International Journal of Engineering Trends and Technology (IJETT) – Volume 10 Number 13 - Apr 2014
#1
,
*2
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Student, School of Electronics and Communication, Lovely Professional University,
Phagwara, Punjab, India
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Assistant Professor, Department of Electronics and Communication, Lovely Professional University,
Phagwara, Punjab, India
Abstract—This paper introduces a simple ray propagation
MATLAB package for the visualization of ray path for wave propagation. The package can be used to model wave propagation through complex wave-guiding environments, either as an education tool for understanding electromagnetics (EM), or as a design tool to visualize Geometrical Optics (GO) ray propagation, reflection, and refraction.
Earth’s radius. Extra terms are also added to model various super- or sub-refraction propagation cases. Because of the closeness of n to unity it is customary to use the refractivity, N, defined as
N = (n-1)*10
6
. (1)
Index Terms—Geometrical optics; electromagnetic propagation; ground wave propagation; ray propagation; land mobile radio propagation factor; atmospheric propagation.
I.
I NTRODUCTION
The topic of EM wave propagation through complex environments has been and will continue to be of great interest for wireless communication propagation engineers, service planners, site-surveyors, etc., in their offices desire to have access in real time to the propagation characteristics on a digital map on a computer, when they select the location of a transmitter/receiver pair. The modelers are still away from satisfying this requirement directly in three dimensions; the field strength can be simulated for two dimensional path
(projection from the ground profile between those two selected points: see figure 1). Even propagation along these twodimensional projection poses problem need further investigation. This paper introduces a simple MATLAB package for visualization of ray paths through two-dimensional complex environments heaving obstacles and variable refractivity profiles.
The matched coordinates for the problem of threedimensional wave propagation over a spherical earth with a non-flat terrain profile, plus impedance boundary conditions and a radially dependent atmosphere, excited by a vertical electric dipole located near the earth’s surface, are spherical coordinates (the earth’s radius is r=a; the finite conductivity of the Earth’s surface at r=a is described by surface impedance).
For the relevant source (r’) and observation (r) heights – which satisfy the inequalities [(r’- a)/a]<<1 and [(r-a)/a]<<1, respectively – the three-dimensional problem in spherical coordinates can be approximated in “earth-flattening” nondimensional two-dimensional rectangular (x,z) coordinates.
The Earth’s curvature and the standard atmosphere condition can be included by using n=n
0
+x/a e
, where n
0 is the refractivity value at the surface, and a e
=4a/3= 8504km is the effective
Figure 1. Two dimensional propagation of an urban propagation path.
N is dimensionless, but is measured in “N units” for convenience. N depends on the pressure, P (mbar), absolute temperature, T ( o k), and the partial pressure of water vapor, e
(mbr), as follows[1]:
N = 77.6*(P/T) + 3.73*10
5
*(e/T
2
), (2)
Which is valid in earth-troposphere waveguides, and can be used in ground-wave propagation modeling. If the refractive index was constant with height, ratio wave are bent downward toward the earth, so that the ratio horizon lies further away than the optical horizon account either by using N with a e
, or by introducing a fictitious medium, where N is replaced by the modified refractivity, M:
M=N+(x/a)*10
6
= N+157x, (3)
With the height, x, given in kilometers. In equation (2), a=6378km,
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and
International Journal of Engineering Trends and Technology (IJETT) – Volume 10 Number 13 - Apr 2014
10
6
/a=157, (4) the edges and tips of obstacles. Reflections are accounted for by using the reflection coefficient for an impedance boundary:
R=(sin ɸ ’ – sin ɵ )/(sin ɸ ’ + sin ɵ )
∂M/∂x=∂N/∂x+157.
Sin ɵ = 1/n for soft boundary (7)
For the standard atmosphere (i.e., for a vertical, linearly decreasing refractive index), N decreases by about Nunit/km, while M increases by about 117 Nunit/km. sub-refraction
(super-reflection) occurs when the rate of change in N with respect to height (i.e., ∂N/∂x) is less (more ) than 40 Nunit/km.
= u for hard boundary
Where n is the normalized surface impedance, and ɸ ’ is the angle between the surface and ray-incidence direction (R= -1 for a perfect electrical conductor – PEC – boundary).
