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MEE02:08
A Deterministic Channel Model for Simulation ofMobile Radio
Communications
Daniel Martinsson Tobias Johnsson
September, 2002
Degree of Master of Science in Electrical Engineering
Supervisor: Ronnie Gustafsson
Department of Telecommunications and Signal Processing
Blekinge Institute of Technology
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Abstract
The last decade there has been an enormous expansion in the area
of wirelesscommunication. As new services and devices are
introduced, and more informa-tion is sent between an increasing
number of users, more bandwidth is requiredand the spectrum becomes
more limited. To increase capacity in cellular net-works, cells can
be made smaller and smaller. To be able to plan picocells, suchas
an indoor environment, in an ecient manner, it is important to have
a moredetailed understanding about the channel characteristics.
Further ways to im-prove radio communication is to make use of more
ecient encoding and receivingtechniques, such as spread spectrum.
Also when testing new techniques, knowl-edge about the channel
characteristics and limitations are of interest.
This thesis models the channel characteristics of an indoor
deterministic environ-ment with a simulator using ray tracing
techniques. To make the environment asrealistic as possible, the
physical properties of construction materials are takeninto
account. The simulator is able to track each individual radio wave,
makingit possible to calculate interesting parameters such as
received power, phase, anddirection-of-arrival. The simulator
operates in 2D-environments. A lot of workhave been done to extend
the simulator to a 3D-version, although some problemsstill remains
to be solved.
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Contents
1 Introduction 71.1 Wireless communication system . . . . . . .
. . . . . . . . . . . . 7
1.1.1 The channel . . . . . . . . . . . . . . . . . . . . . . .
. . . 71.2 Multipath . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 81.3 Indoor environments . . . . . . . . . . . . .
. . . . . . . . . . . . 91.4 Simulation . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 10
2 Finding paths 112.1 Ray tracing . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 11
2.1.1 Types of ray tracing . . . . . . . . . . . . . . . . . . .
. . 122.1.2 Acceleration techniques . . . . . . . . . . . . . . . .
. . . . 14
2.2 Ray tracing model . . . . . . . . . . . . . . . . . . . . .
. . . . . 152.2.1 Environment description . . . . . . . . . . . . .
. . . . . . 162.2.2 Options . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 172.2.3 Finding paths . . . . . . . . . . . . . . .
. . . . . . . . . . 172.2.4 Adding transmission to found paths . .
. . . . . . . . . . . 202.2.5 Results . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 20
3 Electric eld calculations 213.1 Electromagnetic elds . . . . .
. . . . . . . . . . . . . . . . . . . 21
3.1.1 Plane waves . . . . . . . . . . . . . . . . . . . . . . .
. . . 213.2 Antenna characteristics . . . . . . . . . . . . . . . .
. . . . . . . . 22
3.2.1 Polarization . . . . . . . . . . . . . . . . . . . . . . .
. . . 233.3 Dielectric materials . . . . . . . . . . . . . . . . .
. . . . . . . . . 233.4 Electric eld calculation . . . . . . . . .
. . . . . . . . . . . . . . 25
3.4.1 Free space propagation . . . . . . . . . . . . . . . . . .
. . 253.4.2 Reection . . . . . . . . . . . . . . . . . . . . . . .
. . . . 263.4.3 Transmission . . . . . . . . . . . . . . . . . . .
. . . . . . 293.4.4 Electric eld calculation . . . . . . . . . . .
. . . . . . . . 323.4.5 Diraction . . . . . . . . . . . . . . . . .
. . . . . . . . . . 32
3.5 Calculating results . . . . . . . . . . . . . . . . . . . .
. . . . . . 363.5.1 Power and phase . . . . . . . . . . . . . . . .
. . . . . . . 363.5.2 Direction-of-Arrival . . . . . . . . . . . .
. . . . . . . . . . 37
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4 Verication 394.1 Verication sources . . . . . . . . . . . . .
. . . . . . . . . . . . . 39
4.1.1 Radiowave Propagation Software (RPS) . . . . . . . . . .
404.2 Model verication . . . . . . . . . . . . . . . . . . . . . .
. . . . . 40
4.2.1 Basic scenarios . . . . . . . . . . . . . . . . . . . . .
. . . 404.2.2 2D-scenarios . . . . . . . . . . . . . . . . . . . .
. . . . . . 42
4.3 Model extensions . . . . . . . . . . . . . . . . . . . . . .
. . . . . 474.3.1 3D-model . . . . . . . . . . . . . . . . . . . .
. . . . . . . 484.3.2 Diraction . . . . . . . . . . . . . . . . . .
. . . . . . . . . 50
5 Conclusion and future work 535.1 Conclusion . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . 535.2 Future work . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . 53
A User manual 55A.1 Starting a simulation . . . . . . . . . . .
. . . . . . . . . . . . . . 55
A.1.1 Dening an environment . . . . . . . . . . . . . . . . . .
. 56
B Software description 59
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List of symbols
h(t) Channel impulse responseAi Magnitude of a multipath
component of a channel impulse response A delayi A phase
() Dirac pulsen(t) A noise process
Wavelengthc Speed of light 3 108 m/sf FrequencyE Electric eldH
Magnetic eld
DG Directivity of an antennaDG(, ) Directive gain
Permittity0 Permittivity of free space = 8.854 1012F/m
Conductivity0 Conductivity of free space Permeability0 Permeability
of free space = 4 107H/m Intrinsic impedance0 Intrinsic impedance
of free space = 120 Complex permittivity
Lfs Free space lossk Relative wave number of materialk0 Wave
number of free space Grazing angle of incidencer Fresnel reection
coecientt Fresnel transmission coecient Intersection coecient
, d, d Diraction angles
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h Height of diraction pointD Diraction coecientL Distance
parameter, plane waves
F (x) Fresnel transition functionA Spreading factorP Power
Aem Maximum eective aperture of an antennaS Power densityE Phase
of received electric eld
, Spherical angles
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Chapter 1
Introduction
This chapter introduces the basic concept of how a communication
system is builtup, and why simulation is interesting to apply when
studying communicationchannels. Further, signal behavior depending
on the construction of environmentsare explained.
1.1 Wireless communication system
The main parts in a communication system consists of a source, a
transmitter,a channel, a receiver and a destination. The
transmitter takes information fromthe source and converts it to a
form suitable for transmission. During transmis-sion the signal is
aected by the environment in the channel. A channel is
theenvironment in which the signal propagates, e.g. a building. If
the channel char-acteristics are known, then it is possible to
predict the eects on the transmittedsignal at the receiver.
Source Transmitter Channel Receiver Destination
Figure 1.1: A simple communication system
1.1.1 The channel
To determine a channel model, mathematical descriptions of the
transmitter, re-ceiver and the eect of the environment (walls etc.)
to the signal must be known.If the channel characteristics are
known, tests can be made to evaluate whichencoding techniques are
most proper to achieve minimum error rate. A linearchannel can be
totally described by its impulse response, i.e. what the
receivedsignal would look like if the transmitted signal was an
impulse. In environments
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where dierent paths from transmitter to receiver exists, the
channel impulseresponse h(t) for a deterministic time-invariant
channel is given by
h(t) =i=0
Aieji(t i) , (1.1)
where Ai is the magnitude of the impulse response at delay i
with phase i and() is the Dirac pulse.
The impulse response can be used to obtain the response y(t) of
the channelfor the transmission of any signal x(t) by convolving
x(t) by h(t) and addingnoise.
y(t) = x(t) h(t) + n(t) =
=0
h()x(t ) + n(t) , (1.2)
where represents the convolution operation and n(t) is a noise
function, oftenassumed to be a zero mean Gaussian process [1].
To allow for a realistic modelling of the mobile radio channel
it is also possi-ble to use other parameters such as power and
direction-of-arrival (DoA) of theincoming signals.
1.2 Multipath
In a wireless environment the transmitted signals are aected in
dierent ways,illustrated in g. 1.2. Reection occurs when the signal
hits a surface that is largerelative to the wavelength of the
signal. Diraction occurs at edges where thesurface is large
compared to the wavelength of the signal. The diracted signalare
divided into many new weaker signals that propagates in dierent
directionswith the edge as the source. Scattering occurs when the
size of an obstructedsurface is equal to or less than the
wavelength of the signal. The scattered signalpropagates into
several weaker outgoing signals [2]. The result of these eectsis
that the signal often reaches the receiver by more than one path,
resultingin a phenomenon called multipath propagation. The signal
that is received bythese multiple paths is a distorted version of
the transmitted signal. The receivedsignal is further corrupted by
other eects, such as noise, co-channel interferenceand
non-linearities. Multipath propagation seriously degrades
performance ofcommunication systems. It is hard to eliminate
multipath disturbances, but ifthe medium is well characterized it
is possible to design transmitter and receiverto t the channel, and
in this way reduce the eect, or even take advantage ofthese
disturbances. Detailed characterization of radio propagation is
therefore amajor requirement for successful design of indoor
communication systems.
