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20 Ansichten179 SeitenPropagation model for radio signal in tunnels

© © All Rights Reserved

Propagation model for radio signal in tunnels

© All Rights Reserved

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20 Ansichten179 SeitenPropagation model for radio signal in tunnels

© All Rights Reserved

Sie sind auf Seite 1von 179

aus

IHE dem Institut

für Höchstfrequenztechnik

und Elektronik

der

Universität Karlsruhe

Herausgeber:

Prof. Dr.-Ing. W. Wiesbeck

Dirk Didascalou

Ray-Optical

Wave Propagation Modelling

in Arbitrarily Shaped Tunnels

Band 24

Copyright: Institut für Höchstfrequenztechnik und Elektronik

Universität Karlsruhe (TH), 2000

Tel. 035841-36757

ISSN: 0942-2935

Forschungsberichte aus dem

Institut für Höchstfrequenztechnik und Elektronik

der Universität Karlsruhe (TH)

Zu Beginn des 3. Jahrtausends überschlagen sich die Berichte in den Medien zur Zu-

kunft der Kommunikation. Daten, Audio und Video werden in den Industrienationen

flächendeckend verfügbar sein. Das Internet wird allgegenwärtig. Wenn man diesen

Erwartungen vertraut, und ich bin der Überzeugung, dass sie die Zukunft noch un-

terschätzen, überfällt einen ein etwas beklemmendes Gefühl ob der bevorstehenden

technischen, gesellschaftlichen und sozialen Revolution. Bringt man es als Ingenieur

technisch gesehen auf den Punkt, dann resultiert hieraus, dass diese gesamte Zu-

kunft in einem begrenzten, nicht vermehrbaren und nicht transportierbaren Spektrum

ausgetragen wird. Die entscheidende Frage für die Unternehmen wird sein: was ist

der Preis für Spektrum? Wird ein Hertz soviel wert sein wie ein Gramm Gold? Welche

regionale Ausdehnung wird damit erkauft? Wird der Preis vielleicht 1000mal so hoch

sein? Diese sicherlich nicht einfach zu beantwortenden Fragen werden die Regulierer,

der Staat und der Markt zu lösen haben. Als Ingenieure können wir durch Technologi-

en, Methoden und Verfahren dazu beitragen, dass das Spektrum so effizient wie nur

irgend möglich genutzt wird. Schlagworte werden heute kaum benutzte Begriffe sein

wie friendly coexistence, sharing rules, adaptability und heterogeneity.

Hiermit ist zum Originalton der Dissertation von Herrn Dirk Didascalou übergeleitet.

Mit seiner hervorragenden Arbeit zur Wellenausbreitung in Tunnels leistet er einen we-

sentlichen Beitrag zur effizienten Planung von Kommunikationsnetzen in dieser sehr

schwierigen Umgebung. Anders als im ländlichen und städtischen Raum wird die

Wellenausbreitung in beliebig im Querschnitt und im Längsschnitt geformten Tunnels

durch Mehrfachreflexionen und bewegte Fahrzeuge dominiert. So führen die Refle-

xionen bei konkaven Krümmungen z.B. zur Wellenfokussierung. Technisch gesehen

resultieren daraus keine Probleme, theoretisch ist es jedoch äusserst schwierig zu be-

handeln. Zur Lösung dieses Problems ist von Herrn Didascalou eine sehr wirkungsvol-

le Normierung in einem strahlenoptischen Wellenausbreitungsmodell eingeführt wor-

den. Damit wird erstmals eine derartige Umgebung effizient und ausreichend genau

berechenbar. Ich wünsche dem entstandenen Planungswerkzeug und den Ideen eine

effiziente Verbreitung.

- Institutsleiter -

Forschungsberichte aus dem

Institut für Höchstfrequenztechnik und Elektronik

der Universität Karlsruhe (TH)

Modellierung und meßtechnische Verifikation polarimetrischer, mono-

und bistatischer Radarsignaturen und deren Klassifizierung

Band 2 Eberhardt Heidrich

Theoretische und experimentelle Charakterisierung der polarimetri-

schen Strahlungs- und Streueigenschaften von Antennen

Band 3 Thomas Kürner

Charakterisierung digitaler Funksysteme mit einem breitbandigen

Wellenausbreitungsmodell

Band 4 Jürgen Kehrbeck

Mikrowellen-Doppler-Sensor zur Geschwindigkeits- und Wegmessung

- System-Modellierung und Verifikation

Band 5 Christian Bornkessel

Analyse und Optimierung der elektrodynamischen Eigenschaften von

EMV-Absorberkammern durch numerische Feldberechnung

Band 6 Rainer Speck

Hochempfindliche Impedanzmessungen an Supraleiter / Festelektro-

lyt-Kontakten

Band 7 Edward Pillai

Derivation of Equivalent Circuits for Multilayer PCB and Chip Package

Discontinuities Using Full Wave Models

Band 8 Dieter J. Cichon

Strahlenoptische Modellierung der Wellenausbreitung in urbanen Mi-

kro- und Pikofunkzellen

Band 9 Gerd Gottwald

Numerische Analyse konformer Streifenleitungsantennen in mehrlagi-

gen Zylindern mittels der Spektralbereichsmethode

Band 10 Norbert Geng

Modellierung der Ausbreitung elektromagnetischer Wellen in Funk-

systemen durch Lösung der parabolischen Approximation der Helm-

holtz-Gleichung

Band 11 Torsten C. Becker

Verfahren und Kriterien zur Planung von Gleichwellennetzen für den

Digitalen Hörrundfunk DAB (Digital Audio Broadcasting)

Band 12 Friedhelm Rostan

Dual polarisierte Microstrip-Patch-Arrays für zukünftige satelliten-

gestützte SAR-Systeme

Forschungsberichte aus dem

Institut für Höchstfrequenztechnik und Elektronik

der Universität Karlsruhe (TH)

Vektorkorrigiertes Großsignal-Meßsystem zur nichtlinearen Charakte-

risierung von Mikrowellentransistoren

Band 14 Andreas Froese

Elektrochemisches Phasengrenzverhalten von Supraleitern

Band 15 Jürgen v. Hagen

Wide Band Electromagnetic Aperture Coupling to a Cavity: An Integral

Representation Based Model

Band 16 Ralf Pötzschke

Nanostrukturierung von Festkörperflächen durch elektrochemische

Metallphasenbildung

Band 17 Jean Parlebas

Numerische Berechnung mehrlagiger dualer planarer Antennen mit

koplanarer Speisung

Band 18 Frank Demmerle

Bikonische Antenne mit mehrmodiger Anregung für den räumlichen

Mehrfachzugriff (SDMA)

Band 19 Eckard Steiger

Modellierung der Ausbreitung in extrakorporalen Therapien eingesetz-

ter Ultraschallimpulse hoher Intensität

Band 20 Frederik Küchen

Auf Wellenausbreitungsmodellen basierende Planung terrestrischer

COFDM-Gleichwellennetze für den mobilen Empfang

Band 21 Klaus Schmitt

Dreidimensionale, interferometrische Radarverfahren im Nahbereich

und ihre meßtechnische Verifikation

Band 22 Frederik Küchen, Torsten C. Becker, Werner Wiesbeck

Grundlagen und Anwendungen von Planungswerkzeugen für den di-

gitalen terrestrischen Rundfunk

Band 23 Thomas Zwick

Die Modellierung von richtungsaufgelösten Mehrwegegebäudefunk-

kanälen durch markierte Poisson-Prozesse

Band 24 Dirk Didascalou

Ray-Optical Wave Propagation Modelling in Arbitrarily Shaped Tunnels

À Michèle et à Florent

Ray-Optical Wave Propagation Modelling

in Arbitrarily Shaped Tunnels

DOKTOR-INGENIEURS

Elektrotechnik und Informationstechnik

der Universität Fridericiana Karlsruhe

genehmigte

DISSERTATION

von

aus Hamburg

Hauptreferent: Prof. Dr.-Ing. Werner Wiesbeck

Korreferent: Prof. Dr.-Ing. habil. Friedrich M. Landstorfer

Acknowledgements

I would like to thank the Director of the Institut für Höchstfrequenztechnik und

Elektronik (IHE), Prof. Dr.-Ing. Werner Wiesbeck, for his support in enabling

me to carry out a Ph.D. degree (Dr.-Ing.) at the Universität Karlsruhe (TH)

in Germany. My appreciation is extended to Prof. Dr.-Ing. habil. Friedrich M.

Landstorfer, the second examiner of this thesis, for the interest in my work.

I acknowledge the invaluable suggestions, corrections and constructive cri-

ticisms of the reviewers of this thesis: my colleagues and friends Dipl.-Ing.

Martin Döttling, Dr.-Ing. Norbert Geng, Dipl.-Ing. Jürgen Maurer and Dipl.-

Ing. Ralph Schertlen. I am particularly indebted to Dipl.-Ing. Gunther Auer

and Dr. Keith Palmer, who tried hard to make my pidgin English more read-

able.

Among the many students and colleagues, who have helped me during

my work on tunnels at the IHE, many thanks are due to Matthias Dübon,

Thomas Kösterkamp, Alexander Widmann, Juan Manuel Canabal Muñoz,

Carles Galdón Durá, Bernd Ruthmann, Dipl.-Ing. Yan Venot and Dr.-Ing.

Frederik Küchen. A special tribute is owed to Dipl.-Phys. Thomas Schäfer

and Dipl.-Phys. Frank Weinmann, who denitively contributed most to the

development of the simulation tool.

I am grateful to Prof. Dr. rer. nat. Manfred Thumm, the Director of the

Institut für Hochleistungsimpuls- und Mikrowellentechnik, at the Forschungs-

zentrum Karlsruhe, Germany, for the provision of the millimeter-wave meas-

urement equipment and to Dipl.-Ing. Andreas Arnold, Richard Kunkel and

Dipl.-Ing. Oliver Schindel for their assistance.

I would like to thank Dipl.-Ing. Jörg Borm, E-Plus Mobilfunk GmbH, Düs-

seldorf, Germany, for the provision of the GSM1800 measurement equipment

and his assistance, as well as Dipl.-Ing. Michael Feistel and Dipl.-Ing. Udo

Straub, both Mannesmann Mobilfunk, Niederlassung Stuttgart, Germany, and

Dipl.-Ing. Jürgen Köttnitz and Dipl.-Ing. Hanns-Michael Müller, both Mannes-

mann Mobilfunk, Niederlassung Nord-Ost, Germany, for the provision of the

GSM900 measurement equipment. Special thanks to Dipl.-Ing. Hans-Jürgen

Liesegang, Berliner Verkehrsbetriebe (BVG), Berlin, Germany, for his kind co-

operation and to all the sta of his company for their unfailing professionalism

and friendliness, making the measurements in the Berlin subway possible.

Finally, I wish to thank my wife, Michèle, for the encouragement and the

patience during the numerous week-ends and evenings of typing. Without her,

this work would have hardly been possible.

