IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS
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Two-Way Propagation Modeling in Waveguides With Three-Dimensional Finite-Element and Split-Step Fourier-Based PE Approaches Gökhan Apaydin, Senior Member, IEEE, and Levent Sevgi, Fellow, IEEE
Abstract—Two-way, three-dimensional finite element and splitstep Fourier-based parabolic equation (PE) wave propagation prediction algorithms are developed, and MATLAB-based simulators are introduced. The simulators are calibrated against analytical exact data derived from modal summation through tests inside rectangular waveguides. Index Terms—Finite element method (FEM), parabolic equation (PE), radio-wave propagation, rectangular waveguides, split-step parabolic equation (SSPE), wave equation.
I. INTRODUCTION
algorithms are compared to each other as well as calibrated against analytical reference data. II. 3-D RECTANGULAR WAVEGUIDE MODEL Consider 3-D propagation inside a rectangular waveguide determined by its width ( ) and height ( ) and located longitudinally along -direction. The initial source distribution inside a rectangular waveguide ( , ) can be represented in terms of modal summation of and modes of the directions and , respectively
P
ARABOLIC equation (PE) models, since their first introduction [1], have long been used in propagation through complex environments. They are based on the assumption of slow longitudinal variations and are derived from the wave equation that yields one-way, forward-marching solutions. A reference list is never complete because of the size of the literature on PE, but [2]–[4] might be a good start. Some of our recent studies and references therein may also be useful [5]–[7]. Finite-difference, finite-element, and fast Fourier transform (FFT)-based algorithms have been used in two-dimensional (2-D) (range-height) space, and effects of ground losses/irregularities and environmental/atmospheric inhomogeneities have been successfully modeled. Three-dimensional (3-D) PE algorithms have also been introduced and applied in propagation modeling through open and closed environments [8]–[11]. Moreover, two-way PE models have been developed for the propagation problems where backward scattering is significant [12]–[15]. Two-way split-step PE (SSPE) (2W-SSPE) and finite element method PE (FEMPE) (2W-FEMPE) algorithms, which were developed, tested, and calibrated for the ground-wave propagation in the presence of knife-edge obstacles [14] as well as arbitraryshaped, irregular terrain profiles, are modified and used in modeling propagation inside rectangular waveguides having obstacles at various ranges. The novel 2W-SSPE and 2W-FEMPE
Manuscript received July 21, 2011; accepted August 19, 2011. Date of publication September 08, 2011; date of current version October 03, 2011. G. Apaydin is with the Department of Electrical-Electronics Engineering, Faculty of Engineering, Zirve University, 27260 Gaziantep, Turkey (e-mail:
[email protected]). L. Sevgi is with the Department of Electronics and Communications Engineering, Dogus University, 34722 Istanbul, Turkey (e-mail:
[email protected]. tr). Color versions of one or more of the figures in this letter are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/LAWP.2011.2167492
(1) where is the longitudinal propagation constant, is the free-space wavenumber, and is the modal excitation coefficient derived from orthonormality condition. The initial field profile is generated from a Gaussian antenna pattern as (2a) (2b) , beamwidth , and specified by its position . is the same with tilt (elevation) angle as in (2b), except , , and . is either an electric or magnetic field and Here, shows sine or cosine function starting from either or for horizontal (TM) and vertical (TE) polarizations, respectively. Terminating the waveguide completely at any range along -direction yields total reflections inside, and two-way interference can be modeled by using image method in order to test numerical PE models. This procedure works by removing the . obstacle at range and putting an image source at range III. 3-D SPLIT-STEP AND FINITE-ELEMENT-BASED PARABOLIC EQUATION MODELS The 3-D parabolic wave equation for rectangular waveguides homogeneously filled with air uses the standard PE and under time-dependence is given as
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(3)
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IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS