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Figure 1.2: Sketch of three important multipath eects; Reection
(R), Dirac-tion (D) and Scattering (S). At the reection point only
one outgoing ray iscreated, while at the diraction- and scattering
points several outgoing raysare created. The gure only shows paths
propagating towards the receiver.(Reprinted from [2])
1.3 Indoor environments
In outdoor environments usually basestations are located above
rooftops, yieldingthat most of the multipath eects occur close to
the mobile. Inside buildings,both transmitter- and receiver
antennas are close to surfaces where multipatheects takes place.
Therefore, inside buildings many paths exists to the receiver.If
the communicating devices are in line-of-sight (LoS), then
diraction and scat-tering have less eect, since most of the energy
is transmitted by the LoS- andreection paths. If no LoS path exist,
diracted and scattered paths plays a moreimportant role to the
received signal.
When modelling an indoor environment, respect can also be taken
to furniture,surface materials etc. Walls, windows and doors have
xed locations and arereasonable to include in the model. However
furniture is more complicated toinclude, since they have irregular
shapes and their positions are often changed.Depending on the
electrical properties of obstructed surface materials, the extentto
which a signal will penetrate, be reected from, or be diracted
around thesurfaces can be determined. This is further discussed in
chapter 3.
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1.4 Simulation
Testing radio communication systems in the eld is time consuming
and expen-sive, and it is dicult to represent all conditions likely
to be encountered inpractice. This make it attractive to test
systems in a simulation environment,since it is easy to dene the
environment conditions. Once the simulator is imple-mented, it is
possible to test existing systems as well as using it as a design
toolin the development of new systems. To test dierent systems in
practice, hard-ware have to be implemented. In simulation the best
system can be evaluatedbefore implementation. One of the techniques
used when implementing a radiosimulator is ray tracing. This method
is widely used in computer graphics [3],and since radio- and
lightwaves have similar properties the simulator can
trackradiowaves in the same manner as lightwaves. Ray tracing
methods are presentedin chapter 2.
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Chapter 2
Finding paths
To describe a channel model earlier methods have been based on
empirical meth-ods, which means that the environment of the model
are considered in a statisticalsense (e.g. average height of walls,
width of corridors, etc.) [4] and do not describevariations in
signal strength around any particular object. Also it does not
helpsystem designers to position the antenna for optimal
performance. When dealingwith indoor environments, the statistical
assumptions of the empirical methods donot work any more. Since
there are not many obstacles shadowing the receiver,the properties
of each individual obstacle, such as exact position, orientationand
dielectric properties, are important. To solve this, deterministic
simulationconsiders a realistic geometrical and physical model of
the environment. One de-terministic approach is ray tracing,
discussed in this chapter. Geometrical optics[5] is used to compute
the strengths of the reected and transmitted rays omittedfrom ray
tracing, and diracted rays can be calculated using dierent
methods,commonly based on Uniform Theory of Diraction (UTD)
[4][6][7].
2.1 Ray tracing
Ray tracing is a technique which have been used for long in
computer graphics,when tracing light waves emitted from a light
source. The idea is to trace radiowaves in the same way, from a
transmitter to a receiver. Once all possible pathshave been
identied, electromagnetic techniques are applied to the rays to
com-pute interesting parameters, such as signal strength. The
electrical length of thedierent ray paths give the amplitudes and
phases of the component waves. Thesignals are also aected in
amplitude and phase when being transmitted, reectedor diracted on
obstacles during propagation. Those eects are accounted for inthe
calculations.
By using site-specic information such as building databases and
antenna char-acteristics ray models can deterministically describe
a complete propagation sce-
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nario. It is possible to implement the model in two- or three
dimensions. Though,a three-dimensional model is preferable since
oor and ceiling also can be takeninto account, giving more
realistic models. Of course ray tracing has some dis-advantages and
limitations. Many calculations of paths are made, making
thesimulation a time consuming process. However, todays computers
are fasterthan earlier ones, and together with dierent acceleration
techniques they cangive reasonable simulation times. It is dicult
to have a precise dened environ-ment, and since objects in the
environment are often moved it is hard to have anup to date
building database.
2.1.1 Types of ray tracing
Two basic methods are widely used: the ray launching method
(direct-, bruteforce-, shoot and bounce method) [8][9] and the ray
tracing method (inverse-,image method) [4][10]. In ray launching
rays are cast in many directions andthen traced. The image method
considers all obstacles as potential reectors andtakes into account
their eect on the ray path using the method of images.
Ray launching
The ray launching method, illustrated in g. 2.1, starts with
checking if LoS existbetween transmitter and receiver. After this,
a ray is launched in a specied di-rection from the transmitter and
further traced to determine if it intersects withan obstacle. If it
does not, the process stops and launches a new ray from thesource
in another direction. If the ray was obstructed the program divides
thesource ray into a transmitted and reected ray, which are then
treated in a simi-lar fashion to the source ray. This recursive
process continues for each ray untilthe ray reaches the receiver,
until a specied number of intersections is exceeded,until the ray
energy falls below a predened threshold or until the ray is lost.In
the ray launching technique it would be unrealistic to consider the
receivinglocation as innitely small. Therefore a reception sphere
around the receiver withsmall radius is used to capture rays
passing by. If the ray intersects this sphere,it is received and
contributes to the total received signal, otherwise the signal
isnot received. The number of launched rays must be large enough to
be able toget a good characterization of the channel, i.e. there
must be a small constantangle separation between launched rays.
Ray launching has a few disadvantages. To get accurate results,
many rays haveto be launched, and only a fraction of these reach
the receiver. The accuracy alsodepends on the radius on the
reception sphere. If it is to small, rays will pass by.If it is to
large, paths might be duplicated. It is dicult to include
diractionin the ray launching method, since the large spread of the
diracted rays will behard to trace further.
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Rx
Tx
Reception sphere
Figure 2.1: Ray launching: The transmitter (Tx) launches rays in
many direc-tions separated by a constant angle. Only rays crossing
the sphere of the receiver(Rx) are considered as received.
The image technique
The image technique overcomes some of the limitations of ray
launching. Theimage technique is an analytical method and therefore
only exact paths are re-ceived by the receiver. Hence the receiver
antenna size is assumed to be innitelysmall. This makes the image
technique reliable and more accurate compared tothe ray launching
technique which give approximate results.
The idea with the image method is to compute only exactly
reected paths to thereceiver by using transmitter images (g. 2.2).
At rst, the transmitter creates animage point with use of the rst
reecting surface. The image point is a mirror ofthe transmitter
position on the opposite side of the surface. If another
reectiontakes place after reecting the rst surface, a new image
point is calculated inthe same way, but now the previous calculated
image point is considered as thetransmitter.
When all image points have been created, a line is drawn from
the receiver lo-cation to the last image point. Then a check is
made if the surface belongingto the image point of interest
intersects with that line. If not, the path is notvalid. If it
does, a new line from the intersection point to the next image
point isdrawn and a new check if intersection occurs is made on the
current surface. Forsuccessful intersections this procedure
continues with new intersection tests untilthe transmitter has been
reached. The image points that have been created can
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be used to test dierent paths for dierent number of reections
before it reachesthe receiver. Consequently many images and paths
have to be calculated andtested, resulting in an exhaustive
ray-path searching, especially in complex envi-ronments with many
surfaces. This results in high computation times. Thereforemany
acceleration techniques have been developed in order to make
simulationmore ecient. However for simple environments, the image
technique is fasterthan ray launching. The image algorithm is in
general more complicated than aray launching algorithm, because all
possible paths connecting transmitter andreceiver must be checked.
However, the image method is well suited to moreaccurately compute
diraction, phase and polarization.
Rx
TxIm 1 (Tx-wall 1) Im 2 (Im 1wall 2)
wall 1
wall 2
Figure 2.2: Example of the image technique: First and second
order images arecreated and used to nd the paths to the receiver.
Only received rays are shown.
2.1.2 Acceleration techniques
As the number of surfaces included in an environment increases,
the numberof intersections tests which must be performed tends to
increase exponentially.This means that the simulation times
increases dramatically. To get reasonablesimulation times for
complex environments, dierent acceleration techniques canbe
applied. Many types of acceleration techniques appear in the
literature, bothfor ray launching- and image techniques
[10][11].