Contents

Glossary v

1 Introduction 1

1.1 Background and motivation . . . . . . . . . . . . . . . . . . . . 1

1.2 Scope and objective of the thesis . . . . . . . . . . . . . . . . . 2

1.3 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1 The ray concept . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Geometrical optics and propagation modelling . . . . . . . . . . 9

2.3 Propagation phenomena . . . . . . . . . . . . . . . . . . . . . . 11

2.3.1 Free space propagation and absorption . . . . . . . . . . 12

2.3.1.1 Free space propagation . . . . . . . . . . . . . 12

2.3.1.2 Absorption . . . . . . . . . . . . . . . . . . . . 12

2.3.2 Reection at smooth surfaces . . . . . . . . . . . . . . . 13

2.3.2.1 Planar surfaces . . . . . . . . . . . . . . . . . . 13

2.3.2.2 Curved surfaces . . . . . . . . . . . . . . . . . 16

2.3.3 Scattering from rough surfaces . . . . . . . . . . . . . . 17

2.3.3.1 Roughness criteria . . . . . . . . . . . . . . . . 18

2.3.3.2 Slightly rough surfaces: the modied Fresnel

reection coecients . . . . . . . . . . . . . . . 18

2.3.3.3 Stochastic scattering approach . . . . . . . . . 19

2.4 Multipath propagation and analysis . . . . . . . . . . . . . . . 23

2.4.1 Formal description of a propagation path . . . . . . . . 24

2.4.2 Multipath propagation . . . . . . . . . . . . . . . . . . . 26

2.4.3 Broadband analysis and channel parameters . . . . . . . 27

2.4.3.1 Frequency and impulse response of the trans-

mission channel . . . . . . . . . . . . . . . . . 28

i

ii CONTENTS

2.4.3.3 Delay spread . . . . . . . . . . . . . . . . . . . 30

2.4.3.4 Doppler spread . . . . . . . . . . . . . . . . . . 31

2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.1 Methods of ray tracing . . . . . . . . . . . . . . . . . . . . . . . 34

3.1.1 Direct approach: image theory . . . . . . . . . . . . . . 34

3.1.2 Indirect approach: ray launching . . . . . . . . . . . . . 35

3.1.2.1 Discrete ray tubes . . . . . . . . . . . . . . . . 35

3.1.2.2 Reception spheres . . . . . . . . . . . . . . . . 37

3.1.2.3 Identication of multiple rays . . . . . . . . . . 38

3.2 Ray launching in curved geometry . . . . . . . . . . . . . . . . 39

3.2.1 Ray tubes: approximate solution for 2D-curvature . . . 39

3.2.2 Reception spheres: the concept of ray density normaliz-

ation (RDN) . . . . . . . . . . . . . . . . . . . . . . . . 40

3.2.2.1 Determination of the number of multiple rays . 40

3.2.2.2 Prerequisites of the RDN . . . . . . . . . . . . 42

3.3 Application of the ray density normalization . . . . . . . . . . . 42

3.3.1 Relationship between divergence of a ray and ray density 43

3.3.2 Field trace . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.3.3 Power trace . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.4 The method of power ow . . . . . . . . . . . . . . . . . . . . . 48

3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.1.1 Straight sections . . . . . . . . . . . . . . . . . . . . . . 52

4.1.2 Curves and clothoids . . . . . . . . . . . . . . . . . . . . 52

4.1.2.1 Clothoid approximation . . . . . . . . . . . . . 53

4.2 Stochastic ray launching . . . . . . . . . . . . . . . . . . . . . . 56

4.3 Coupling into the tunnel . . . . . . . . . . . . . . . . . . . . . . 57

4.4 Moving obstacles . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.5 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.6 Additional remarks on the implementation . . . . . . . . . . . . 60

4.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5 Theoretical verication 62

5.1 The rectangular waveguide . . . . . . . . . . . . . . . . . . . . 63

5.1.1 Geometry of the rectangular waveguide . . . . . . . . . 63

5.1.2 Green's function in PEC rectangular waveguide . . . . . 63

5.1.3 Rectangular waveguide with dielectric boundaries . . . . 66

CONTENTS iii

5.2 The circular waveguide . . . . . . . . . . . . . . . . . . . . . . . 69

5.2.1 Choice of the canonical (reference) geometry . . . . . . 69

5.2.2 Corrugated circular waveguide . . . . . . . . . . . . . . 69

5.2.3 Comparison of the dierent techniques in circular wave-

guide with dielectric boundary . . . . . . . . . . . . . . 71

5.3 Stochastic scattering approach . . . . . . . . . . . . . . . . . . 73

5.3.1 Arrangement and procedure . . . . . . . . . . . . . . . . 73

5.3.1.1 Radar cross section per unit area . . . . . . . . 74

5.3.1.2 Determination of the RCS matrix by ray tra-

cing using the stochastic scattering approach . 75

5.3.1.3 The Kirchho models . . . . . . . . . . . . . . 76

5.3.2 Comparison of the methods . . . . . . . . . . . . . . . . 77

5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

6.1 Measurement setup and procedure . . . . . . . . . . . . . . . . 83

6.2 Comparisons in a straight concrete tube . . . . . . . . . . . . . 85

6.3 Comparisons in a bent stoneware tube . . . . . . . . . . . . . . 88

6.4 Comparisons in a model tunnel . . . . . . . . . . . . . . . . . . 92

6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

7.1 The Berlin subway . . . . . . . . . . . . . . . . . . . . . . . . . 97

7.1.1 Measurement setup and procedure . . . . . . . . . . . . 98

7.1.1.1 Measurement equipment . . . . . . . . . . . . 98

7.1.1.2 Measurement environment . . . . . . . . . . . 99

7.1.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

7.1.2.1 Achievable accuracy in the rectangular straight

tunnel (U5) . . . . . . . . . . . . . . . . . . . . 101

7.1.2.2 Path loss in the curved arched-shaped tunnel

(U8) . . . . . . . . . . . . . . . . . . . . . . . . 103

7.1.2.3 Inuence of curves on the simulation accuracy 105

7.1.2.4 Inuence of the cross-sectional shape on the

simulation accuracy . . . . . . . . . . . . . . . 105

7.1.2.5 Fast fading characteristics . . . . . . . . . . . . 107

7.1.2.6 Distribution of the propagating power in curves 110

7.2 Exterior antenna placement . . . . . . . . . . . . . . . . . . . . 112

7.2.1 Dierence of interior and exterior positions . . . . . . . 112

7.2.2 Exterior antenna locations . . . . . . . . . . . . . . . . . 113

7.2.2.1 Description of the scenario . . . . . . . . . . . 113

7.2.2.2 Antenna height . . . . . . . . . . . . . . . . . . 114

iv CONTENTS

7.2.2.4 Lateral placement . . . . . . . . . . . . . . . . 115

7.3 Moving vehicles and broadband analysis . . . . . . . . . . . . . 116

7.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

8 Conclusions 120

A Radii of curvature after reection 123

B Intersection algorithms 125

B.1 Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

B.2 Rectangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

B.3 Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

B.4 Circular cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . 127

B.5 Elliptical cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . 128

B.6 Elliptical torus . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

C.1 Rectangular cross section . . . . . . . . . . . . . . . . . . . . . 133

C.2 Elliptical (arched) cross section . . . . . . . . . . . . . . . . . . 135

D.1 D-band horn and waveguide probe . . . . . . . . . . . . . . . . 137

D.2 Kathrein K73226X (LogPer) . . . . . . . . . . . . . . . . . . . . 140

D.3 Jaybeam J7360 (Yagi) . . . . . . . . . . . . . . . . . . . . . . . 141

D.4 =4-monopoles with circular ground plane . . . . . . . . . . . . 142

Bibliography 143

Glossary

iD i-dimensional

AS angular spread

BWO backward wave oscillator

CDF cumulative distribution function

D-band frequency band between 110GHz and 170GHz

DAB digital audio broadcasting

DCS1800 digital cellular system (at 1800MHz), now GSM1800

DVB digital video broadcasting

FD nite dierence

FDTD nite dierence time domain

FM frequency modulated/modulation

FT eld trace

GO geometrical optics

GTD geometrical theory of diraction

GSM global system for mobile communications (formerly groupe spé-

cial mobile) at 900MHz, 1800MHz or 1900MHz

GSM900 cf. GSM

GSM1800 cf. GSM, formerly DCS1800

HF high frequency ( 3MHz30MHz)

HP Hewlett Packard

IMR identication of multiple rays

IS-95 interim standard 1995

IT image theory

LMS least mean square

LogPer log-periodic

LOS line of sight

MoM method of moments

PDF probability density function

vi GLOSSARY

PE parabolic equation (method)

PF power ow

PO physical optics

PT power trace

PVC polyvinyl chloride

RAM random access memory

RCS radar cross section

RDN ray density normalization

RMS root mean square

Rx receiver

SBR shooting and bouncing ray (approach)

TEM transverse electromagnetic

TETRA terrestrial trunked radio

Tx transmitter

UHF ultra high frequency ( 300MHz3GHz)

UTD uniform geometrical theory of diraction

UMTS universal mobile telecommunications system

VHF very high frequency ( 30MHz300MHz)

VNWA vector network analyzer

Table of symbols and

variables

The additional symbols and variables used in the appendices are not listed

in the following tables. They are introduced and explained in the respective

appendices.

Mathematical notation

F real scalar

F complex scalar

F~ vector

F^ vector with unit length

F~

F

complex vector

complex dyadic

[] real or complex matrix

F~ approximate value

F averaged value

<fg real part of complex quantity

F~1 F~2 cross product of two vectors

jj absolute value

hi expectation operator

Æ() dirac function

O(x) of the order of x

p() probability density function (PDF)

S (); C () real Fresnel integrals

Greek letters

attenuation constant (linear)

viii TABLE OF SYMBOLS AND VARIABLES

angular separation of launched rays

tan Æe eective electric loss tangent

; resolution of discretized solid angle

" permittivity, " = "0 "r

zero mean gaussian random variable

; polar (spherical) coordinates

angle of incidence/reection/scattering

# surface slope

tan # surface gradient

wavelength, = c=f

permeability, = 0 r

M mean (error) between measurement and prediction

doppler shift

any random variable

radius of curvature of wave front (astigmatic ray tube)

M 2D correlation coecient between measurement and prediction

[ ] polarimetric radar cross section (RCS) matrix

[ 0 ] RCS matrix per unit area

h standard deviation of surface roughness

e eective conductivity

M standard deviation between measurement and prediction

doppler spread

delay spread

delay time of multipath component

hh autocorrelation coecient of surface heights

! angular frequency, ! = 2f

solid angle

Latin letters

a amplitude of multipath component

a horizontal half axis of ellipse

A area

A attenuation (in dB)

Ae = 40 GjC~ j2

2

Ae eective area of antenna,

b proportionality factor of clothoid

a vertical half axis of ellipse

C directional antenna pattern

TABLE OF SYMBOLS AND VARIABLES ix

d diameter

d distance

D maximum dimension of an aperture

e^ unit vector for eld decomposition at reection point

E electric eld

f frequency

G antenna gain

h surface height, height

h impulse response

H magnetic eld

H frequency response

hh; vv vv: from vertical to vertical)

co-polarization indices (e.g.

hv; vh hv: from vertical to horizontal)

cross-polarization indices (e.g.