where and stand for the transverse coordinates, is the longitudinal coordinate, and is the reduced function since the direction of wave propagation is predominantly along the -axis. The idea of the SSPE model is to solve (3) in the transverse and spectral domains. In the numerical computations, this is achieved by using FFT. The boundary conditions (BCs) are , imposed manually. The choice of discrete step size ( , ), the injection of transmit antenna, implementation of and boundary losses, as well as irregularities are well known and can be found in many sources (e.g., see [2], [4], [6], and [15]). The standard SSPE algorithm marches longitudinally with the help of the given 2-D transverse initial antenna pattern (4a) (4b)
where is either 0 or 1 for narrow and wide angle cases, respectively. Here, and show the one-sided discrete sine/cosine transform (DST/DCT) and inverse transform for the given BC. Considering perfectly electric conductor (PEC), Dirichlet (DBC), and Neumann (NBC) type BCs correspond to horizontal (TM) and vertical (TE) polarizations, respectively. The idea of FEM-based PE model is dividing the 2-D transverse domain into subdomains first. Then, the initial field profile, generated from a Gaussian antenna pattern as in (2), is injected. The approximated field values at discrete nodes are then propagated longitudinally by using the Crank–Nicholson approach, which is based on improved Euler method with DBC (NBC) at each range for horizontal (vertical) polarization, respectively [6], [15]. The matrix form of the FEMPE propagator is (5) where is the unknown coefficients matrix related to the field profile at each range with the help of stiffness ( ) and mass ( ) matrices [6]. The modified PE models can handle two-way propagation effects. The modification is based on the generation of backward-reflected waves when an obstacle is met. In other words, the wave is marched out in the forward direction until it hits an obstacle and is separated into two components, forward propagating and backward propagating, by imposing the appropriate BCs. The forward wave continues after setting the field profile of obstacle region zero. Then, the backward field is obtained by providing the BCs for obstacle. Since the total tangential field must be zero for PEC case, the backward function can be found . Finally the total field is as obtained by superposing both fields at each range. For multiple obstacles and when resonances occur, multiforward and multibackward generation is applied. The number of reflections in both directions is then determined from the stated accuracy.
Fig. 1. 2-D cross-sectional field distributions inside PEC rectangular waveguide terminated at 200-m range: Rows: Reference, FEMPE, and SSPE simulations. Columns: 50-, 100-, and 150-m ranges, respectively (Source position; m, m, ), TM polarization. GHz. Beamwidth: . Tilt angle: . , , ).
IV. TESTS AND CALIBRATION A square waveguide (4 4 m ) terminated at the 200-m range is used for the tests and calibration. The wave function inside the PEC waveguide with TM waves is represented in terms of modal superposition using (1). The frequency is chosen as 3 GHz (well above the cutoff frequency). The transverse discretization values are and , and the longitudinal step size is . A 2-D untilted-Gaussian spatial source is used with a beamwidth of . The necessary number of modes for the given Gaussian source is found to be 19 along both - and -directions (for a relative maximum error of ). The relative maximum error is defined as the maximum difference in absolute values of the Gaussian field and the modal summation divided by the maximum absolute value of the Gaussian field. Perfect agreement is obtained among three models. Then, tilted-antenna with is applied. Fig. 1 shows the 2-D cross-sectional field distributions for this case. The number of modes in the transverse domain increases to and for the same error value. As observed, very good agreements are clearly observed. Transverse field patterns in Fig. 1 are suitable for visualization purposes only. Therefore, results should be compared in a different way: Field versus range variations or the root mean square (RMS) error (defined as the square root of the sum of squares of the differences between approximated field and the modal solution, divided by the number of nodes on the transverse domain) with respect to modal solution may be used for this purpose. Table I lists RMS errors for two untilted and tilted antennas at ranges 100 and 150 m. As seen, the RMS error increases with
APAYDIN AND SEVGI: TWO-WAY PROPAGATION MODELING IN WAVEGUIDES WITH 3-D FE AND SPLIT-STEP FOURIER-BASED PE APPROACHES
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Fig. 2. 1-D longitudinal field distributions inside PEC rectangular waveguide terminated at 200-m range. Solid line: reference. Dashed line: SSPE. Dashm, m, . Redotted line: FEMPE (Source position: m, m). TM polarization. GHz. Tilt angle: ceiver: ; (bottom) . Beamwidth: (top) . , , ).