Bounding volumes
Bounding volumes is a simple but ecient technique [9]. It can be
implementedwith dierent algorithms, but the basic idea is presented
in g. 2.3.
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As the environment is dened, each surface is associated with a
larger boundingvolume. Intersection tests are rst performed on
bounding volumes, rather thanon every existing surface. If a test
fails (no intersection), then intersection testson the group of
surfaces associated with that bounding volume does not needto be
performed. If an intersection with a bounding volume is detected,
furthertests are performed on the surfaces within the volume.
By further grouping surfaces and, by extension, grouping nearby
bounding vol-umes into a hierarchy of bounding volumes, the number
of intersection testswhich must be performed can be signicantly
reduced. Other more advancedimplementations detects areas in the
environment where many surfaces exists,and associates those areas
with bounding boxes of custom sizes.
Im(Tx)Tx
Boundingvolume
Wall
reflectionarea
Figure 2.3: Basic concept of bounding volumes. Surfaces
belonging to boundingvolumes not intersected by the possible
reection area of the wall do not have tobe considered as potential
reectors.
2.2 Ray tracing model
The ray tracing model in this thesis uses the image technique to
nd paths. Thismethod was chosen due to its many advantages that has
been stated earlier. This
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part of the chapter shows the main structure of the ray tracing
software, and howthe software has been implemented with use of the
image technique.
Environmentdescription
Options
Create trees
Find reflection paths
Find diffraction paths
Add transmission to found paths
Results
Figure 2.4: Structure of the ray tracing software
2.2.1 Environment description
The model is able to describe a typical indoor environment (e.g.
a blueprint ofa building) for both 2D- and 3D-environments. In 3D
it is possible to includewindows and doors.
Walls are considered to consist of only one material with
various thickness, andare dened with 2D-coordinates. For a more
realistic model, the material of eachfacet are dened and taken into
account. Walls are assumed to be perpendicularto the oor, with a
common height. These assumptions are possible since themodel do not
include furniture objects with dierent heights and shapes.
Furni-ture is time-consuming to describe and update for the user,
and more complicatedto implement in the simulator. Floor and
ceiling are dened using the knowl-edge of which walls are outer
walls. Since the model is concentrated on indoorcommunication, the
environment outside the building is of no interest and notpossible
to dene.
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Figure 2.5: Example of an indoor environment with windows and
doors
2.2.2 Options
Besides dening the environment, the inclusion of dierent types
of paths isoptional. The direct path and diraction paths can be
included or not, andthe maximum number of reections can be set. The
model handles only onediraction in the same path, since
measurements [8][12][13] have shown thatthere are no signicant
dierence in the accuracy of results when simulating withan
increasing number of diractions. As the number of reections
increases or ifdiraction is included, the simulation time
increases.
2.2.3 Finding paths
Use of image trees
Since the model uses the image technique, it is suitable to use
tree structures,illustrated later in this chapter, to organize the
potential paths from transmitterto receiver. As a tree consists of
images, a method to calculate the imagepointsis needed.
To nd the imagepoint S belonging to point P , (2.1)-(2.3) can be
used, where A
is an arbitrary point in the plane, Q lies in the plane and on
the lineSP and
denotes the dot product:
u =AP (2.1)
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u =QP =
u n|n|2 n (2.2)
S =OP 2u (2.3)
The surface normal n is equal to the cross product between the
two vectorsdescribing the surface. O is dened as origo.
S
A P
u
u
Q
Figure 2.6: Illustration of an imagepoint S
For cases when a point is projected onto a facet that is roof or
ceiling, the im-agepoint has the same position as the point except
that the height is changed.When it comes to diraction no
imagepoints are calculated. The geometry ofdiraction is presented
in chapter 3.
Reection paths
Fig. 2.7 is a simple room with belonging three structure. There
are four facets,where all possible paths with up to two reections
are considered. Consecutivereections from the same facet are not
possible. By traversing the image tree, allpaths are found and
tested, and valid paths are saved for further treatment. Foreach
level, the receiver can start new paths towards the
transmitter.
Diraction paths
Fig. 2.8 includes a diraction edge. Diraction edges are treated
as facets andincluded in the creation of the usual image tree with
the transmitter as thestartpoint. However, the image tree never
continues to expand from a diractionedge. This is because the
height of the diraction point is unknown, and toget this, it is
necessary to know where the transmitter and receiver are located.To
know where the sender should be considered to be located if
reections hasoccurred before the diraction edge, the image point
before the diraction edgein the tree is used as a virtual
transmitter.
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T1 2 3 4
1 3 4 1 2 4 1 2 32 3 4
R
Rx
Wall 3
Wall 1
Wall 4
Tx
Wall 2
LOS
LOS
Figure 2.7: Simple room with belonging image tree illustrating
possible reec-tion paths. Each number in the image tree corresponds
to the wallnumber in theblueprint. In the image tree, the dashed
lines indicates the valid paths in theroom.
d
Rx
Wall 3
Wall 1
Wall 4
Tx
Wall 2
T
2 3 4 d1
d d dd
R
2 3 41
d
Figure 2.8: Simple room illustrating how diraction paths are
found by combin-ing two image trees; one starting from transmitter
and one starting from receiver.The lines with arrows shows found
paths and the direction they are traversed inthe trees.
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The receiver position can be considered to be at various
locations since theremay be many reections after the diraction
edge. These positions are unknown,and to nd these, an image tree
starting from the receiver also is created. Ifthe diraction edge
now acts as an end point traversing this three to reach
thereceiver, all possible paths to the receiver and their virtual
receiver positions arefound. Now the exact diraction points for all
possible paths can be calculatedusing the virtual transmitter and
the virtual receiver locations. When the exactdiraction point is
known, intersection points for each path is found by traversingthe
trees from the diraction point to the transmitter and in the other
case fromthe diraction point to the receiver.
2.2.4 Adding transmission to found paths
Between every pair of intersection points occurring in a path, a
check is madewhether this part of the ray also transmits through
other facets. The found trans-mission intersections are added to
the path. If many transmissions occur, it isimportant to arrange
them in correct order, i.e. by checking where on the line,that is
created by the two intersection points, the intersections
occur.
It is possible to choose how the signal behaves when it hits an
outer wall; ifit is transmitted or not. When transmission is
allowed through outer walls andan intersection on a oor or ceiling
occurs outside the building, the path is notvalid and is discarded
since the environment outside the building is not dened.This can be
checked by counting the number of outer walls that has been
passedbetween two intersection points.
2.2.5 Results
The results consists of all found ray paths. Each path is
described by its inter-sections, and each intersection contains
geometric information that is needed forcalculations of reection-,
transmission- and diraction coecients described inchapter 3.
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Chapter 3
Electric eld calculations
When the ray paths from transmitter to receiver has been found,
the geometricalproperties of each ray can be used to calculate the
total received electromagnetic(EM) eld. The electromagnetic eld is
aected by several parameters such ascharacteristics of the antenna
and dielectric properties of the surface materialsintersected in
the propagation path. This chapter outlines how this process
worksand how the electric eld can be used to obtain dierent
results.
3.1 Electromagnetic elds
An electromagnetic eld is generated when charged particles
(electrons) changevelocity. All electrically charged particles are
surrounded by electric elds, andcharged particles in motion produce
magnetic elds. Electromagnetic elds aretypically generated by
alternating current (AC) in electrical conductors [14].
Thewavelength () of an electromagnetic eld is related to the
frequency (f) by = c/f , where c 3 108 m/s (speed of light in free
space).
3.1.1 Plane waves
The space surrounding an antenna can be divided into three
dierent regions.These are named the reactive near eld, the
radiating near eld region and thefar eld region. In those regions
the electromagnetic elds behaves in dierentways. To reduce
complexity when making a simulation model, it is assumed thatthe
radiation occurs in the far eld. The far eld is entered when ( r),
wherer denotes the radius of the antenna [15]. This thesis consider
antennas to beinnitely small points in space. With this
consideration radiation always occurin the far eld. The main
characteristic of the far eld is that the electric eld (E)and
magnetic eld (H) components are transverse to the direction of
propagationand to each other, resulting in plane waves (g.
3.1).
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EH
Direction of propagation
Figure 3.1: Denition of a plane wave: The electric eld (E) and
the magneticeld (H) are transverse to the direction of propagation
and to each other.
3.2 Antenna characteristics
An antenna is an electrical conductor that radiates and receives
electromagneticenergy [2]. The antenna transforms radio frequency
electrical energy from thetransmitter into electromagnetic energy
and radiates it into the surrounding en-vironment. For a receiving
antenna the same procedure is done in reversed order.