j index indicating number of reection/interaction (1 : : : m)

j 2

imaginary unit, j = 1

k index indicating number of rays on same physical propagation

path ( 1 : : : M)

k wave number, k = 2=

l index indicating the number of (physical) waves/rays reaching a

receiver ( 1 : : : n)

l distance between two points on rough surface

l length

L correlation length of (random) surface heights

Lx;y dimensions of rectangular plate

m number of reections

M (theoretical) number of multiple rays

n^ normal vector

n number of rays reaching receiver due to multipath propagation

nd ray density

N number of launched rays

N number of rays

P power

P0;1;2 points in space

Pprop total time averaged propagating power through a tunnel

QR reection point

r distance (from transmitter), total unfolded path length

r radius, radius of curvature (e.g. of curve)

rR radius of sphere receiver

x TABLE OF SYMBOLS AND VARIABLES

larization

s distance

s RMS gradient (mean slope) of a surface

s arc length (clothoid, intermediate arc)

S radiation density (or time averaged Poynting vector)

S complex polarimetric scattering (matrix)

t time

T transfer factor

v speed (of obstacle, Tx or Rx)

V induced voltage at output/input port/terminals of antenna

w width

x; y location, cartesian coordinates

X weighting factor

Z impedance

p

Zw wave impedance, Zw = "

Subscripts

0; 1; 2 indices

0 free space (or vacuum) case

0 reference

k; ? parallel, perpendicular

doppler (shift/frequency)

delay (time)

a (intermediate) arc

A area

c curve or curved

cloth. clothoid(al)

cs cross section

d (ray) density

dB in decibel

D divergence

DCS in the DCS1800/GSM1800 band

e eective

exit re exit

F eld trace

GSM in the GSM900 band

h height

i incident

TABLE OF SYMBOLS AND VARIABLES xi

i isotropic

j; k; l indices

m number of ray interactions/reections

max maximum (number of reections)

M measurement (compared to)

n normal

n normalized

p reception plane

pq polarization indices

prop propagating

P plane receiver

P power trace

P propagation

r reected

r relative

rp rectangular plate

rt ray tube

R reception sphere

R receiver or received

s surface

s scattering

tot total (i.e. by all rays)

t transmitted power per ray (power trace approach)

T transmitter or transmitted

U 5; U 8 subways U5 and U8 in Berlin

w wave

x; y location

Superscripts

c; i coherent, incoherent

i; r; s incident, reected, scattered

LP lowpass

mod modied

Constants

"0 permittivity of vacuum, "0 = 01c20 = 8:85418810 12 361 10 9 Vm

As

0 permeability of vacuum, 0 = 4 10

7 Vs

p Am 8m

c0 speed of light in vacuum, c0 = 1= "0 0 = 2:997925 10

sq

free space wave (or characteristic) impedance, Zw 0 =

Zw 0 0

"0

120

Chapter 1

Introduction

Mobile radio communications and broadcasting have become common place.

A variety of established (FM-radio, GSM, IS-95, DAB etc.) and upcoming

(UMTS, TETRA, DVB etc.) systems and services coexist and share one of

the most valuable resources, the electromagnetic spectrum. As a result, almost

all these systems operate in the UHF frequency range and above ( > 300MHz)

in order to meet their spectrum requirements. The motto the freedom to

communicate at any time and at any place implies a complete and seamless

coverage for the oered services. Although in mobile communications over land

and in urban areas this is mainly achieved, the one Achilles heel is propagation

in tunnels. For example in Germany, there are more than 200 road tunnels with

a total length of more than 150km [Str96], and about 4% of the rail way consists

of tunnels. Furthermore there is an increased trend in the construction of new

tunnels, e.g. in and near settlements for noise prevention, and in fast rail tracks

for security reasons. To plan and provide ecient mobile services in tunnels,

knowledge of the transmission channel properties is obviously required.

+

a subject of research for several decades [DD 70, GDW71, MW74b, Wai75,

ELS75, Del82, etc.]. Initially this was mainly intended for professional usage

in mines for control, signalling, and emergency applications [MP78]. Tradi-

tionally these systems operate in the HF and VHF frequency ranges and be-

low (< 300MHz). For these frequencies, guidance of electromagnetic waves is

normally supported by special-purpose cables, e.g. leaky feeders (or radiating

cables).

2 CHAPTER 1. INTRODUCTION

1

At higher frequencies, however, natural propagation becomes an alternat-

ive due to the decreasing attenuation of electromagnetic waves in tunnels for

increasing frequency [Lan57, Der78]. In fact, at frequencies above 2GHz, leaky-

feeder systems are infeasible because of the opposite eect, namely, the atten-

uation constant in such cables increases drastically with increasing frequency

[Del82, Del87]. Furthermore, natural propagation has the great advantage

of lower installation and maintenance costs [Kle93]. In addition to economic

reasons, there is another argument in favour of natural wave propagation. Re-

cently, several major tunnels in the Alps were aicted by disastrous accidents.

The enormous heat produced by res inside the tunnels destroyed all install-

ations. The only means to communicate in such an extreme situation is given

by mobile communications via natural propagation.

The overall goal of this thesis follows from the above considerations:

netic wave propagation and the determination of the propagation

channel characteristics in highway or railway tunnels in the UHF

frequency range and above ( > 300MHz).

Realistic tunnel geometries are generally of rectangular cross section or arched

shape, i.e. of elliptical cross section with a raised oor and eventually an ad-

ditional ceiling. Furthermore, they are mostly curved [Str96]. In order to

calculate electromagnetic wave propagation in such tunnels, several modelling

approaches can be followed.

The ideal solution would be to solve Maxwell's equations for the bound-

ary conditions imposed by the tunnel's materials, its geometry and eventually

enclosed vehicles. Unfortunately, this eld-theoretical solution can only be ob-

tained by numerical methods. Full-wave solutions, like nite-dierence (FD)

[Yee66, Taf95] or method-of-moments (MoM) [Har68, Mit75] techniques, are

not feasible due to the enormous computational burden. Already a relatively

short rectangular tunnel section of 100m length, 8m width and 5m height would

require more than 1 billion voxels at 2GHz due to the necessary discretization

of at least =10 = 1:5cm. For each voxel, a minimum of six values, represent-

ing the complex vector eld, need to be stored in double precision, resulting in

6 8 = 48 bytes per voxel. Thus, only for the discretization about 50 gigabyte of

random access memory (RAM) would be required, indicating the almost inn-

ite computer resources required to solve the problem, regardless the numerical

1 I.e. no special-purpose cables are strung in the tunnel to guide electromagnetic waves

[Del82].

1.2. SCOPE AND OBJECTIVE OF THE THESIS 3

ginal computational costly boundary-value problem can be reduced to a more

ecient initial-value problem, which can be solved by the parabolic equation

(PE) method [Foc65, GW96, Gen96]. However, a fully three-dimensional PE

solution is still not available so that the actual modelling can only be performed

+

by further simplications [ZL 99].

A more conventional way to tackle the problem is given by mode the-

ory. Here, the tunnel is treated as a hollow waveguide with perfectly electric

conducting (PEC) or dielectric boundaries. Unfortunately, analytical expres-

sions for the wave impedance, propagation constant, cuto frequency, guide

wavelength etc. exist only for very few types, e.g. rectangular or circular cross-

section waveguides [Col91, Bal89, Mah91]. In practice, often the rst propagat-

ing mode is considered solely to characterize wave propagation in dierent

tunnels [ELS75, Del82], [Mah91, Chap. 6]. If one needs to compute the total

eld in a waveguide, the concept of Green's functions [Bal89, Tai93] or the

method of fast mode decomposition [MT99] are available for some canonical

geometries and usually for only PEC boundaries. Obviously, realistic bent or

arch-shaped tunnels with dielectric or even lossy boundaries can therefore only

be treated in a simplied or empirical way by mode theory [MW74b, YA 85].

+

2

All of the above stated frequency domain approaches have in common that

they only predict the electric or magnetic eld in a tunnel or waveguide. Wide-

band channel parameters, like the power delay prole (PDP), delay spread,

Doppler spread, or the angular spectrum (AS) cannot be obtained directly.

These parameters, however, are mandatory for modern digital communica-

tions systems design and evaluation [Lee89, Pro89, Ste92]. A means to predict

wave propagation in a complex environment with an adequate precision in

nite time, which also allows to obtain wideband channel characteristics, is

given by ray-optical modelling approaches [Kür93, Cic94, vD94, Gsc95, Riz97,

Küc98, GW98]. They can also be adopted for tunnels because generally all

obstacles in a tunnel are large compared to the wavelengths considered in this

thesis [LCL89].

Recently, several ray based methods have been proposed to model the

electromagnetic wave propagation in tunnels. Irrespective of their ray-tracing

technique (ray launching [ZHK98b], imaging [Mar92, Rem93, Kle93, MLD94]

or a combination of both [CJ95, CJ96a]), they all have in common that they

can only treat reections at plane boundaries. As a consequence, either they

only look at rectangular (piecewise) straight tunnel sections [Kle93, MLD94,

ZHK98b], or they tessellate more complex geometries into multiple plane facets

[CJ96a, HCC98].

In contrast, a novel ray-optical method is presented in this work, which

4 CHAPTER 1. INTRODUCTION

is not restricted to planar surfaces. For the rst time it allows a suciently

accurate ray-tracing based coherent calculation of the electromagnetic eld in

tunnels of arbitrary shape. The modelling is based on geometrical optics (GO)

[Bal89]. Contrary to classical ray tracing, where the one ray representing a

locally plane wave front is searched, the new method requires multiple repres-

entatives of each physical electromagnetic wave at a time. The contribution

of each ray to the total eld at the receiver is determined by the proposed

ray density normalization (RDN). This technique has the further advantage of

overcoming one of the major disadvantages of geometrical optics, the failure

at caustics. Additionally, wideband channel signatures like the power delay

prole, or parameters like the delay and Doppler spread, are predicted. The

model also handles moving vehicles inside tunnels so that sets of time series can

be generated automatically. These sets can for example be used to evaluate

the performance of dierent transmission schemes. Furthermore, deterministic

modelling is extended by a stochastic scattering component for rough surface

scattering, taking into account the non-deterministic nature of the scattering

process. These approaches are veried theoretically with canonical examples,

by various measurements at 120GHz in scaled tunnel models, and in real sub-

way tunnels at mobile communications frequencies.

The methods and techniques described in this thesis are based on ray-optical

wave propagation modelling. The chapter 2 is devoted to the foundations

and concepts of this modelling approach, leading from the ray concept to geo-

metrical optics and wave propagation modelling. The relevant propagation

phenomena are reviewed in section 2.3 and the notation used throughout this

thesis is introduced. A new method to include stochastic rough surface scatter-

ing into deterministic ray-optical propagation modelling is derived in section

2.3.3.3. Furthermore, multipath propagation is dealt with and the determina-

tion of broadband channel parameters with respect to the modelling results is

presented (cf. section 2.4).

Then the treatment of ray tracing in tunnels is discussed in chapter 3. First,

the existing methods of ray tracing are presented and evaluated in terms of

their ability to cope with the special environment of curved tunnels. It was

found that all of the commonly used techniques, i.e. imaging, ray launching

with reception spheres, or ray launching with discrete ray tubes, are only

suited for planar geometries. To circumvent this problem, a novel technique

termed ray density normalization (RDN) is derived in section 3.2.2, allowing a

suciently accurate ray-tracing based prediction of the electromagnetic eld

in curved geometries. As a side eect, the failure at caustics, which is inherent

1.3. OUTLINE OF THE THESIS 5

3.3.3). Additionally, a computational eective, approximate method is presen-

ted in section 3.4, which determines the mean received power in a tunnel as a

function of distance from the transmitter.