II.
R AY PROPAGATION APPROACHES
Ray propagation models have gained special attention for the last decade, parallel to the introduction of personal communication system [2-7] (propagation models based on powerful frequency-time and frequency-domain techniques are beyond the scope of this article; therefore, they are not included here; examples may be found in [2,8]).
The most difficult part of propagation simulation in terms of ray summation is to predict ray path between the transmitter and receiver. The reflected and refracted ray paths are constructed by either of two methods: eigen ray search (i.e., by finding characteristic rays between the transmitter and receiver through a specified propagation path), or by brute force
(shooting and bouncing rays, SBR, i.e., by considering a bundle of transmitted rays that may or may or may not reach the receiver). The challenge for a ray-propagation model is to find a computationally fast way to specify the dominant ray paths that account for the field-strength prediction.
An EM field contribution of a ray at a distance R=R(x,y,z) for a poin source (x’,y’,z’) (in a three-dimensional environment, with the assumption of exp(jwt) time dependence) is
E(R) = r
-jkR
/R, (5)
At a distance ρ = ρ(x,z) for a line source (x’,z’) (i.e., in two dimensional environment), this is
E(ρ)= e
-jk ρ
/ ρ
1/2
, (6)
Where
R = [(x-x’)
2
+(y-y’)
2
+(z-z’)
2
]
1/2
ρ=[(x -x’)
2
+(z-z’)
2
]
1/2
,
K=w/v is the wavenumber, and w = 2*pi*f is the radial frequency. As mentioned above, once a ray is shot (i.e., leaves the transmitter), there are three kinds of mechanisms that change the direction of propagation of a ray: reflection from hard boundaries, refraction because of the refractivity variations of the propagation environment, and diffraction from
Figure 2. Two dimensional propagation environments.
Different kinds of diffraction may be accounted for, from introducing a simple knife-edge diffraction coefficient to double-wedge diffraction, etc. (any time a ray is incident on an edge, a cone of diffracted ray; any time a ray is incident on a tip, a circle of tip-diffracted rays are created).
III.
A
RAY PROPAGATION MATLAB PACKAGE
A short and very simple MATLAB package has been prepared for visualization purpose by using a ray shooting technique. The simulator shoots a number of rays, the angles of departure of which are specified by user, through a propagation medium characterized by various linear vertical refractivity profiles (range-independent cases is assumed here, but its extension to range-dependent case is straightforward). The use selects the maximum range (zlast), the maximum height (xlast), the horizontal layer thickness (dx), the source height (xsource), refractivity parameters, and may choose to locate one or two
PEC obstacles along the propagation path. Figure 2 shows a schematic diagram of core of the package, which is based on direct application of snell’s law between adjacent horizontal layers, the refractive indexes of which are assumed constant.
The ray angle is measured from vertical (x) axis.
As long as the ray coordinates are less than the maximum range and height, the package shoots ray emanating from the source, one-by-one, and stores the points on the ray path as
(x,z) pairs. Five different procedures are included in the package. The first procedure belongs to consecutive application of snell’s law through a multi-horizontally-layered propagation medium. The height of each layer (dx) is user-specified and
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International Journal of Engineering Trends and Technology (IJETT) – Volume 10 Number 13 - Apr 2014 constant. The initial layer is the layer that corresponds to the user-supplied source height. The first ray emanates from the source with the first ray angle, and snell’s law, n i
sin ɸ i
=n i+1
sin ɸ n+1
, (8) gives the refracted ray angle in next layer(here, I is the index of the source layer). The horizontal distance, dz, and the ray segment, ds, of the next (i+1) layer are then calculated by using the trigonometric relations dz i+1
= |dx| sin ɸ i+1
, (9) ds i+1
= [(dx)
2
+(dz i+1
)
2
]
1/2
, (10)
The ray propagates through the layer in the same way until it reaches the maximum range or maximum height. The ray segments’ coordinates, z(i+1) = z(i) +dz i+1
(11) x(i+1) = x(i) ± dz i+1
, (12)
(Where the ± in equation (12) corresponds to upward and
Figure 3. Flowchart of MATLAB simulator downward-propagating rays, respectively) are stored, and the ray loci is then plotted.