All kinds of antennas does not radiate equally well in all
directions. Dierentantenna types are characterized by their
radiation- and receiving patterns, whichare illustrated in a
graphical way (g. 3.2). The patterns can be coupled to thethe
directive gain DG(, ), which is a measure of the power radiated in
a cer-tain direction. The directivity DG is the maximum value of
the directive gain.The simplest antenna pattern is produced by an
idealized antenna known as theisotropic antenna. This kind of
antenna radiates power in all directions equally,yielding a
radiation pattern that is a sphere, with unity gain (DG = 1) and
withthe antenna in the middle. This is a suitable antenna model to
use when simulat-ing indoor environments and when examining ideal
theoretical models [16]. Anisotropic radiator is ideal and not
physical realizable, and is taken as referencefor expressing
directive properties of other types of antennas [15].
Figure 3.2: Antenna patterns for an isotropic antenna (left) and
a directiveantenna (right).
22
-
3.2.1 Polarization
When choosing antennas it is very important taking the
polarization into account.Communication systems use linear- or
circular polarization. This thesis is onlyfocused on linear
polarization. Linear polarization can be divided into vertical-and
horizontal polarization, and radiates in the direction of
propagation. The E-and H-eld decides the polarization of the wave.
Vertical polarization occurs whenthe electric eld is vertical
(perpendicular) to the surface and the magnetic eld ishorizontal
(parallel) to the surface (g. 3.3). Accordingly, horizontal
polarizationhas the electromagnetic eld horizontal to the surface
and the magnetic eldvertical to the surface (g. 3.4). The
orientation of a plane wave can be describedby a horizontal- and
vertical polarized part. At the receiving antenna, onlythe part
with the same polarization as the antenna is considered.
Maximumsignal strength is reached when both transmitter and
receiver uses the samepolarization. Most base stations use vertical
or horizontal antennas, and due tothis linear antennas are modelled
in this thesis. Vertical polarization is oftenused for two-way
communication and horizontal polarization for broadcast suchas
television and radio [17].
Plane of Incidence
H
E
H
E
E
H
Surface
Figure 3.3: Vertical polarization (Reprinted from [16])
3.3 Dielectric materials
Depending on the dielectrics of the material of a surface, the
intersected rayis aected in dierent ways. To be able to calculate
eects on incoming raysproperly, a few assumptions are made:
Both air and surfaces in the environment are homogenous, i.e the
charac-teristics of the material do not depend on position
23
-
HE
Plane of Incidence
HE
HE
Surface
Figure 3.4: Horizontal polarization (Reprinted from [16])
The surface between two dierent medias are perfectly smooth,
i.e. noroughness exists
The parameters that describes the dielectric properties of a
material are permit-tivity, conductivity and permeability [14].
The denition of permittivity is a constant of proportionality
between elec-tric displacement and electric eld intensity. In
free-space the permittivity is0 = 8.854 1012 Farad per meter (F/m).
Permittivity is often expressed in rel-ative permittivity, r = /0,
which is the permittivity of the medium () relativeto
free-space.
Conductivity is a measure of how easy an electric current ows
through a medium,measured in Siemens per meter (S/m), equal to the
inverse of the resistance. Theconductivity of free-space is 0 = 0.
When = , the medium is dened as aperfect conductor.
Permeability diers from the denition of permittivity in the
sense that it isa constant of proportionality that exists between
magnetic induction and mag-netic eld intensity instead of the
electric eld. In other words, it tells howmuch of the
electromagnetic wave the medium absorbs. This constant is equalto 0
= 4 107 Henry per meter (H/m) in free space [15], and can be taken
as = 0 for non-magnetic materials [16].
With use of these three parameters it is possible to express the
intrinsic impedance(equal to wave impedance of a plane wave) of the
medium. The intrinsic impedanceis given by
24
-
=
j
, (3.1)
where = 2f . The intrinsic impedance is equal to the ratio
between the electriceld and the magnetic eld, as seen in g. 3.1. In
free space it reduces to [15]
0 =
00
= 120 . (3.2)
To express the dielectric properties between two dierent medias
a term calledcomplex permittivity () is used,
=(12
)2. (3.3)
The complex permittivity is frequency dependent, especially the
imaginary part.In this work 1 is always the intrinsic impedance of
free space = 120. With useof and , the complex permittivity can be
calculated using
= r j60 , (3.4)when 1 is the intrinsic impedance of free space
[12].
Type of material Complex permittivity ThicknessWood 2.5 0.03j
0.05 mConcrete 5.0 0.4j 0.20 mGlass 6.0 0.05j 0.01 m
Table 3.1: Dielectric properties of dierent materials at
frequency 1 GHz [18]
Thickness of the surface is also an important parameter taking
into account whencalculating the eect of the media on the
electromagnetic wave.
3.4 Electric eld calculation
To calculate the total received electric eld the free space loss
and the intersectioncoecients have to be known.
3.4.1 Free space propagation
For all types of radio propagation the signal strength decreases
with distancetravelled. This form of attenuation is known as free
space loss. The basic trans-mission loss for a unity gain and
lossless antenna is
Lfs =P0PR
=(4r
)2, (3.5)
25
-
where P0 is the power radiated by the transmitting antenna and
PR the powerreceived by the receiving antenna [15].
The strength1 of the electric far-eld can be calculated
using
E0 =
30P0DG(, )
rejk0r , (3.6)
where k0 = 2/ is the wave constant of free space, DG(, ) the
directive gainand r the path distance.
3.4.2 Reection
In the case when a ray propagates trough air, consideration only
has to be takento free space loss. When it comes to paths that
involves reections, loss arisingfrom intersections into other
medias has to be added to the free space loss. Thisremaining
electromagnetic eld is determined by calculating reection
coecientsthat are dependent of incident angles to the medium, the
dielectric properties ofthe medium and the thickness of the
surface. The reection coecient is dierentdepending on the
polarization of the incoming wave.
The geometry of a reection is in some way quite simple. Using
Snells lawof reection [5] it is shown that the grazing angle of
incidence () of the incom-ing ray is the same as the angle of the
outgoing ray (g. 3.5).
The grazing incidence can easily be calculated using the surface
normal n,
= arcsin(i n|i|
), (3.7)
where i is the vector in the direction of propagation, |i| is
normalized, and denotes a vector dot product [16].
Figure 3.5: Geometry for reection
1The sign of the exponential part is dierent from [15].
26
-
Reection coecient
The most simple case is the scenario when the intersected
surface is a perfect con-ductor ( =).This means that the medium is
perfectly reecting the incomingelectromagnetic wave. In this case
the reection coecient is 1 for horizontalpolarization and +1 for
the case of vertical polarization. This relationship mightbe easier
to understand when looking at g. 3.3-3.4 where, in the horizontal
case,the electric eld (E-eld) is rotated 180 degrees after an
intersection. In the verti-cal case the E-eld remains in the same
direction after the intersection comparedto the direction of
propagation.
A widely used formula for calculating the reection coecient is
the Fresnel for-mula [4][16]. In the calculations the Fresnel
formula takes into consideration thecomplex permittivity and the
angle of incidence (). For horizontal polarization,the coecient is
calculated by
rh =sin cos2 sin +
cos2 (3.8)
and for vertical polarization
rv = sin cos2 sin +
cos2 . (3.9)
When a ray propagates from a horizontally polarized antenna and
intersects witha vertical surface, the vertical coecient is used.
Subsequently, the horizontalcoecient is used when it intersects
with a horizontal surface. The scenario isreversed when the
transmitting antenna is vertically polarized. In g. 3.6 and3.7, it
is shown for horizontal and vertical polarization respectively, how
the re-ection coecient behaves for dierent types of materials and
angle of incidence.To make the model more realistic it is possible
to introduce the thickness of theintersected surface. Intersected
surfaces can consist of several layers with dierentmaterial and
thickness. Reection coecients for various thickness are presentedin
g. 3.9 and 3.10.
The formula for calculating the reection coecient () according
to g. 3.8 isgiven by2 [19]
h = v =r0 + rde
2jkd
1 + r0rde2jkd, (3.10)
where r0, rd are the Fresnel coecient when z = 0 and z = d. k is
the relativewave number of the intersected material, given by
2The sign of the exponential part is dierent from [19]
27
-
0 10 20 30 40 50 60 70 80 900.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Incident angle in degrees
Ref
lect
ion
coef
ficie
nt
(a)
(b)
(c)
Figure 3.6: Reection coecient for horizontal polarization as a
function of inci-dent angle. Material: (a) Wood ( = 2.5 0.03j) (b)
Concrete ( = 5.0 0.4j)(c) Glass ( = 6.0 0.05j).