Chapter 4 deals with common geometries of tunnels and their modelling

with a computer. The approach to allow for transmitter placement outside

a tunnel is explained in section 4.3. Also, the inclusion of moving obstacles

like vehicles, the concept of stochastic ray launching, and the dierent types

of analysis are presented.

The next two chapters are devoted to the validation of the proposed mod-

elling schemes. In chapter 5, the models are compared to theoretical reference

solutions in canonical geometries. The method of Green's functions and image

theory are used to validate the schemes in a rectangular waveguide (cf. section

5.1). This is followed by a validation in a PEC corrugated circular waveguide.

The reference solution for this type of geometry is based on fast mode decom-

position (cf. section 5.2). After the theoretical validation, in chapter 6 com-

parisons to measurements are conducted. The measurements are performed in

scaled model tunnels at 120GHz with various constellations: a straight con-

crete tube with and without a vehicle present in section 6.2, a bent stoneware

tube in section 6.3, and a comprehensive model tunnel made of a straight and

a curved stoneware tube with a concrete road lane in section 6.4. The almost

perfect agreement of theoretical reference solutions and predictions, as well as

the very good agreement with measurements validate the presented modelling

techniques.

Finally, the applicability and performance of the modelling in real scenarios

is tested in chapter 7. For this purpose, measurements have been conducted in

the Berlin subway in two dierent tunnels with various antenna constellations

at 945MHz and 1853:4MHz. In contrast to the laboratory-like conditions of

the model tunnels in chapter 6, these measurements represent a type of worst-

case scenario, because of the dicult and varying geometry of the dierent

tunnel sections. It is shown that a correct modelling of the tunnel's cross sec-

tion and course is mandatory to obtain a suciently accurate wave propagation

prediction. Furthermore, the ability of the model to generate time series and

broadband channel parameters is presented in section 7.3, together with some

general results on electromagnetic wave propagation in curved tunnels.

Chapter 2

modelling

In this chapter the concepts of ray-optical wave propagation modelling are

presented as they are used throughout this thesis. The underlying assump-

tions and preconditions are briey reviewed, leading from the ray concept (cf.

section 2.1) to geometrical optics (GO) and wave propagation modelling (cf.

section 2.2). The formulations to describe the dierent propagation mech-

anisms are collected and reproduced in a uniform notation (cf. section 2.3).

Emphasis is placed on curved surfaces, because they appear in most tunnel geo-

metries (cf. section 2.3.2.2). Furthermore, a new method to include stochastic

rough surface scattering into deterministic ray-optical propagation modelling

is derived (cf. section 2.3.3.3). Finally, the handling of multipath propagation,

which is always present and particularly severe in a tunnel, together with the

possible methods of analysis based on the outcome of ray-optical propagation

modelling, conclude this chapter (cf. section 2.4).

For source-free, isotropic and homogenous media, the well-known (vector) wave

equations [Bal89, chap. 3] take the form

assumed, where ! denotes the radial frequency. This is justied, because any

2.1. THE RAY CONCEPT 7

a known spatial distribution of the complex amplitudes E~ (~x) and ~ (~x),

H the

instantaneous quantities are represented by

n o

E~ (~x; t) = < E~ (~x)ej!t (2.2a)

n o

~ (~x; t) = < H

H ~ (~x)ej!t ; (2.2b)

where <fg is the real part operator. The underbar denotes complex values,

whereas a vector is represented by an arrow. In the following only complex spa-

tial quantities are used and the explicit harmonic time dependency is omitted.

The complex wave number k is given by

k=

2 p" = k0p" ;

0 r r r r (2.3)

where k0 denotes the wave number in vacuum, "r the relative (eective) com-

plex permittivity [Bal89, chap. 2]

"

"r = = "r j e = "r (1 j tan Æe ) "r j 600 e AV ; (2.4)

"0 !"0

and r is the relative complex permeability. Furthermore, 0 = c0 =f rep-

in vacuum, and f is the frequency. In (2.4) " is the complex permittivity,

"r the relative permittivity of the medium, e is its equivalent conductivity,

"0 = 8:854 10 12As=Vm is the permittivity of vacuum, and tan Æe is the

1

eective electric loss tangent of the medium.

The simplest solution of the vector wave equations (2.1a) and (2.1b), ob-

tained by the method of separation of variables [Bal89, chap. 3], is given by

the travelling uniform plane wave

~ (~x) = H

H ~ 0 e j~k~x : (2.5b)

E~ 0 and H

~ 0 = k E0 ;

~ ~

H (2.6)

k Zw

1 The eective electric loss is the sum of the static and the alternating electric loss [Bal89,

chap. 2].

8 CHAPTER 2. RAY-OPTICAL WAVE PROPAGATION MODELLING

that is both the electric and the magnetic eld are perpendicular to each other

and to the direction of propagation, referred to as a transverse electromagnetic

(TEM) wave. The ratio of the complex amplitudes of the electric to magnetic

eld is known as the wave impedance

r

E0 r

Zw = = Zw 0 : (2.7)

H0 "r

In free space ( "r 1, r 1) the wave impedance (also named free-space

characteristic impedance) is given by

r

0

Zw0 = 120

; (2.8)

"0

where 0 = 4 10 7Vs=Am denotes the permeability of vacuum.

ray ray

wavefronts wavefronts

The uniform plane wave of (2.5) is now totally characterized by one of its

complex vectors E~ 0 or H

~ 0 , its wave number k, and its direction of propagation

k^, with ~k = k k^. Therefore, the wave may be seen as a ray (cf. Fig. 2.1(a))

propagating perpendicular to the wavefronts. However, such a plane wave

cannot be generated in reality. Provided that the point of observation is

suciently distant from a point source, the resulting spherical wave can be

approximated by a locally plane wave on a small portion of the sphere [GW98,

chap. 2], i.e. it still may be interpreted as a ray (cf. Fig. 2.1(b)). This approx-

imation is generally valid for distances greater than 2D2= from any source , 2

where D denotes the largest dimension of the source's aperture [Bal97, chap. 2].

2 Additionally, the extension of the area of observation has to be limited, so that the

phase dierence between its borders is suciently small.

2.2. GEOMETRICAL OPTICS AND PROPAGATION MODELLING 9

However, the amplitude of this locally plane wave decreases with distance in

contrast to the genuine uniform plane wave. The dependency of the amplitude

and the phase of the wave (or the ray) from the distance to the source is given

by geometrical optics and the Luneburg-Kline high-frequency expansion, as

indicated in the following section. In the remainder of this thesis, solely the

formalism for the electric eld will be stated, since the corresponding magnetic

eld can be obtained at any time by (2.6).

modelling

Geometrical optics (GO) is an approximate high-frequency method for determ-

ining wave propagation for incident, reected, and refracted elds. Because

it uses ray concepts, it is often referred to as ray optics [Des72, Bal89]. In

classical GO, the transport of energy between any two points in an isotropic

lossless medium is accomplished using the conservation of energy ux in a tube

of rays. The rays between them follow a path according to Fermat's principle

[Bal89, chap. 13]. If the background medium is homogenous (like the air in

a tunnel), the ray trajectories are straight lines, perpendicular to the wave

fronts (cf. Fig. 2.1). Within a tube of rays, also called pencil of rays [Des72],

the power energy ux has to remain constant

where

areas of the tube at two dierent locations separated by a distance s. For

TEM waves in a lossless medium S and the electric eld strength are related

as follows

S=

1 jE~ j2;

2Zw (2.10)

such that

jE~ j2 = dA0 :

jE~ 0 j2 dA

(2.11)

For an astigmatic tube of rays depicted in Fig. 2.2, being the most general

conguration of a ray tube, (2.11) leads to

r

jE~ j = dA0 = r 1 2

;

jE 0 j

~ dA ( 1 + s)(2 + s)

(2.12)

10 CHAPTER 2. RAY-OPTICAL WAVE PROPAGATION MODELLING

where 1;2 are the radii of curvature of the wave front at s = 0, whereas

(1;2 + s) are the radii at distance s from the reference point. For a positive

radius of curvature the wave is diverging, for a negative radius it is converging,

i.e. energy is focused. It is apparent from Fig. 2.2 that all rays of the astigmatic

tube of rays pass through the same lines PP0 and QQ0, which are called

caustics. The GO eld in such caustics is in principle innite, because an

innite number of rays pass through it. This behaviour is also reected in

(2.12): at s = 1;2 the denominator vanishes and hence the eld becomes

innite. In these areas/points, a quantitative evaluation of the actual eld is

therefore not possible by means of GO, although qualitatively, this focusing of

energy can be experimentally veried (cf. chapter 6). A means to overcome this

problem is presented in section 3.3.3. If the wave front is spherical, cylindrical

and a plane wave the radii are 1 = 2 = , 1=2 = 1, 2=1 = , and

1 = 2 = 1, respectively.

dA

dA0

caustic lines

P

s

P'

ρ1

Q

s=0

Q' ρ2

reference magnitude at s = 0. In order to obtain an expression for the actual

complex eld vector, i.e. including phase and polarization, the Luneburg-Kline

high-frequency expansion [Lun64, Kli51, Bal89] is adopted. By restriction to

rst-order solutions this approach in combination with classical GO leads to

r r

~ (s) = 1 2 ~ 0;

E e jk0 s E

1 + s 2 + s

(2.13)

2.3. PROPAGATION PHENOMENA 11

where ~0

E is the electric eld at the reference point (s = 0). Contrary to

common usage [MPM90, vD94, Bal89], the square root must be split in order

to correctly predict +90Æ phase jumps each time a caustic is crossed in the

direction of propagation [LCL89].

The previous equation is the basis for all following considerations. It is

only valid for high frequencies, if the eld is to exhibit a ray-optical behaviour,

i.e. if the assumption of locally plane waves holds, and all obstacles are large

compared to the wavelength. At mobile radio frequencieswhich are of matter

in this thesiswith corresponding wavelengths of 0 < 0:5m, this assumption

is naturally fullled.

In (2.13) the free space dependency of the electric eld of a locally plane

wave front, i.e. a ray in free space, is stated. This ray is now interacting

with its surrounding by means of reection, scattering etc. The actual total

eld at a particular point of observation is given by the complex and vectorial

superposition of all present rays (or wave fronts) at that point. The overall

aim in ray-optical wave propagation modelling is now to determine all relevant

rays from a source to the point of observation, to calculate the eld strength of

each single ray, and to superimpose all rays in order to obtain the total eld.

In the following section, the formalisms to calculate the ray propagation

phenomena free space propagation, absorption by the atmosphere, specular

reection, and scattering are presented. Diraction is neglected due to its

minor inuence compared to reection in realistic tunnel scenarios, although

it can be included in ray-optical wave propagation modelling [Leb91, Cic94,

vD94, Küc98] by the geometrical theory of diraction (GTD) [Kel62] and its

extension to the uniform geometrical theory of diraction (UTD) [PBM80,

Jam86, BU88, Bal89, MPM90]. Then multipath propagation is treated and

how the dierent rays are actually combined.

In this section the phenomena free space propagation, attenuation through the

atmosphere, specular reection at planar and curved surfaces, and scattering

at rough surfaces are treated in a way that they can easily be used in con-

junction with the ray-optical approach of the previous sections. The goal is

to successively take into account the dierent propagation mechanisms after

a ray trajectory is known from a source to a point of observation, in order to

determine the eld strength of the ray.