The second procedure belongs to the full reflection from the
PEC’s bottom surface, and this is realized by just bending the ray upward with the same angle with which it hits the bottom surface. In MATLAB it is achieved by just replacing dx with – dx. Third process is same as second, except that the reflection occurs on the top of the user specified obstacles. Procedure four is reserved for the case when a ray hits the left wall of the obstacle; in this case, ray ends there, and the package starts shooting the next ray. (It should be noted that the package presented here is designed only for visualization purpose and only for forward propagating rays, but it is quite straightforward to add a procedure to visualize backwardpropagating rays. This is especially essential to account for rays and fields between nearby buildings, where multiple forward and backward reflections, resonances occur).
An important procedure for rays that reach almost 90 o ray angle is the treatment of the ray causics. In terms of programming, this is very similar to procedure two. The rays that reach 90 o
ray angle and that are propagating upward are bent downward by replacing dx with –dx once again (the same applies for rays that
Figure 4. Front end design
Figure 5. Ray propagation through homogeneous environment
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International Journal of Engineering Trends and Technology (IJETT) – Volume 10 Number 13 - Apr 2014 reach a ray angle of -270 and are propagating downward, which are bend upward by replacing –dx with dx ). The flow
IV.
CONCLUSION chart explaining all of these procedures is given in figure 3. In figure, the thick arrow loop is for an individual ray trace. The
A ray propagating MATLAB package has been designed and was discussed in this paper. The package may be used as thin arrow loop shows the loop is for a complete bundle of an educational tool in various EM lectures (e.g., “EM wave theory,” as being used as a research tool to predict ducting rays.
The tri-linear vertical refractivity of the environment is specified via and/or anti-ducting conditions in the presence of buildings and under various atmospheric-refractivity conditions. The user may improve the MATLAB code to include more obstacles, as well as subroutines to accommodate ray fields. n(x)= 1-a o x x<=X
D1
(13)
= 1-a o
(2X
D1
-x) X
D1
<x<=X
D2
= 1-2a o
(X
D1
- X
D2
)+a o x X
D2
<x
[1]
R EFERENCES
M.P.M HALL, L. W. Barclay, and M.T. Hewitt (ed),
Propagation of Radiowaves, London, IEEE press, 1996.
Where a o
is the slope of the refractivity, and X
D1 and X
D2 are ducting and anti-ducting heights, respectively (all of which are user specified).
[2] L. Sevgi, F. Akleman, and L.B. Felsen, “Ground wave propagation Modeling: Problem-Matched Analytical
Formulations and Direct Numerical Techniques, “ IEEE
It should be noted that the package presented here is designed to draw ray paths through a complex environment,
Antennas and Propagation Magazine, 44, 1, February
2002, pp. 55-75. with a variable refractivity profile along a path that may or may not include obstacles. The EM field contributions of rays at a
[3] K. Rizk, J.F. Wagen, and F.Gardiol, “Two-Dimensional chosen observation range or height (i.e., ray field as a function of height at a specified range, ore as a function of range at a
Ray-Tracing Modeling for Propagation Prediction in
Microcelluler enviorments,” IEEE translations on constant height) may be added by using equation (5)-(7), and vehicular technology, VT-46, 2, 1997, pp. 508-518. accumulating ray’s contribution that pass the same observation point:
[4] J.h. tarng, W. R. chang, and b. j. hsu “three-dimentional modeling of 900- MHz and 2.44-GHz radio propagation in corridors,” IEEE transactions on vehicular technology, vt-46, 2, 1997, pp. 519-527.
[5] G.liang and H. L. bertoni,”A new approach to 3D ray tracing for propagation prediction in cities,” IEEE transactions on antenna and propagation, AP-46, 6, june
1998, pp.853-863.
[6] G. Durgin, N. Patwari, and T.S.Rappaport,”improved 3D
Ray Launching Method for wireless propagation prediction,” electronics letters, 33, 16, july 1997, pp.
1412-1413.
Figure 6. Rays through atmosphere with typical refractivity with obstacle in path and tunneling effect.
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