0 10 20 30 40 50 60 70 80 900
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Incident angle in degrees
Ref
lect
ion
coef
ficie
nt
(a)
(b)
(c)
Figure 3.7: Reection coecient for vertical polarization as a
function of inci-dent angle. Material: (a) Wood ( = 2.5 0.03j) (b)
Concrete ( = 5.0 0.4j)(c) Glass ( = 6.0 0.05j).
28
-
z=dz=0
1
23
Figure 3.8: Example of three dierent medias. 1 and 3 are dened
as innitelywide.
k =2f
sinc
. (3.11)
In this thesis, intersected surfaces are assumed to consist of
only one material,yielding three dierent medias: airdielectric
materialair. This results in aspecial case where the wave
impedances in material 1 and material 3 are thesame (1 = 3). In
this case
3, r0 = rd, yielding
h = v = r01 e2jkd1 r20e2jkd
. (3.12)
3.4.3 Transmission
Transmission intersections are considered in a similar way as
reection intersec-tions, and the same parameters are taken into
account. As for reection thetransmission coecients are calculated
separately for horizontal- and vertical po-larization. Due to the
higher permittivity in materials compared to the permit-tivity of
free space, the velocity of electromagnetic waves is lower and
createsdelays. However, the delays in walls are so small that they
can be neglected [18].In reality reections occur within the medium.
They bounce and creates new re-ected and transmitted outgoing rays
from the media. Still, the main contributoris the rst transmitted
ray. When a ray enters a new media, it is bent towards the
3The sign of the exponential part is dierent from [19]
29
-
0 10 20 30 40 50 60 70 80 900
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Incident angle in degrees
Ref
lect
ion
coef
ficie
nt
(b)
(c)
(a)
Figure 3.9: Reection coecient, depending on thickness, for
horizontal polar-ization as a function of incident angle. Material
wood ( = 2.5 0.03j) withthickness: (a) 0.05 m (b) 0.20 m (c) 0.50
m
0 10 20 30 40 50 60 70 80 900
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Incident angle in degrees
Ref
lect
ion
coef
ficie
nt
(a)
(b) (c)
Figure 3.10: Reection coecient, depending on thickness, for
vertical polar-ization as a function of incident angle. Material
wood ( = 2.5 0.03j) withthickness: (a) 0.05 m (b) 0.20 m (c) 0.50
m
30
-
iEr
tE
Et
E
I II
2
III
Figure 3.11: Inner reections (Reprinted from [16])
normal. In this thesis, the changes of the direction at the
dielectric boundariesare ignored, because of the complexity of
implementation.
Transmission coecient
The Fresnel coecients for transmission are
th =2 sin
sin +
cos2 (3.13)
for horizontal polarization, and
tv =2
sin sin +
cos2 (3.14)
for vertical polarization.
When taking thickness into consideration, the procedure is the
same as for thereection case. The transmission coecient according
to g. 3.8 are calculatedby4 [19]
h = v =t0tde
jkd
1 + r0rde2jkd, (3.15)
where t0, td are the Fresnel transmission coecients when z = 0
and z = d.
In the special case when 1 = 3 the formula reduces to4
4The sign of the exponential part is dierent from [19]
31
-
h = v =(1 r20)ejkd1 r20e2jkd
. (3.16)
An interesting observation of the resulting formula is that the
transmission coef-cient can be calculated with use of the Fresnel
reection coecient instead ofthe Fresnel transmission coecient.
3.4.4 Electric eld calculation
The resulting reection- and transmission coecients contains a
magnitude- anda phase part that is described by a complex value.
When the refracted eldhas been determined at the rst intersected
surface, it is used as the incidentwave eld on the next
intersection. This continues until the ray has reached thereceiver.
In a direct path it is only the magnitude that is aected, while
fora path including any type of intersection, there are changes in
both magnitudeand phase [16]. The received electric eld for a path
including i intersections iscalculated by
E =
30 P0 DG(, )
r
( ni=1
i
)ejk0r , (3.17)
where is the coecient for the current type of intersection.
Equation 3.17 isvalid, no matter how many intersections or what
kind of intersections (reectionor transmission) there has been
along the path.
At the receiver the received eld from all intersecting paths are
added with theelectromagnetic eld from the direct path, if such
exist.
ETOT = E0 +n
j=1
Ej , (3.18)
where j denotes the intersected path.
3.4.5 Diraction
Diracted rays are produced by incident rays which hit edges or
corners. Thiskind of multipath eect can play a signicant role when
there is no line-of-sightand many reections are required to reach
the receiver.
Geometry of Diraction
As in the reection case with Snells law, there is a geometrical
connection be-tween the incident ray and outgoing diracted rays.
Seen from the elevationview, the angle is equal to the angle of the
incident ray relative to the edge, as
32
-
illustrated in g. 3.12. When the incident ray intersects with a
corner it is splitinto several outgoing rays. These diracted rays
propagates with the same angle as the incident ray relative to the
edge. This creates a hollow cone of rays,known as the Keller cone
[6].
Figure 3.12: Elevation view of diraction
Seen from above, the geometrical properties of diraction is
described by theangle
d for the incoming ray, and by d for the outgoing diracted ray,
both
with respect to the 0-face (g. 3.13).
These geometric properties must be known to calculate the
diraction coecient D.Since in must be equal to out, the correct
height where this property is fullledat the diracting corner must
be calculated. By using knowledge of the trans-mitting point (T)
and receiving point (R), the height h of the intersection pointat
the corner is found using eq. 3.19 [20].
h =hTdR + hRdT
dR + dT, (3.19)
where hT and hR are the height of the sending- and receiving
point respectively.dT is the distance vector from the corner to the
sending point when their heightsare set to zero. Accordingly, dR is
the distance vector from the corner to thereceiver point at zero
height.
33
-
d
Shadowboundrary
d
s
s
0-face
n-facen=3/2
Figure 3.13: Top view of diraction
Diraction coecient
Several dierent methods how to calculate the eects of diraction
are presentedin the literature. Historically, the rst diraction
coecients were only workingfor perfectly conducting obstacles. The
original Geometric Theory of Diraction(GTD) was developed by Keller
[6]. GTD overcame the shortcomings of geo-metrical optics, where
rays in the shadow region were considered as a zero eld,see g.
3.13. However, GTD had some limitations in the vicinity of the
shadowregion, where it predicted only a singular diracted eld.
Uniform Geometri-cal Theory of Diraction (UTD) overcame those
limitations. This is the rstray based method that successfully
describes wave scattering from a perfectlyconducting wedge, valid
at all spacial locations. A widely used UTD extension,presented
below, that makes it possible to take lossy materials into account
hasbeen developed by Luebbers [7].
The diraction coecient D is calculated from
D =ej 4
2n2k sin
{D1 + D2 + R0D3 + RnD4} , (3.20)
where k is the wave constant. R0 and Rn are the reection
coecients for the0-face and n-face respectively. The factor n is
found knowing the interior angle of the corner: n = 2
, as illustrated in g. 3.13. For a 90-degree corner =
2
and n = 3/2.
The diraction components are:
34
-
D1 = cot[ + (d d)
2n
]F (kLa+) (3.21)
D2 = cot[ (d d)
2n
]F (kLa) (3.22)
D3 = cot[ (d + d)
2n
]F (kLa) (3.23)
D4 = cot[ + (d +
d)
2n
]F (kLa+) (3.24)
where F (x) is the Fresnel transition function,
F (x) = 2j
xejx
xej
2
d . (3.25)
For large values of x, F (x) can be approximated by [16]
F (x) 1 + j 12x
34
1
x2 j 15
8
1
x3+
75
16
1
x4(3.26)
and for small values of x,
F (x) [
x 2xe j4 23x2e
j4
]ej(
4+x) . (3.27)
For plane waves, the distance parameter L is equal to
L = s sin2() , (3.28)
where s is the distance from the receiver- to the diraction
point (g. 3.13).
The a function used in calculation of the diraction coecient is
dened as
a(d d) = 2 cos2[2nN (d d)
2
], (3.29)
where N+ and N are the integers that most nearly satisfy the
equations
2nN+ (d + d) = (3.30)2nN (d d) = . (3.31)
(a+, N+) are associated with the n-face and (a, N) are
associated with the0-face.
The received electric eld for diracted rays are calculated in
another way thanfor reection- and transmission rays,
35
-
Ed = E0DAejks . (3.32)
A is the spreading factor with a form depending on the type of
wave beingconsidered. For plane waves, A = 1/
s.