12 CHAPTER 2. RAY-OPTICAL WAVE PROPAGATION MODELLING

2.3.1.1 Free space propagation

In a tunnel, the surrounding medium is air, which at a rst approximation can

be taken as lossless with "r = r = 1. In free space, the dependency of the

electric eld is then already given by (2.13). Of further interest is the special

case of a spherical wave, which is radiated by a point source. For a point source

the initial values of the radii of curvature are theoretically zero. Therefore, it

is convenient to take the reference value ~0

E not at the point source, but at a

reference distance. Throughout this thesis the reference distance s0 = 1m is

taken, resulting in

~ (r) =

E

1m e jk0 (r 1m)E~ 0 ; (2.14)

r

where r = 1m + s denotes the distance from the point source. In (2.14) it

is assumed that the phase at the input port/terminals of the antenna equals

0 = 0Æ. If the transmitting antenna is modelled by a point source with

complex vector directional pattern

3

C~ T and gain GT , the reference value is

given by [GW98, chap. 2]

r

~0 = PT GT Zw0 e jk0 1m ~

E

2 1m C T ; (2.15)

where PT denotes the input power of the transmitting antenna. One should

note that (2.14) in conjunction with (2.15) is only valid in the far eld of the

antenna, but not in the vicinity of the source.

2.3.1.2 Absorption

At certain high frequencies, additional attenuation of the electromagnetic

waves in the atmosphere may occur due to resonance absorption of specic

gaseous molecules. In the frequency range up to 100GHz this is mainly the

case for the self-resonances of water (H O) at 2 22GHz and of oxygen (O2) at

60GHz [Hal79, ITU676]. This gaseous absorption results in an exponential

decay of the eld strength with the geometrical optical path length

jE~ (s)j = e s jE

~ 0 j; (2.16)

3 The complex vector directional pattern denes the phase, polarization and directivity

of an antenna [Bal97, GW98].

2.3. PROPAGATION PHENOMENA 13

where denotes the attenuation coecient with dimension 1=m. Usually the

attenuation is given in dB=km yielding the following relations

!

A

= 20 lg jE~ 0 j = s 20 lg(e) 8:69s

dB jE~ (s)j

(2.17)

or

1=m 8:169 A=

s=m

dB = 1 dB

8:69 10 dB=km

3 1

:15 10 4 dB :

dB=km (2.18)

communication link [SL90] or mobile broadband systems [COST231], the at-

tenuation has a maximum of dB 15dB=km [ITU676].

Thus far, propagation in unbounded media has been treated. In a tunnel,

however, the rays (and therefore the elds) encounter boundaries, such as the

tunnel walls or the ground, and other obstacles like cars or trains, at which

the energy is reected or scattered in dierent directions. In the next section,

reection at smooth surfaces is treated, followed by some considerations on

scattering from rough surfaces.

Reection is obviously one of the dominant propagation mechanisms in a tun-

nel environment. A general solution for the eld must satisfy both Maxwell's

equations and the boundary conditions. For the special case of two dier-

ent homogeneous innite halfspaces, which are separated by a smooth planar

interface, the solution is given by the so-called Fresnel reection and trans-

mission coecients. The total eld is split into an incident, a reected and a

transmitted portion. At the tunnel walls, only the incident and reected parts

are of importance, since it is assumed that the transmitted energy will be

4

completely absorbed by the building materials . The derivation of the Fresnel

coecients are given in almost any textbook on electromagnetic eld theory

[UMF83, Bal89, Leh90]. Thus, only the results together with the convention

used throughout this thesis are stated in the following.

In a tunnel, reection normally occurs at boundaries between the air and

e.g. the tunnel's walls or ground. Therefore, the original problem with two

4 Also, as will be seen in section 4.4, the vehicles in the tunnel are supposed to be metallic

and hence purely reective.

14 CHAPTER 2. RAY-OPTICAL WAVE PROPAGATION MODELLING

vacuum (or air) and a non-magnetic material ( r;1 = 1) with relative eective

complex permittivity "r;1 . The incident plane wave has to be decomposed into

two orthogonal polarizations which are treated separately. The total eld is

then given by the vector sum of the two components. Figure 2.3 depicts the

convention used for this decomposition: the electric eld perpendicular to the

plane of incidence, i.e. parallel to the interface, is referred to as perpendicular

polarization (index: ?); when the electric eld is parallel to the plane of

incidence, it is referred to as parallel polarization (index: k). The plane of

incidence is spanned by the normal vector of the interface between the two

media n^ and the unit vector in the direction of incidence of the plane wave (or

ray) k^i . The superscripts i and r refer to the incident and the reected values,

respectively.

i

E Er Hi H

r

k^ k^r k^i k^

i r

Hi θi θr Ei θi θr

^ Hr ^ Er

ε0,µ0 n ε0,µ0 n

ε1,µ0 ε1,µ0

Figure 2.3: Reection of an incident plane wave (or ray) at a smooth plane

boundary between air ("r; 0 = 1; r;0 = 1) and a homogenous dielectric innite

halfspace with "r;1 and r;1 = 1

The reection coecients Rk;? , relating the incident and the reected elds

according to

Er r

Rk;? = ki ;?

E k;?

= HH ki ;? ; (2.20)

k;?

are given by

q

"r;1 cos i "r;1 sin2 i

Rk (i ; "r;1 ) = q ; (2.21)

"r;1 cos i + "r;1 sin2 i

2.3. PROPAGATION PHENOMENA 15

q

cos i "r;1 sin2 i

R? (i ; "r;1 ) = q : (2.22)

cos i + "r;1 sin2 i

As previously indicated, the transmitted portion of the wave is assumed

to be totally absorbed and hence not further treated in this thesis. If a more

conned model for vehicles than the one of section 4.4 is used, transmission

needs to be considered. Transmission can be described in analogy to the

reection problem according to Snell's law of transmission and by transmission

coecients, which are similar to those in (2.21) and (2.22) [UMF83, Bal89,

Leh90].

The following special cases are needed in parts of the thesis. First, at

an ideal PEC metallic surface with ! 1, the reection coecients are

independent of the angle of incidence

R k = R k = 1; R? = R? = 1: (2.23)

5.2.2) [Kil90], with

Rk = R? = R = 1: (2.24)

The previous equation is also valid for a non-PEC halfspace at grazing incid-

ence ( i ! 90Æ), which is important at large distances from the transmitter.

In this case the coecients are independent of the electrical properties of the

dielectric material.

The separation of the incident eld into a parallel and a perpendicular

component allows to treat reection by (2.19)(2.24). This separation, how-

ever, depends on the incidence direction k^i and the orientation of the reecting

n^, and is therefore a local

R so that

surface given by transformation. A compact for-

mulation is achieved by introducing the dyadic reection coecient

the reected eld at QR can be obtained from the incident eld at QR via

E~ (QR ) = R E~ (QR ):

r i

(2.25)

R = Rk e^rk e^ik + R? e^r? e^i? ; (2.26)

with

n^ k^i

e^i? = e^r? =

jn^ k^i j

(2.27a)

16 CHAPTER 2. RAY-OPTICAL WAVE PROPAGATION MODELLING

Strictly speaking, the aforementioned formulas are only valid for an uniform

plane wave incident on an innite smooth planar boundary, but they may still

be adopted for locally plane waves (or rays), if the dimensions of the interface

(and therefore obstacles) are large compared to the wavelength.

If the interface is curved, like arched cross sections or bends of a tunnel, re-

ection can still be treated by ray-optical methods provided that the radii of

curvature of the surface are large compared to the wavelength [KP74, Bal89].

The incident ray, representing a tube of rays or a locally plane wave front, is

simply reected at the tangential plane in the intersection point of the incident

ray trajectory and the surface. The angle of reection is again given by Snell's

law of reection and equals the incident angle determined by the normal n^ of

the tangential plane and the direction of propagation of the incident ray. In

contrast to the reection of a plane wave at a planar boundary, however, one

must consider that the principal radii of curvature of the reected tube of rays

are in general dierent from the ones of the incident tube of rays due to the

curvature of the surface. In other words, the tube of rays may be focused or

defocused by the reecting surface.

Figure 2.4 depicts the geometry of the reection at a curved surface, where

i1;2 and r1;2 denote the principal radii of curvature of the incident ray tube

and the radii after reection, respectively. The reference point in both cases

i

is the point of reectionQR . For a given electric eld E~ of the incident wave

at the point of reection QR , the reected eld at a distance s from the point

of reection becomes

s s

r1 r2

~r

E (s) = e jk0 s R E

~ i (QR ):

1 + s 2 + s

r r (2.28)

If the electric eld vectors are decomposed in parallel and perpendicular po-

larization referring to the plane of incidence (spanned by

reection coecient R reduces to a diagonal matrix with components Rk and

R? . The principal radii of curvature r1;2 are related to the principal radii of

the incident wave front i

1;2 , the angle of incidence, and the curvature of the

reecting surface at QR in a non-trivial way. The equations for calculating

r1;2 are taken from [KP74, Bal89] and are summarized in appendix A. For the

special case of a planar surface, the principal radii of the tube of rays remain

unchanged, i.e. r1;2 = i1;2 .

Thus far, perfectly smooth surfaces were assumed. The impact of surface

roughness on wave propagation and its classication is treated in the following

section.

2.3. PROPAGATION PHENOMENA 17

reflected

tube of rays

ρi2

ρi1

^

n

s

QR

ρr1 surface

ρr2

Figure 2.4: Ray optical reection of an astigmatic tube of rays from a curved

surface (cf. [Bal89])

For innite perfectly smooth surfaces, the total energy of a ray is reected in

the specular direction, given by Snell's law of reection as indicated in the

previous section. In reality however, some portion of the energy is scattered

in other directions than the specular one due to the roughness of the surface.

The scattered energy is generally split in the so-called coherent and incoherent

+

components [RB 70, UMF83, GW98]. The coherent part is dened by the

mean value of the scattered eld, while the incoherent part characterizes the

deviation of this mean. The coherent component decreases with increasing

surface roughness, whereas the incoherent (or diuse) component becomes

more signicant. Before attempting to tackle the problem of scattering, one

must quantify the roughness of a surface.

18 CHAPTER 2. RAY-OPTICAL WAVE PROPAGATION MODELLING

θr

θi

L

σh

surface height and by its correlation length L

of surface heights, a geometrical measure of the roughness is given by the

standard deviation h of the surface height and by its correlation length L

(cf. Fig. 2.5). The rst is a measure for the variations of the height against

its mean value, the second is a simple measure for the statistical dependence

of the heights at two dierent points on the surface. The greater h and the

smaller L, the rougher the surface in mechanical terms. For the scattering

of an electromagnetic wave, however, not the absolute value but the relation

between h and the wavelength 0 matters. A heuristical but widely accepted

criterion for the roughness of a surface with respect to the wavelength is given

by the Fraunhofer criterion, namely that a surface can be considered smooth

for [RB

+ 70, BS63, UMF83]

0

h <

32 cos i : (2.29)

Equation (2.29) results from the demand that the root mean square (RMS)

phase dierence between two rays reected at two dierent heights on the

surface must be smaller than =8 in the far eld in order to combine coherently,

i.e. the rays are almost in phase, as in the case of a perfectly smooth surface.

cients

If the surface has only a slight roughness, i.e. h 0 , the diuse component

can be neglected and the scattering of the coherent component may be well

2.3. PROPAGATION PHENOMENA 19

direction is reduced due to the partial cancellation of the statistically varying

phases, compared to the pure specular reection discussed in section 2.3.2. In

this case, the so-called modied Fresnel reection coecients [BS63, UMF83,

LFR96] can be adopted, which approximate the reduction of the magnitude

of the eld by

k;? = Rk;?e

Rmod (2.30)

According to (2.29) the ratio of the modied to the original Fresnel reection

coecients at the Fraunhofer limit is Rmod=R = 0:926 or 0:67dB. As a rule

of thumb, (2.30) is applicable up to roughly 4 times the roughness allowed by

(2.29), known as the Rayleigh criterion .

