional signal distribution for an idealized urban scenario [ 7 ] in order to visualize the effect of the wave components direct, diffracted, etc on the final solution. Finally, Section IV summarizes the conclusions. The modeling of wave propagation adopts the two-dimensional approach vastly found in literature [ 1 ]-[ 12 ], where the environment is assumed invariant in one direction.

Then, the solution can be found for TE or TM modes based on the chosen signal polarization. In this work, the same geometry arrangement for the FDTD propagator developed by the authors in [ 8 ],[ 9 ] is used. In the same manner, it is adopted a vertical polarized signal modeled by a TE z E x , E y , H z solution.

The computation domain mesh is truncated with PML layers. Then, general information of the propagation environment Earth curvature, inhomogeneous irregular terrain, urban constructions, atmospheric refractive index, etc are added to the mesh via electrical parameters. The initial point of the FDFD formulation development is done as in [ 14 ] and [ 16 ]. It starts from the Helmholtz equation instead of the traditional approach to use the separate one order derivative Ampere's and Faraday's law.

Although it becomes more complicated to obtain the final FDFD expressions, this procedure minimizes the number of field components of the problem and consequently the unknown variables of the linear system. In the present paper, different from the previous mentioned works, it is applied the SCPML [ 18 ] structure in order to achieve higher wave absorption and numerical performance [ 19 ]. For vertical polarization, i. Hence, Eq. Now, Eq. So, after some extensive mathematical manipulations, the final algorithm for the determination of H z is. Here we adopt the work of Marengo et al.

The problem solution is achieved by applying Eq. Therefore, the field values of all points must be found simultaneously. The matrix A is highly sparse, has an unsymmetrical structure and for typical propagation situations is very large. For such class of linear systems, iterative methods are preferred to direct methods due the memory required.

Besides, they often produce the solution faster than direct methods [ 13 ]. In this work, the CGS method was adopted, which belongs to the Krylov subspace methods [ 17 ]. In this Section, three case studies are presented and preliminary results are generated. The simulations were performed in a personal computer with a 3. First, it is investigated a canonical problem of an infinite magnetic line source placed at a height h above an infinite perfectly electric conductor PEC plane.

The analytical solution can be obtained by Image Theory, where the plane is replaced by a virtual source to account for the reflections from the surface.

The total field is the sum of the direct ray from the magnetic line and the field generated by the virtual source reflected ray from the ground , given in terms of Hankel functions [ 20 ]. The line source radiates a 1 GHz sinusoidal field located at 1 m height. The background medium is vacuum. The results computed by the analytical solution [ 20 ] are exhibited in a and b presented the data estimated by the FDFD method. Here both the background medium situations are analyzed. One can see a very good agreement between the analytical solution and the FDFD results. This indicates that the proposed formulation has a valuable potential.

Moreover, the results from Fig. The FDFD propagation analysis in a real urban environment is performed through a cellular system measurement campaign carried out in Belo Horizonte, Brazil. The base station is placed at the Federal University of Minas Gerais UFMG in an irregular terrain region, and the environment is mainly urban with several buildings and constructions. The vertical polarized signal was generated at 2. The received power were recorded at 60 points distributed around the base station, whose distances from the source vary from 40 to m.

The mobile unit used an omni-directional antenna at 1. The campaign detailed information can be found in [ 9 ]. In order to compare 2D propagation methods directly to measurements or 3D formulations, the spherical wave attenuation with distance is added likewise in [ 9 ] and [ 24 ]. So, the finite-difference fields are multiplied by a correction factor CF :. The receiver-transmitter distances vary from 40 to m. Computational Aeroacoustics. Christopher K. Designing Quiet Structures.

Gary H. Smart Cities.

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