3.5 Calculating results
So far calculations of the electromagnetic eld has been
presented. Togetherwith the geometry of the paths, it is possible
to calculate several results for bothnarrowband- and wideband [18].
Since narrowband results are focused on in thisthesis, they are
presented below. Parameters that allow for a realistic modellingof
the mobile radio channel are power, phase and
direction-of-arrival.
3.5.1 Power and phase
By placing several receivers in dierent positions, it is
possible to get an under-standing of the coverage (signal power) in
the environment. This can be usefulwhen placing base-stations.
The power (P ) is equal to the product of the power density (S)
of the incidenteld and the maximum eective aperture of the antenna
(Aem).
P = SAem (W ) (3.33)
The maximum eective aperture of an antenna is related to the
physical size ofthe antenna and to its shape. Knowing the
directivity of the antenna, this canbe expressed as
Aem =2DG4
(m2) . (3.34)
The power density of a plane wave in free space is
S =|E|2120
(W/m2) . (3.35)
Usually power is presented in dB-scale.
The phase of the received electromagnetic eld (E) is obtained
from
E = arctanIm(E)
Re(E). (3.36)
36
-
3.5.2 Direction-of-Arrival
Every ray hits a receiver with a specic angle of incidence,
called direction-of-arrival (DoA). DoA are described in spherical
coordinates (, ), which are foundwith aid of the vector P , created
by the receiver point and the last intersectionpoint of the
path.
= arccos(
Pz|P |
), (3.37)
= arcsin(
Py|P | sin
), (3.38)
where the length of P is given by |P | =
P 2x + P2y + P
2z .
DoA is an important measure in for example SDMA
(space-division-multiple-access) techniques, where the information
that is to be transmitted just are sentin the angular section where
the users signals are impinging [21].
x
y
P
z
Figure 3.14: Spherical coordinates
37
-
38
-
Chapter 4
Verication
This chapter presents dierent scenarios to verify that the
2D-version of the im-plemented model is working as expected. The
model takes into account reection-and transmission intersections,
and respect is taken to the electromagnetic prop-erties of
construction materials and antenna polarization. Problems occur
whenintroducing the 3D-version and propagation by diraction. These
are presentedand discussed at the end of the chapter.
The theoretical statements from earlier chapters are discussed
and comparedwith simulation results. Dierent parameters are used to
illustrate their aectson the received power, phase etc. All
testcases are intended to be well specied,making it possible for
the readers to make their own verications.
4.1 Verication sources
There are a lot of articles with simulated and physical
measurements presentedin the area of radio propagation and ray
tracing. However, those have beenfound not to be fully described.
Since it is very important to have all parameterswell dened
(environment description, frequency, polarization etc.),
vericationresults of articles have not been possible to use.
Instead, the verication models inthis thesis are based on a
commercial software, Radiowave Propagation Software(RPS) [18],
where dierent parameters can be dened exactly as in the ray modelof
this thesis. RPS uses the ray launching technique while the ray
model ofthis thesis is based on the image technique. However, since
it is possible todene simulation scenarios where dierences between
the image- and launchingtechniques are neglectable, it is not
considered as a problem. Instead, this can beseen as a possibility
to sort out the properties of the dierent techniques. Someof the
simulation environments are dened in order to create special cases
thatare of interest to discuss.
39
-
4.1.1 Radiowave Propagation Software (RPS)
RPS is developed by a company called Radioplan [18]. Radioplan
has a strongcooperation with leading cellular network operators,
system vendors and plan-ning tool suppliers. Their software, RPS,
is an advanced radio planning tool fordetermining the site-specic
radio channel in 2D- and 3D-environments, with useof deterministic
ray launching prediction models. RPS is able to calculate
allresults that are of interest in the model presented in this
thesis. This assures forRPS as a valid verication tool for
measurement predictions.
4.2 Model verication
The simulations in this chapter are based on the parameters
presented in table 4.1.As long as no other parameters are dened
together with the gures, these areused.
Antenna:Type IsotropicPolarization HorizontalFrequency 1 GHz
Material:Complex permittivity, 2.5 0.3jThickness 0.05 m
RPS Specic:Stepsize , 0.1
Noise oor -110 dB
Table 4.1: Data for simulation models
4.2.1 Basic scenarios
To start with it is important to show that the fundamentals of
the ray tracerare working properly. Basic scenarios with LoS,
single reection and a singletransmission are presented. With use of
those scenarios it can be shown that thegeometry and the
electromagnetic calculation of single paths are working in thesame
way as for RPS.
LoS and reection paths
In g. 4.1 a scenario including a LoS path and a single reection
on a wall aresimulated.
40
-
14 m
7 m
Tx Rx
Figure 4.1: Scenario with LoS path and reection path
In this scenario the received power- and phase results for the
raytracer com-pared to RPS agrees with high accuracy: power
dierence < 0.01 dB and phasedierence < 0.01. The simulation
results are presented in table 4.2.
LoS magnitude -55.36 dBLoS phase 120Reection magnitude -71.67
dBReection phase 2.63
Table 4.2: Simulation results
The agreement of this scenario veries that the calculations with
the electromag-netic elds, the reection coecient, free-space loss
and the basic geometry ofthe ray tracer are working correctly.
Transmission paths
The scenario in g. 4.2 veries the electromagnetic calculation of
a ray propagat-ing through a wall.
Tx Rx
10 m 15 m
Figure 4.2: Single transmission path
The power result of this simulation is well in accordance with
the result of RPS(< 0.01 dB), see table 4.3, but there is a 60
degree phase dierence between
41
-
Raytracer magnitude -61.35 dBRaytracer phase 25.73
RPS magnitude -61.35 dBRPS phase 85.73
Table 4.3: Simulation results
the results. Further simulations have shown that this oset is
added at everytransmission intersection. An explanation to why the
oset exists have not beenfound. In the verication of further
scenarios this oset will be added to everytransmission coecient to
adjust for the dierence: adjusted = e
j 3 .
4.2.2 2D-scenarios
Until now, single ray paths have been veried. The rst step into
the real worldscenario, is to introduce a model where paths can
include several reection- andtransmission intersections.
Limited number of rays
The aim with this simulation is to prove that the summation of
electric elds arecorrect. Therefore it is important that the number
of rays reaching the receiverare limited and have the same
geometrical paths as in RPS. To fulll this, thesimulation takes
place in a strategically designed environment, see g. 4.3.
Rx
Tx [10 10]
[0 13] [7 13]
[0 7]
[0 2]
[8 0]
[4 4] [6 4]
Figure 4.3: The receiver is moving along the path dened in the
gure
42
-
A plot of the total received magnitude in each receiver position
is used to verifythe simulation (g. 4.4).
0 0.25 0.5 0.75 1 1.25 1.5 1.75 256
55
54
53
52
51
50
49
48Coverage vs receiver positions
Distance from first receiver position [m]
Mag
nitu
de [d
B]
RaytracerRPS
Figure 4.4: Simulation results for the environment in g. 4.3
As seen in g. 4.4 the results of the total received power at
each receiver positionoverlaps each other. This result conrms that
the summation of electric eldsworks correctly.
Comparison between the image- and the launching techniques
Certain demands has to be fullled when a model based on the
image techniqueis veried with use of a simulator based on the ray
launching technique. As men-tioned in earlier chapters the image
technique is an analytical method, makingthe found paths to be the
exact ones. A simulator using the launching techniquelaunches rays
with a constant angle separation. When the angle separation isbig (
1) the simulation is rather fast, but the accuracy decreases
comparedto results received by an analytical method. When the angle
separation is small( 0.1) the simulation is more time consuming,
but the found paths approachesanalytical paths, yielding high
accuracy. Fig. 4.5 shows the power plot of theenvironment in g.
4.3, with the dierence that the constant angle separationis set to
1 instead of 0.1 (default). As seen, there are dierences between
the
43
-
image and launching method when the angle separation is big.
What constantangle separation to use is a trade-o between adequate
results and computationtime.