5

At the Rayleigh limit R

mod =R =

0:291 or 10:7dB.

The roughness h of common building materials in tunnels is in the lower

millimeter range (e.g. 1mm for tarred road surfaces [Sch98, LS99] and for

smooth concrete surfaces [LS99]). Hence, the modied reection coecients

are well applicable for standard mobile communications frequency ranges. For

millimeter waves or if the roughness exceeds the millimeter range, the dif-

fuse scattering can no longer be neglected. An (almost) exact solution for the

scattering of a statistically rough surface can only be obtained numerically

by computationally costly Monte-Carlo methods [AF78], [WSC97, with fur-

+

ther references]. Analytical scattering models [BS63, RB 70, UMF83, Fun94],

using the statistical parameters of the surface h , L etc., provide only mean

values for the coherent and/or the diuse components. An approach, which

combines these two methods and which suits well with the stochastic ray-

launching approach of section 4.2, is presented in the following section.

The novel stochastic scattering approach (SSA) is based on the same assump-

tions as the Kirchho or the physical optics (PO) formulation for surface

scattering. These theories are applicable to surfaces with gentle undulations,

whose horizontal dimensions are large compared with the incident wavelength.

Hence, the total eld at any point on the surface can be computed as if the

ray (or the wave) is impinging at the tangential plane in that point [UMF83].

This tangential plane approximation is the basic assumption for the Kirchho

methods. The dierence to specular reection, where all elementary waves

of an incident uniform plane wave are reected at the same ideally smooth

planar boundary, lies in the dierent orientation of the tangential planes in

5 (2.30) actually results from the scalar Kirchho approximation, which is valid for the

conditions given in (2.44) and (5.12).

20 CHAPTER 2. RAY-OPTICAL WAVE PROPAGATION MODELLING

space and the varying heights at the respective points of reection for each

elementary wave. Thus, depolarization and scattering in directions other than

the specular one may occur.

nt s

t fro y

ron s ave le ra

a vef e ray e w tip

ne

w tip

l

l

p lan mul

p l a m u

ally by

ally by loc ented

loc ented e s

s r

rep

r e rep

tangential planes for each discrete ray (here only a 2D representation is shown

for simplicity)

The main idea of the stochastic scattering approach is now based on the

stochastic ray launching of sections 3.2.2 and 4.2, where each locally plane wave

front is actually represented by multiple discrete rays instead of only one ray

(cf. Fig. 2.6). Instead of reecting all these discrete rays at the same boundary

plane, the orientation of the plane (its normal vector) and its position (its

height) are varied statistically for each discrete ray and for each reection.

In that sense each discrete ray is seen as a representative of an elementary

6

wave for the locally plane incident wave front. The variations of the local

tangential planes naturally have to be related to the properties of the rough

surface, which will be derived in the following.

The surface height of most man-made materials is (at least approximately)

normally distributed [BS63, LS99]. Let the height h be normally distributed

with zero mean and variance h , i.e. the distribution of the height is given by

ph (h) = p

1

1 h 2

2h e

2 h : (2.31)

hh (l) = e l =L ;

2 2

(2.32)

6 The tangential planes are local in the sense that for each discrete ray a dierent

tangential plane is generated.

2.3. PROPAGATION PHENOMENA 21

which gives the correlation between the random height h at two points on

the surface separated by a distance l. L is the correlation length, for which

(2.32) equals1=e. Then, according to [BS63, appendix D], [Hag66], [UMF83,

# of a normally distributed one-

appendix 12F], the distribution of the slopes

dimensional (1D) surface h(x), dened by

tan # = h0 = dh

dx

; (2.33)

h20 = 2 2

h2 00hh (0) = 2h : (2.34)

L

Hence the distribution of h0 equals

L Lh0 2

p (h0 ) =

2hp e :

2h

h0 (2.35)

becomes

L L tan # 2

p# (#) = p

2h cos2 # e :

2h (2.37)

For any random variable , the second central moment (variance) is given by

h i 2 = h 2 i h i2 ;

(2.38)

where hi is the expectation operator. With (2.38) and hh0 i = 0, it follows that

the root mean square (RMS) gradient (often misleadingly called RMS slope)

is given by the root of (2.34)

p

= 2Lh :

q

hh0 2 i = h0 (2.39)

For a two-dimensional (2D) isotropic rough surface with h = h(x; y), which

is described by (2.31) and (2.32), a measure for the slope magnitude of the

irregularities forming the rough surface is therefore given by [BS63, RB 70]

+

s = h0x 2 + h0y 2 2

1

= 2Lh : (2.40)

22 CHAPTER 2. RAY-OPTICAL WAVE PROPAGATION MODELLING

through the origin of random orientation and the same mean slope having the

same statistical properties as the considered Gaussian rough surface is shown

p

p

in Fig. 2.7. It is determined by three points P0 (0; 0; 0), P1 (L= 2; 0; x) and

P2 (0; L= 2; y ) in cartesian coordinates. x;y are two independent Gaussian

random variables with zero mean and variance h . Equivalently the plane is

dened by P0 and its normal vector

p p

~n =

2x ; 2y ; 1T =

h0x; h0y ; 1 T : (2.41)

L L

The rst two components of the normal vector represent the negative gradients

of the plane h0x and h0y in the x- and y-directions, respectively. It is easily

veried that these gradients are distributed according to (2.35). The RMS

n P2

ζy y

ϑn

L/√2 P1

P0 ζx

x

L/√2

Figure 2.7: Local tangential plane of random orientation with the same statist-

ical properties as a Gaussian rough surface with a given mean gradient

tan2 #n 2

1

=

2x 2 + p2y 2 21 = p2

2 + 2 12 = 2h :

L L L x y L

(2.42)

distribution

12 + 22 = 22 ; (2.43)

where the 1=2 are statistically independent, normally distributed random vari-

ables with zero mean and variance . It should be noted that the absolute

2.4. MULTIPATH PROPAGATION AND ANALYSIS 23

p

p

value of the gradient j tan #n j = tan2 #n is Rayleigh distributed with mean

L h . The agreement of (2.42) and (2.40) proves the equivalence of the stat-

istical properties of the slopes of the randomly oriented plane in Fig. 2.7 and

a Gaussian rough surface. This means that the variation of the orientation

of the local tangential planes in the stochastic scattering process is to be per-

formed according to (2.41). The local height variation of the reection point is

naturally performed according to a Gaussian law with zero mean and variance

h2 .

By applying the stochastic scattering approach, the inclusion of random

(but directive) surface scattering into ray-optical propagation modelling be-

comes possible. The purely deterministic GO modelling is expanded by a

stochastic component, resulting in varying prediction results; thus allowing

for the rst time to account for non-deterministic scattering in ray-optical mod-

elling. The results can be used in the generation of time series for the purpose

of system performance evaluation, which otherwise could only be obtained by

the assumption of ergodicity with respect to time and space [Kür93, Bec96].

7

In contrast to the two Kirchho models , which are only valid for either

slightly rough or very rough surfaces, the stochastic scattering approach in-

cludes both the coherent and incoherent components at the same time. Its

validity conditions are only determined by the tangential plane approximation

[UMF83]

L2 6 0:

2:76h > 0 ; and L>

2 (2.44)

The rst condition in (2.44) calls for the mean radius of curvature of the ran-

dom rough surface to be large compared to the wavelength, the second calls

for a large correlation length compared to the wavelength. Multiple scattering

between various parts or the surface is neglected in the stochastic scatter-

ing approach, like in the Kirchho models. A comparison of the proposed

stochastic scattering approach with the Kirchho models is given in section

5.3.

Note that an empirical approach, which uses randomly changed normal

vectors of the walls to take surface roughness into account, was proposed in

[LL98].

In the previous section the dierent (relevant) propagation mechanisms and

their integration into ray-optical propagation modelling was presented. Each

7 The Kirchho model with scalar approximation and the Kirchho model with stationary

phase approximation (cf. section 5.3).

24 CHAPTER 2. RAY-OPTICAL WAVE PROPAGATION MODELLING

ray experiences on its way from the transmitter to the receiver a certain at-

tenuation, depolarization, phase shift, and delay, due to the dierent afore-

mentioned propagation eects. Moving objects, as vehicles or trains, or the

motion of transmitter and/or receiver, additionally introduce a Doppler shift

for each ray. In the complex environment of a tunnel, there are generally

several ray paths from the transmitter to the receiver, each representing a

physical propagation path for the corresponding electromagnetic wave. For

the described multipath propagation, the dierent impinging rays interfere,

contributing to the resulting eld at the receiver. How to perform this combin-

ation of rays and what other consequences result from multipath propagation

is treated in the following.

by the generalization to multipath propagation. Then the broadband channel

signatures frequency response, impulse response, delay-Doppler-spread func-

tion, the parameters delay spread and Doppler spread are given as functions

of the multipath components.

On each propagation path from the transmitter to the receiver, a ray may

experience several propagation phenomena in any order. The starting point

for each ray is the radiated (far) eld of the transmitting antenna, which is

given by (2.14) and (2.15)

r

~ (r) = PT GT Zw0 e jk0 r ~ jk0 (r 1m) T D;0 E

E

2 r

CT =e ~ 0; (2.45)

where T D;0 = 1m=r denotes the divergence transfer factor, which is a measure

for the divergence of the tube of rays. At each interaction of the ray with an

TP;j , which accounts for the actual propagation eect, and by T D;j , which ac-

obstacle, the eld strength is multiplied by a dyadic propagation transfer factor

counts for a change in divergence due to the interaction. For a reection, T P;j

is given by the dyadic reection coecient and T D;j by the possibly complex

square roots in (2.28). Considering the absorption case, T P;j reduces to a

8

scalar, given by the exponential term in (2.16), and T D;j = 1. Cascading all

transfer factorsand therefore all occurring propagation phenomenaleads

8 The square roots may be complex after crossing a caustic (cf. section 2.3.2.)

2.4. MULTIPATH PROPAGATION AND ANALYSIS 25

0

Y

E ~ (r) = e jk0 (r 1m)

~R =E T D;j TP;j E~ 0 ; (2.46)

j =m

| {z }

T=T D TP

r m the number of ray

TP;0 in (2.46) equals

where denotes the total unfolded path length and

interactions. The free-space propagation transfer factor

T = T T

the unit dyadic. As indicated in (2.46), all transfer factors can be combined to

the total (dyadic) transfer factor D P , which, together with the phase

factor ejk0 r ( 1m)

, completely characterizes the propagation of the ray.

