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

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).

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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

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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

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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

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

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

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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) = [];

%-----------------------------------------------------------------------

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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

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

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

62

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