0 0.25 0.5 0.75 1 1.25 1.5 1.75 256
55
54
53
52
51
50
49
48Coverage vs receiver positions
Distance from first receiver position [m]
Mag
nitu
de [d
B]
RaytracerRPS
Figure 4.5: Dierence in received power when RPS uses a constant
angle sepa-ration of 1
Notice that in this scenario the number of rays reaching the
receiver is limited. Asthe number of rays increases, each ray may
produce a small dierence resulting ina non-neglectable dierence of
the total result. The fact that the ray launchingtechnique uses a
reception sphere (discussed in chapter 2) may also aect thetotal
result. The accuracy of the result depends on the size of the
receptionsphere. If it is to small, rays may be missed, and if it
is to large, paths mightbe duplicated and non-exact paths may reach
the receiver. The phase is verysensitive to changes in angle
separation. Since the path length becomes dierentfor dierent angle
separations, the phase can dier several degrees, especiallywhen
simulating at high frequencies. Examples of the phase variation at
1 GHzfrequency for dierent ray paths are shown in table 4.4.
Indoor environment scenario
To verify a scenario with enclosed walls, a full indoor
environment is considered,seen in g. 4.6. In both the raytracer and
RPS, rays that have lower power
44
-
Raytracer RPS 0.1 RPS 1
-128.03 -128.01 -126.2785.28 85.56 93.3582.76 82.93 102.40
157.46 157.46 158.80-66.08 -66.03 -52.71
-159.95 -159.94 -158.92-106.17 -106.09 -97.33-166.01 -165.99
-163.56-115.04 -114.91 -102.41
-3.39 -3.62 70.59
Table 4.4: Phase values for some of the rays found at rst
receiver position ing. 4.3. Angle separation 0.1 and 1 are compared
to phase values generated bythe image technique.
than 110 dB are ignored (noise oor). When using the image
technique it ispossible to set the maximum number of reections that
a path can include. Anexamination is made of the maximum number of
reections that is needed toachieve accurate results. The simulation
times are discussed and compared.
[83,55][0,55]
[0,0] [83,0]
[35,35]
[0,30] [25,30] [25,25]
Tx = [20,35]
Rx = [30,30] [80,30]
[50,25] [83,25]
[83,35]
[50,0] [25,0]
[25,55] [35,55]
Figure 4.6: Blueprint of indoor environment
RPS oers two dierent algorithms: the 2.5D- and the 3D-algorithm.
Both canbe used for simulation of a 2D-scenario. The power plot in
g. 4.7 shows theresults obtained from the ray tracer and RPS. Phase
and DoA results from theindoor environment are presented in the end
of the chapter.
45
-
0 5 10 15 20 25 30 35 40 45 5085
80
75
70
65
60
55
50Coverage vs receiver positions
Distance from first receiver position [m]
Mag
nitu
de [d
B]RaytracerRPS 3DRPS 2.5D
Figure 4.7: Power plot of indoor environment. Rays that
undergoes 110 dBare ignored (noise oor).
As seen in g. 4.7 there is good agreement in the results of the
ray tracer com-pared to the RPS 3D-algorithm. However, there are
bigger dierences betweenthe raytracer and the 2.5D-algorithm of
RPS, in some positions over 4 dB. Afteranalysis, the dierence
between the simulation results can be explained by thatthe
2.5D-algorithm of RPS are not able to nd more than one path in a
specicoutgoing angle, i.e. additional paths with the same outgoing
angle are not found.In g. 4.8 paths found by the ray tracer but
lost in RPS are displayed. Theposition of the receiver is where the
result in the power plot diers at most, i.e.38 meters away from rst
receiver position. The 2.5D-algorithm is optimized forfast
simulations, but apparently the accuracy decreases.
In the image technique, simulation times can be reduced by
decreasing the max-imum number of reections, but it is still
important to keep the accuracy ofthe results. As seen in g. 4.9 the
dierence in power between an increasingnumber of reections gets
smaller and smaller. Already between three and fourreections the
dierence is small. It seems meaningless to include more than
vereections since the contribution from higher order of reections
are neglectable.This conclusion agrees well with several
publications [12][13].
46
-
Tx
Rx
Figure 4.8: Examples of correct paths not found by the
2.5D-algorithm of RPS.Tx = [20 35 1] and Rx = [68 30 1].
When increasing the number of reections, the received number of
paths and thesimulation time increases. In the case of three
reections about 60 paths are foundfor each receiver position, about
200 paths for four reections and 550 paths forve reections. For
complex environments, consisting of many surfaces, the
imagetechnique gets inecient. Although, in a 3D environment with
few surfaces,simulations have shown that the image technique is
more time-ecient comparedto the launching technique. The image
technique is not dependent on the numberof dimensions, just the
number of surfaces. For ve reections the model in thisthesis needs
24 hours to simulate the indoor scenario in g. 4.6, while the
3D-algorithm of RPS needs about 2 hours. When using three and four
reectionsthe simulation time can be reduced to 1 and 25 minute(s)
respectively. Sincethe results of three and four reections are
close to the result of ve reections,it is possible to save a lot of
simulation time with only a minor decrease inaccuracy. To reduce
simulation times further, acceleration techniques needs tobe
introduced, as discussed in chapter 2.
4.3 Model extensions
So far the 2D-model has been successfully veried. To make the
model morerealistic, 3D-models can be introduced. The ray tracer of
this thesis has a fully
47
-
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44
46 48 5080
75
70
65
60
55
50Coverage vs receiver positions
Mag
nitu
de [d
B]3 reflections4 reflections5 reflections
Figure 4.9: Comparison of received power for dierent number of
reections
implemented geometry for nding paths in the 3D-environment. For
some rea-sons problems occur when making electric eld calculations
of found 3D-paths.
To further approve the model, the eects of propagation by
diraction can beincorporated. The eects of diraction are especially
important when no LoS ex-ists, and paths including reection and
transmission intersections have problemsreaching the receiver.
However, when those paths exists, diraction contributionsare of
minor importance. As in the 3D-case, the geometry for nding paths
in-cluding diraction is implemented. Although, for certain angles,
problems withthe electric eld calculations appears.
A lot of work have been done with 3D and diraction as presented
in the followingsections.
4.3.1 3D-model
When introducing oor and ceiling into the model, 3D-paths are
created. Thismeans that the paths change direction both
horizontally and vertically, i.e. both
48
-
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44
46 48 503
2
1
0
1
2
3
Phase vs receiver positions
Distance from first receiver position [m]
Phas
e [ra
d]
Figure 4.10: Phase plot of the indoor environment
110 dB
100 dB
90 dB
80 dB
70 dB
60 dB
180 0
270
90
Figure 4.11: Direction-of-Arrival of the indoor environment at
Rx = [68 30 1]
49
-
- and -angles varies. In this case the ray is aected in another
way than for the2D-case. The polarization of a wave can consist of
a horizontal- and a verticalpart. To nd the size of the dierent
parts, simulations with a single reectionagainst a wall was
analyzed in RPS. By dening the wall as a perfect conductor,the
received ray is aected only by the free space loss (which is known)
and thedivision into dierent polarization parts. Only the part of
the ray matching thepolarization of the receiving antenna is
received. Measurements for a transmitterwith several receivers, at
the same distances but with dierent - and -angles,results in a
relationship depending on the -angle. Dierent -angles do not
af-fect the relationship. By multiplying the electric eld with
cos(2), the part withthe polarization of interest is found. This
relationship holds for equally polarizedantennas. For antennas with
dierent polarization, the relationship changes tosin(2). In
simulations where the reection instead takes place on a oor or
ceil-ing, there are no division into dierently polarized parts.
Therefore no changeshave to be done to the received electric eld.
This dierent polarization behaviorbetween a horizontal- and
vertical surface seems very strange. Further similarrelationships,
by extending the simulations to paths with several intersections
onboth horizontal- and vertical surfaces, have not been found. It
gets even morecomplicated when introducing a non-nite conducting
material to the simulationscenario including a single reection
against a wall. Then the previous foundrelationship cos(2) do not
work anymore. This means that in some way the di-electric
properties of the material aects the size of the polarization
parts. This isstrange, since the polarization eects due to the
material characteristics alreadyare incorporated into the used
reection coecients.
A complete explanation how to handle polarization eects of
3D-paths has notbeen found. Most of the documentation do not
mention anything on how to treatthis. Though, there are authors
presenting potential solutions [15][16], but theuse of those
methods have not given any correct results compared to RPS.
4.3.2 Diraction
The formulas for the diraction coecient based on Luebbers method
have beencarefully implemented. Several articles, based on this
method, contains simula-tion results fully dened for a perfectly
conducting corner. These are suitable fora rst verication of
diraction, as no reection coecients aects the result.
The implementation proved to be functional with very high
accuracy for mostof the angles, but in a certain region the results
dier, see g. 4.134.14. If theimplementation is assumed to be
correct, this certain region has to be treated inanother way.
However, articles do not present anything about this at all.