The electric eld can only be determined indirectly via a receiving antenna

by its induced voltage V R at the output port/terminals of the antenna and/or

the amount of power PR delivered to its load. The eective area of the receiving

antenna is given by [Bal97, GW98, chaps. 2]

2

AeR = 0 GR C~ R 2 :

4 (2.47)

voltage V, and the internal impedance of a generator Z under complex con-

jugate matching is

jV j :2

P = 8<f Zg

(2.48)

9

rary orientation of the receiving antenna one obtains

s

20 4<fZ R g ~ ~

VR =

4 GR Zw0 C R E R; (2.49)

where GR , C~ R and ZR denote the gain of the receiving antenna, the complex

vector directional pattern and the impedance, respectively. Consequently, the

power delivered to the load equals [Bal97, GW98]

2

20 1 2 2

0

PR = GR C R E R =

~ ~

GR GT PT C R T D T P C T :

~ ~

8 Zw0 41m | {z }

T

(2.50)

26 CHAPTER 2. RAY-OPTICAL WAVE PROPAGATION MODELLING

matched to its load. In chapter 3.3, a modied form of (2.50) is used to

simplify the notation

0 2 2 2

PR =

41m GR GT PT T| D{zTP} : (2.51)

T2

TD TP

TP

In (2.51), the real equivalents and of the complex divergence trans-

fer factor TD and the complex dyadic propagation transfer factor after

reception are introduced with

TD = jT D j (2.52a)

TP = C~ R T P C~ T :

(2.52b)

For the direct or line-of-sight (LOS) path from the transmitter to the receiver

with TD = TD;0 and TP = jC~ R C~ T j, (2.51) (respectively (2.50)) reduces to

the well-known Friis transmission equation [Bal97]. Equations (2.46)(2.52)

are valid for any propagation path between the transmitting and receiving

antennas.

In the previous section, the propagation path of a single ray (or wave front)

at a given frequency was studied. In general, a large number of rays n reach

the receiver on dierent propagation paths and interfere, especially in conned

spaces as tunnels. The overall electric eld at the receiver E~ R;tot is therefore

~ ~

given by the vector sum of all E R;l . The individual elds E R;l of the multipath

components l , with l = 1; : : : ; n, are given by (2.46). The total induced voltage

V R;tot at the output port/terminals of the receiving antenna is obtained by

the summation of all contributing complex voltages V R;l from (2.49), i.e.,

n

X

V R;tot = V R;l : (2.53)

l=1

Equivalently, the total power delivered to a matched load is

2 n 2

c

PR;tot =8

V

R;tot 1

= 8<fZ g

X

V R;l

<fZ R g R

l=1

(2.54a)

n 2

0 2 X

= 41m GR GT PT e

jk0 rl C R;l T l C T;l :

~ ~ (2.54b)

l=1

2.4. MULTIPATH PROPAGATION AND ANALYSIS 27

by the superscript c), which requires perfect knowledge of the amplitude and

phase of each single V R;l . However, already small variations in the geometry

of the propagation environment on the order of 0 =2, like the dimension of

a tunnel's cross section or a receiver location, generally lead to a signicant

10

change in the phase of each multipath component . Therefore, it is often

useful to average the received power over a certain range. This averaging can

be easily approximated, if the complex voltages of the multipath components

are substituted with the respective received powers PR;l , yielding

n 2

P R;tot =

1 X

V R;l 8<f1Z g V R;l 2 = PR;l = PR;tot

n

X n

X

i :

8<fZ Rg

l=1 R l=1 l=1

(2.55)

termed power sum, equals the averaged received power P R;tot , if the phases

of the voltage components are completely uncorrelated, and if the amplitudes

are constant. Using the simplied formalism of the previous section, one can

write

n

0 2 X

i

PR;tot = GR GT PT Tl2:

41m l=1

(2.56)

Again, (2.56) presents only the mean received power. Locally, the instantan-

eous values may dier, according to the values obtained by (2.54).

One of the major advantages of ray-optical wave propagation modelling is the

ability to obtain broadband channel parameters in addition to the narrowband

eld strength/received power prediction. This is possible due to the distinction

of each ray and its attributes delay, attenuation, phase shift, and Doppler

shift. Additionally, the fully polarimetric modelling allows to investigate the

inuence of dierent antennas and transmission schemes without increase in

computational costs.

Originally, the calculation is performed at one single frequency, i.e. for

harmonic excitation. For non-dispersive propagation paths, i.e. under the

assumption that the phenomena reection, scattering, and attenuation are

similar (if not equal) within the considered bandwidth, the single frequency

result of each ray can be used for the entire frequency band. However, the

10 Whereas the amplitude remains almost constant.

28 CHAPTER 2. RAY-OPTICAL WAVE PROPAGATION MODELLING

the channel, because the absolute phases of the single rays depend on the

traversed path length and the frequency.

Furthermore, the channel is also time-variant. The reason for time-variance

in a tunnel is movement of obstacles, transmitters, and/or receivers. Movement

leads to a Doppler shift for each ray resulting in a Doppler spread (and/or

shift) after multipath reception. Modelling the Doppler shift is incorporated

in the ray-optical approach in a straightforward way: each ray carries its ac-

tual Doppler shift in addition to the eld strength information. Consequently,

the output of a ray-optical propagation prediction is a polarimetric Doppler-

resolved impulse response with additional angle-of-arrival information, from

which the power delay prole (PDP), the Doppler spectrum, and the angu-

lar spectrum (AS) can be derived. In the following, the explicit connection

between ray-tracing results and the channel functions frequency response, im-

pulse response, and delay-Doppler-spread function are pointed out, together

with the general parameters delay spread and Doppler spread.

nel

The frequency response of the transmission channel H (f ) can be dened by

the ratio [GW98]

V

H (f ) = R;tot ;

(f )

V T (f )

(2.57)

the transmitting antenna. Assuming complex conjugate impedance matching,

H follows from (2.53) for positive frequencies f >0

r n n

0 2 o

e j2fl C~ R;l Tl C~ T;l

X

H (f ) = GR GT

41m l=1

(2.58a)

n

X

= al e j2fl : (2.58b)

l=1

multipath component. Except

for the wavelength, the only explicit frequency dependence of the multipath

components in (2.58) results from the delay term. Their complex amplitudes

al are assumed to be independent of f.

Using the notation for bandpass signals and systems with their equivalent

lowpass representation [Pro89, chap. 4], the lowpass impulse response hLP ( )

2.4. MULTIPATH PROPAGATION AND ANALYSIS 29

by f0, leading to

n

X

hLP ( ) = al e j2f0 l Æ( l ): (2.59)

l=1

The real-valued bandpass impulse response is therefore given by

The above equations hold for a channel with innite bandwidth. For band-

limited systems, the dirac function in (2.59) has to be replaced by the actual

lter function of the system, which includes both the transmitting and receiv-

ing blocks.

The aforementioned channel functions reect the frequency-selective behaviour

of the channel. The time-variant nature of the channel is integrated in de-

terministic wave propagation modelling by means of movement and therefrom

resulting Doppler shifts. The Doppler shift j experienced by a ray at the

interaction with a moving obstacle is illustrated in Fig. 2.8.

Assuming a xed direction of propagation k^j , the Doppler shift of the ray

is given by

~vj k^j

j = f;

c0 j

(2.61)

where ~vj denotes the velocity vector of the obstacle, ~vj k^j is the speed in the

direction of propagation, and fj is the frequency before the interaction. In the

case of a moving receiver (or transmitter), (2.61) states the total experienced

Doppler shift. However in a strict sense, if reections at moving obstacles are

considered, the Doppler shift of the incident ( i) and the reected ray (r)

have to be treated separately. For reections at a planar surface moving in the

direction of its normal vector (cf. Fig. 2.8(b)), the Doppler shift is given by

j = ji + jr = fj + j j (fj + ji )

c0 c0

~vj k^ji ~vj k^ji 2 ~vj k^ji (2.62)

= 2 c0 + c0 fj 2

c 0

fj = 2ji :

| {z }

1

30 CHAPTER 2. RAY-OPTICAL WAVE PROPAGATION MODELLING

fj

fj

k^ij

k^j

vj

Rx

vj

k^ j

r

fj+νj

Figure 2.8: Ray interaction with moving receiver or obstacle of speed ~vj , result-

ing in a Doppler shift j for the ray

The overall Doppler shift of a ray is obtained by the sum of all occurring j .

The approximation in (2.62) is equivalent to setting fj = f0, which can be

generalized to simplify the calculation of the overall Doppler shift.

Using the overall Doppler shift l of each multipath component, the delay-

Doppler-spread function [Bel63, Fle90], or Doppler-resolved lowpass impulse

response [Kat97], is given by

n

X

sLP (; ) = al e j2f0 l Æ( l ) Æ( l ); (2.63)

l=1

from which all other general system functions can be obtained via Fourier

11

transforms. The time-variant lowpass impulse response , which is the gen-

eralized form of (2.59), is e.g. obtained by the inverse Fourier transform of

(2.63) with respect to the Fourier relation ( Æ t).

2.4.3.3 Delay spread

In order to characterize the frequency-selective behaviour of a transmission

channel, the integral parameter delay spread plays an important role. It is a

2.5. SUMMARY 31

inuence of the channel. Mathematically it is equivalent to the denition of

the standard deviation of the power delay prole (PDP) of a channel and

is therefore related to its width. The PDP, generally obtained by ensemble

averaging, is proportional to the square of the absolute value of the lowpass

impulse response (2.59) [Ste92, GW98]. The square root of the second central

moment of the PDP, which is the delay spread, can thus directly be calculated

from the output of the ray tracing as

v

2 2

u Pn Pn !

u 2 2

l=1 l Tl l Tl

= t P

n 2 Pl=1

n T2 : (2.64)

l=1 Tl l=1 l

The right part in the radical is the squared mean delay. A non time-dispersive

channel has a delay spread equal to zero. The larger , the stronger is the

frequency-selective behaviour of the channel.

Following the same approach as for the delay spread in the delay domain,

the time-variant behaviour of a channel can be integrally characterized by the

Doppler spread in the Doppler domain. The Doppler spread is a measure of

the broadening of the theoretically innitely narrow spectral line of a harmonic

excitation due to the frequency-dispersive inuence of the channel. Mathem-

atically it is equivalent to the denition of twice the standard deviation of

the Doppler spectrum. The Doppler spectrum is proportional to the square

of the absolute value of the delay-Doppler-spread function (2.63) integrated

over [Ste92, GW98]. Thus, according to the case of the delay spread, can

directly be calculated from the output of the ray tracing as

v

2 2

uP Pn !

u n 2T 2 l Tl

=2 t Pl=1 l l Pl=1 :

n 2

l=1 Tl

n T2

l=1 l

(2.65)

Again, the right part in the radical is the squared mean Doppler. A non

frequency-dispersive channel would have a Doppler spread equal to zero. The

larger , the stronger is the time-variant behaviour of the channel.

2.5 Summary

In this chapter, a ray-optical modelling approach was explained together with

the underlying assumptions. The necessary formulations of the most import-

ant wave propagation phenomena in the environment of a curved tunnel were

32 CHAPTER 2. RAY-OPTICAL WAVE PROPAGATION MODELLING

scattering into deterministic ray-optical propagation modelling was derived.

Finally, multipath propagation was treated, and the radio propagation chan-

nel functions and parameters were stated in the context of the ray-optical

modelling results.