TABLE I RMS ERRORS OF SSPE AND FEMPE MODELS FOR THE RECTANGULAR WAVEGUIDE SIMULATIONS WITH DIFFERENT ELEVATION ANGLES. AT IS ( , ) AT 3 GHz, SOURCE POSITION , TM POLARIZATION, , , , WAVEGUIDE CROSS SECTION:
the increasing tilt angle. Also, more modes are required, which increases the computation time. Field versus range at point m m for tilted and untilted sources are plotted in Fig. 2. As observed, there is a very good agreement among the modal solution, SSPE, and FEMPE solutions. Note that SSPE, compared to FEMPE, gives better results because of the match of the DST and sinusoidal functions of modal summation. The second test belongs to one 4 2-m obstacle at the 100-m range. The region up to 100 m exhibits forward propagating waves interfering with backward waves. Fig. 3(a) shows the cross-sectional field distributions of two methods at three different ranges. The rows correspond to 1W-SSPE, 2W-SSPE, and 2W-FEMPE; the columns belong to ranges of 80, 90, and 150 m, respectively. The difference between one-way and two-way models is clearly seen, and both numerical techniques give similar results. Note that field distributions at the 150-m range are similar because only forward waves contribute in this region. The transmitted power is obtained as 48.38% and 48.34% for SSPE and FEMPE methods, respectively.
Fig. 3. Field distributions inside PEC rectangular waveguide with one 4 2-m obstacle at 100-m range. (a) 2-D cross-sectional field distributions. Rows: 1W-SSPE, 2W-SSPE, and 2W-FEMPE simulations. Columns: 80-, 90-, and 150-m ranges, respectively. (b) 1-D longitudinal field distributions. Solid line: 1W-FEMPE/SSPE. Dashed line: 2W-FEMPE/SSPE m, m, ). TM polarizasimulations (Source position: GHz. Beamwidth: . Tilt angle: tion. . , , , s, s).
Field versus range at m m for one-way and two-way FEMPE and SSPE results are given in Fig. 3(b). As observed, the difference between one-way and two-way propagators is significant before the obstacle. Beyond that, field distributions are exactly the same because only forward waves contribute in this region. The final test belongs to two obstacles with 4 2 m at 100and 150-m ranges. Fig. 4(a) shows transverse field patterns of one-way and two-way SSPE results at three different ranges. The rows correspond to 1W-SSPE and 2W-SSPE; the columns belong to ranges of 90, 110, and 140 m, respectively. Same as
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IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS
FEMPE solves matrices ( ) at each range step. FEMPE matrices are range-independent for PEC boundaries, therefore they can be calculated once. SSPE range steps can be chosen much higher than FEMPE. Range step sizes of FEMPE are based on finite-difference approach longitudinally. FEMPE would be more accurate for lossy and irregular boundaries. Propagation through waveguides with arbitrary shapes and/or impedance boundary conditions (IBC) may also be modeled with SSPE and FEMPE. The staircase discretization or conformal mapping approaches can be applied in arbitrary-shaped waveguides for both SSPE and FEMPE. The IBC condition may directly be applied for FEMPE, while the discrete mixed Fourier transform (DMFT) is required for SSPE. REFERENCES
Fig. 4. Field distributions inside PEC rectangular waveguide with two 4 2 m obstacles at 100- and 150-m ranges. (a) 2-D cross-sectional field distributions. Rows: 1W-SSPE and 2W-SSPE simulations. Columns: 90-, 110-, and 140-m ranges, respectively. (b) 1-D longitudinal field distributions. Solid line: 1W-SSPE. Dashed line: 2W-SSPE simulations (Source m, m, ). GHz. Beamwidth: position: . Tilt angle: . TM , , ). Polarization.
before, the difference between one-way and two-way models is clearly seen. Using the two-way PE program also increases the transmitted power more than 20%. The transmitted power is obtained as 28.35% and 34.05% for one-way and two-way PE models, respectively. Fig. 4(b) shows field versus range variations at m m obtained via one- and two-way SSPE models. As observed, the difference between one- and two-way propagators is obvious, especially between obstacles where resonance occurs. The transmitted and reflected powers are found to be 34.05% and 65.95%, respectively. V. CONCLUSION Two-way, 3-Dl split-step and finite-element-based PE algorithms are developed, applied to propagation inside a rectangular waveguide, and compared to each other. The algorithms are also calibrated against analytical reference data. It is shown that the transverse discretizations in both PE approaches are similar. On the other hand, SSPE is more accurate and faster than the FEMPE for the same discretization parameters. SSPE uses 2-D DST/DCT for PEC boundaries, and
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