Fur-ther, when calculating the diraction coecient for a dielectric
material, RPSuses dierent reection coecients than those presented
in chapter 3. No liter-
50
-
Rx = [-300,7] Rx = [300,7]
45
Tx = [-70.71,-70.71]
[0,0]
Diffracting corner
Figure 4.12: Diraction scenario with Rx moving in the specied
direction
ature states that the reection coecients should be dierent just
because it isused in diraction calculations.
To include operational diraction scenarios, further studies have
to be madefor the behavior in the non-working region. To be able to
verify against RPS,also the reection coecients they use in
diraction calculations must be found.
0 100 200 300 400 500 600160
150
140
130
120
110
100
90
80Coverage vs receiver positions
Distance from first receiver position [m]
Mag
nitu
de [d
B]
RaytracerRPS
Figure 4.13: Power plot of the received diracted rays for a
perfectly conductingcorner at frequency 1.89 GHz
51
-
275 280 285 290 295 300 305 310 315 320 325
115
110
105
100
95
90
Coverage vs receiver positions
Distance from first receiver position [m]
Mag
nitu
de [d
B]RaytracerRPS
Figure 4.14: Zoomed power plot of the region containing dierent
results (25meters from the diracting corner) in g. 4.13
52
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Chapter 5
Conclusion and future work
5.1 Conclusion
In this thesis a simulator for channel modelling of indoor
environments has beenimplemented with ray tracing techniques. With
use of the image technique pathscan be traced in both 2D- and
3D-environments. Compared to the ray launch-ing technique, the
image technique has proved to be accurate but more time-consuming
for complex environments. When calculating results, respect has
beentaken to surface thickness, electromagnetic properties of
surfaces and polarizationof antennas. The model has been
successfully veried in the 2D-case. However,the electric eld
calculations for the 3D-model needs to be further investigated.
5.2 Future work
This section presents some areas for future work on the model
presented in thisthesis.
Finding paths- Acceleration techniques to decrease computation
times
Electric eld calculations- Full solution for electric eld
calculation of 3D-paths
- Identify solution for the erreounus region of diraction
Antennas- Include more types of antennas
53
-
Results- Calculate more results from the electric eld, such as
impulse response
Graphical User Interface Verication
- Use other types of verication sources/perform own
measurements
Environment- Extend to urban areas
54
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Appendix A
User manual
A.1 Starting a simulation
A simulation scenario is started from raytracer.m. For the
raytracer a number ofsimulation options can be dened:
raytracer(direct_path_included, no_of_refl, diffr_included,
outer_transm, S_pos, R_pos_vect, stepsize, f, pol,
antenna_type, power_plot, phase_plot, doa_plot)
direct path included: 1 direct path included, else 0no of re:
maximum number of reection that can exist in a pathdir included: 1
diraction included, else 0outer transm: 1 allows transmission
through walls dened as outer wallsS pos: Position of transmitter
antenna, [x,y,z]R pos vect: Position(s) of receiver antenna(s),
[x1,y1,z1; . . . ]stepsize: If several receivers are dened, this is
the distance between themf: Antenna frequency (Hz)pol: Polarization
of antenna; 1 horizontal, 2 verticalantenna type: 0 isotropic
antennapower plot: 1 Presents power plot when several receivers,
else 0phase plot: 1 Presents phase plot when several receivers,
else 0doa plot: 1 Presents DoA plot for a receiver position, else
0
55
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A.1.1 Dening an environment
Before starting a simulation the environment must be dened in
g.m. Thestructure of g.m is shown below. The arrows indicate rows
that are set by theuser.
%define
materials-------------------------------------------------------
%material parameters: (epsr, d)
materials = [];
>> materials = [materials, material(2.5-0.03*i, 0.05)];
%wood, index 1
>> materials = [materials, material(6-0.05*i, 0.01)];
%glass, index 2
%define
environment-----------------------------------------------------
facet parameters: (p0, p1, h, type, outer, d_facets,
material)
facets = [];
>> height = 3; %general building height
%wall (and inner facets)
>> facets = [facets, facet([0 0], [10 0], height, 0, 0, 0,
1)]; %facet 1
>> facets(length(facets)) =
add_inner_facet(facets(length(facets)), 0, ...
0, 4, 6, 1, 2, 2);
>> facets = [facets, facet([0 0], [0 10], height, 0, 0, 0,
1)]; %facet 2
wall_facets = (1: 1: length(facets));
%floor, ceiling
>> facets = [facets, facet([0 0], [0 0], 0, 2, 0, 0,
1)];
>> facets = [facets, facet([0 0], [0 0], height, 2, 0, 0,
1)];
refl_facets = (1: 1: length(facets));
%diffraction edges
>> facets = [facets, facet([0 0], [0 0], height, 1, 0, [1
2], 1)];
all_facets = (1: 1: length(facets));
diffr_facets = all_facets;
diffr_facets(refl_facets) = [];
%-----------------------------------------------------------------------
56
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Dening a material
material(epsr, d)
epsr: Complex permittivityd: Thickness (m)
Dening a facet
facet(p0, p1, h, type, outer, d_facets, material)
p0, p1: Start- and end point for a surface in 2D-coordinates
[x,y]h: Height, all walls must have a common heighttype: 0 wall, 1
diraction edge, 2 oor/ceilingouter: If a wall is dened as an outer
wall, this should be set to 1, else 0d facets: Facets connected by
a diraction corner, [facet1, facet2]material: Couples the facet to
a material, with use of a material index
Dening an inner facet (i.e. windows, doors etc.)
add_inner_facet(facet, x1, x2, y1, y2, z1, z2, material_no)
facet: Index of belonging facetx1,x2,y1,y2,z1,z2:
3D-coordinatesmaterial no: Couples the inner facet to a material,
with use of a material index
57
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58
-
Appendix B
Software description
This appendix presents a brief description of the main les
together with a graph-ical description of the software
architecture.
raytracer.m Main program; nds paths, performs electric eld
cal-culations and present results.
g.m Includes denitions of environment and materials.
LOS.m Checks if LoS exists. If not, the direct path
containingtransmission intersections is checked.
make tree.m Creates tree of images with a specied number of
levelsand facets (a facet can be dened as a diraction point).
nd re paths.m Finds valid reection paths by traversing an image
treeconsisting of reection facets (walls/oor/ceiling).
nd dir paths.m Finds valid diraction paths by traversing two
imagetrees from the found diraction point. One tree startsfrom the
transmitter and one tree starts from the re-ceiver.
add transm isects.m Checks if transmission(s) occurs in found
paths. If so,the intersections for transmission are added to the
cur-rent path.
calc res.m Uses geometrical information from the found paths
andcalculates the received electric elds used for power-,phase- and
DoA results.
59
-
doa.m Calculates spherical angles for incoming rays and
turnsthem into a suitable form for presentation.
isect coe.m Calculates the reection- and transmission
coecient(s)depending on polarization.
make im Calculates image points.
aoi.m Calculates grazing angle of incidence to a surface. Fora
diraction facet the -angle is calculated.
dir values.m Checks if the ray hits the diraction corner with a
validdirection. Calculates
d- and d angles for the dirac-
tion point.
quadrant.m Determines to which quadrant a point belongs.
nd inner facets.m After nding paths, each wall containing inner
facets,such as windows and doors, are checked. If the inter-section
of a path occurs on an inner facet the materialinformation is
changed.
potential facets.m Returns potential intersection facets. Can be
used whenimplementing acceleration techniques.
calc d coe.m Testle for calculation of the diraction coecient.
In-formation on how to run this is found in the le.
Classes
material.m Material propertiesfacet.m Facet informationim.m
Image informationim tree.m Stores image treeintersection.m
Intersection informationpath.m Path information
60
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fig.m
LOS.m
make_tree.m
find_refl_paths.m
find_diffr_paths.m
add_transm_isects.m
calc_res.m
draw_facets.mdraw_S_R_pos.mdraw_LOS.mdraw_paths.m
material.mget_epsr.mget_d.mdisplay.m
make_im.m
aoi.mdiffr_values.m
quadrant.m
raytracer.m
facet.madd_inner_facets.mget_inner_facets.mget_d_facets.mget_outer.mget_p0.mget_v1.mget_v2.mget_material.mget_type.mget_d.mget_n.mdisplay.m
im.mget_p.mget_parent.mget_facet.mdisplay.m
im_tree.madd_children.mget_child.mget_children.m
intersection.mget_ip.mget_type.mget_facet.mget_aoi.mget_d_values.mget_material.mset_material.mdisplay.m
path.madd_isect.mget_ips.mget_isects.mget_last_isect.mget_dist.mset_distfind_inner_facets.mflip_isects.mdisplay.m
doa.misect_coeff.m
Classes
potential_facets.m
Figure B.1: Software architecture
61
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62
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