The centralbecause computationally expensiveissue of ray-optical propaga-

tion modelling in an arbitrarily shaped tunnel is to determine the ray traject-

ories from the transmitter to the receiver. This problem is dealt with and

solved in the following chapter.

Chapter 3

environment

propagation modelling have been presented. However, one major point has not

yet been considered but implicitly assumed, namely, that the ray trajectories

from the transmitter to the receiver are known. This chapter now deals with

the question of how these trajectories are determined in the curved geometry

of a tunnel environment. In this context, the concept of ray tracing becomes

indispensable. The term ray tracing includes two dierent aspects: rstly,

the determination of a ray trajectory from the transmitter to the receiver;

secondly, once this trajectory is available, the determination of the actual eld

strength of the ray at the receiver by tracing (i.e. following) the ray along

its trajectory, taking into account all occurring propagation phenomena. The

latter was treated in the previous chapter, the former is treated in this one.

A sharp distinction between the above two aspects is not always possible.

Instead, often they are performed in parallel during the process of ray tracing.

The principle methods of ray tracing are rst examined (cf. section 3.1).

In section 3.2, the shortcomings of existing ray-tracing techniques arising in

curved geometries are identied. In the same section, a general solution to the

problem is derived in connection with ray launching. Section 3.3 shows how the

proposed technique is applied to ray tracing and the dierent types of analysis

(i.e. coherent or incoherent analysis). This is followed by an approximate

solution for the power ux through a tunnel in section 3.4.

34 CHAPTER 3. RAY TRACING IN A TUNNEL ENVIRONMENT

The retrieval of valid ray trajectories from the transmitter to the receiver in

a given geometry via ray tracing can be distinguished in direct and indirect

1

methods. Direct methods, like image theory or the rubber band method ,

directly lead to the exact ray paths. For indirect methods, also called ray

launching, a number of rays is launched (i.e. send out) from the transmitter

in arbitrary directions and traced until they eventually hit the receiver or

until they surpass a certain maximum attenuation. The approaches which are

relevant to ray tracing in tunnel environments are discussed in the following.

Obviously, it is most desirable to determine all ray trajectories exactly. This is

possible by means of image theory, but unfortunately only in planar geomet-

ries. The principle of image theory is depicted in Fig. 3.1. The transmitter

Tx Rx

QR

Tx'

an image transmitter Tx' (or image receiver Rx'). The straight line from the

image transmitter to the receiver (or from the transmitter to the image re-

ceiver) intersects the plane of interaction in the reection point, which clearly

denes the ray trajectory for this single reection. For multiple reections, the

process of imaging is repeated accordingly. At curved boundaries, however,

the concept of images becomes ambiguous, because the image of the transmit-

ter (or receiver) is a continuous image line, instead of a discrete point. Image

theory is therefore only applicable to straight tunnels with rectangular or piece-

wise/partially planar cross sections. In such planar geometries it delivers the

most accurate ray-tracing results [Rem93, Kle93, MLD94], thus it is taken as

1 The rubber band method is used to determine diraction wedges and paths in a 2D

vertical plane for rural or urban wave propagation modelling [Leb91, Cic94]

3.1. METHODS OF RAY TRACING 35

(piecewise) planar geometries implies that realistic tunnels can only be dealt

with by image theory in a highly approximate manner. Usually the real cross

section is approximated by an equiarea rectangle [YA

+ 85, ZH98a], or the whole

+

tunnel is tessellated into multiple planar facets [CJ96a, HCC98, TV 99].

It is worth noticing that if a straight tunnel with rectangular cross section

is considered, a direct relation between mode theory and the GO imaging

approach can be established [MW74a, Del82].

In contrast to image theory, where the retrieval of the exact ray trajectory

precedes the actual propagation modelling, ray launching follows a completely

dierent approach. In ray launching, also referred to as the shooting and

bouncing ray (SBR) method [LCL89], a large number of rays is sent out from

the transmitter in arbitrary/specic directions. The initial phasors of the rays

are determined by the radiation pattern of the transmitting antenna including

phase and polarization. Each ray is traced in space and wave propagation is

calculated according to geometrical optics, including reection, scattering etc.

The trace terminates if a ray eventually hits a receiver, or when it surpasses a

certain maximum attenuation. At the receiver, all incoming rays are combined

in order to determine the overall reception level. Generally, only a few of the

launched rays, of which the trajectories have been searched for, actually hit

the receiver. Therefore ray launching is referred to as an indirect ray-tracing

approach. To ensure that all relevant propagation paths are found in the

process of ray launching, a suciently large number of rays has to be sent out

at the transmitter. The major advantage of ray launching is its applicability

even in curved geometries. However, the decision, whether a ray hits a receiver

or not, is actually one of the most severe problems in ray launching. The

two known methods to solve this problemat least in planar geometryare

+

discrete ray tubes [CZL95, SM97] or reception spheres [HB 92, SR94].

Discrete ray tubes of nite width may be seen as a discretization/magnication

of the dierential tube of rays in GO (cf. Fig. 2.2). They consist of typically

three to four limiting rays (or lines) circumscribing the actual ray, which it-

self is often named reference or central ray (cf. Fig. 3.2). Discrete ray tubes

have the purpose to delimit the dierent reference rays to each other, all of

which represent locally plane wavefronts. Therefore, the limiting rays belong

to (dierent) adjacent discrete ray tubes at the same time, in order to make

this delimitation unambiguous. The decision, whether a receiver is reached by

36 CHAPTER 3. RAY TRACING IN A TUNNEL ENVIRONMENT

reference ray

limiting rays

Tx

2

a ray or not, becomes quite simple: it is sucient to determine, if the receiver

lies in the respective discrete ray tube. Figure 3.3 shows the same 2D scenario

as Fig. 3.1 but with discrete ray tubes. It illustrates the approximate nature

of ray launching: the reference rays of the direct and the reected ray tubes

are similar but almost never equal to the exact rays given by image theory.

Rx

Tx

predict wave propagation in indoor environments, which are constituted by

planar surfaces [Zwi94, Cic94, GW

+ 95, COST231]. The disadvantage of dis-

crete ray tubes is their failure at curved boundaries, which will be addressed

in section 3.2.

3.1. METHODS OF RAY TRACING 37

Another tool widely used in ray launching is the reception sphere concept

[HB

+ 92, SR94]. The receivers are assigned to a certain volume in space, typ-

ically a sphere, yielding the termination reception sphere. A launched ray

reaches a receiver, if it intersects the corresponding reception sphere. The dif-

culty with the classical approach of reception spheres is the determination of

their size. If the size is too small, only a few rays reach the receiver and the res-

ults are inaccurate, since important propagation paths may not be considered.

On the other hand, if the size is too large, several physically identical rays

reach the receiver so that the results become faulty without a correct normal-

ization. This multiple ray problem is illustrated in Fig. 3.4, where receiver

Rx 1 is reached by multiple direct rays, each representing a distinct wavefront.

In reality, however, there is only one physical wavefront reaching the receiver

on the direct propagation path and thus only one direct ray. The multiple

rays can also be classied as being too close to be considered as independent

[Des72]. Additionally, Fig. 3.4 shows that the number of registered rays also

Rx1 Rx2

Tx

Figure 3.4: The multiple ray problem using reception spheres with constant size

in ray launching

the reception spheres. An approximate solution to overcome those problems

in planar geometries are adaptive reception spheres. If the rays are launched

with an uniform angular separation at the transmitter, the spacing between

two adjacent rays is proportional

3

to and the unfolded path length r. If the

distance of the receiver to a ray is smaller than half of the ray spacing, the

ray is assumed to contribute to the received signal. For small , the tested ray

will be a good approximation of the ray passing directly through the center

of the receiver [HB 92].

+ An almost identical angular separation between a

ray and its nearest neighbours can be achieved, if e.g. the spherical surface

of the source is subdivided by a geodesic polyhedron with tessellated trian-

2 2 2 r for small .

38 CHAPTER 3. RAY TRACING IN A TUNNEL ENVIRONMENT

4

gular faces, resulting in hexagonal shaped wavefront portions [SR94]. The

wavefronts can be further approximated by circumscribed circles around the

hexagons to make sure that no reception points are missed. Thus the radius

of the reception spheres becomes

rR = p
r ;

3 (3.1)

which is of course dierent for each ray path. Due to the small overlap of the

γr rR

2

γ r

Figure 3.5: Hexagonal shaped ray tubes with circumscribed, slightly overlapping

reception spheres, still leading to multiple received rays (cf. [SR94])

reception spheres (c.f. Fig. 3.5), there may be still some multiple received rays.

To eliminate them, all received rays have to be stored and compared pairwise.

This process is called identication of multiple rays.

The task of identifying multiple rays, which seems obvious for the human being,

cannot be performed exactly on the computer, but has to be done according

to several conditions. One possibility is to test the following three criteria

consecutively:

for two rays, they cannot be the same.

2. Delay time: if the number of reections are equal, the delay time of the

two rays are compared. They are considered to have equal delay times,

if both fall into the same delay interval.

4 Actually, resulting in pentagonal and hexagonal wavefronts on the surface of the poly-

hedron.

3.2. RAY LAUNCHING IN CURVED GEOMETRY 39

angles of transmission of both rays under test are compared. Finally,

if both are launched into the same direction, specied by a solid angle

interval, the rays are considered to be equal.

Although the rst point is unambiguous, the following two require the spe-

cication of a certain test range. The delay interval is less critical and mainly

depends on the accuracy of the computer arithmetic. The solid angle interval,

p3 r

however, also depends on the angular separation of the rays at the transmitter

and has to be chosen to be signicantly larger than R , which follows from

r

(3.1).

The identication of multiple rays (IMR), which has been originally used

in conjunction with adaptive reception spheres in planar geometries, can be

utilized as an independent method even in curved geometries. It delivers reas-

onable results with the exception of the areas near and in caustics [SW98].

However, the approach is only feasible for a small number of receiver locations,

due to the fact that all incoming rays have to be stored and post-processed,

which requires large storage capacity and excessive simulation times.

3.2.1 Ray tubes: approximate solution for 2D-curvature

The advantage of discrete ray tubes is the unique assignment of rays to the

receiver. The test, whether a receiver falls into a ray tube, becomes quite

ecient, if adjacent limiting rays lie in a plane. In this case, fast geometrical

algorithms [Zwi94, Gla95] can be used, which determine on which side of the

plane the receiver is located. For point sources and planar boundaries, the

premise of limiting planes is always fullled.

The situation changes, if curved boundaries are to be considered. The

two obvious methods to proceed with the reection of a discrete ray tube at

a curved surface are either to reect the whole tube at the tangential plane

in the intersection point of the central ray and the surface, or to reect each

limiting ray and the central one at the tangential plane of its own intersection

point with the surface. If the joint tangential plane approach is chosen, the

initial intention of the discrete ray tubes, which is the unique delimitation of

the dierent central rays, is invalidated. For convex surfaces, this approach

leads to dispersedformerly adjacentray tubes, leaving the space between

them uncovered. For concave surfaces, adjacent ray tubes overlap and become

ambiguous. If the separate reection plane approach is chosen, the resulting

tubes can no longer be interpreted as a discretization of the dierential GO

tube of rays. In fact, the central ray may even leave the limiting tube. But

40 CHAPTER 3. RAY TRACING IN A TUNNEL ENVIRONMENT

the most important drawback is the arbitrary shape of the reected ray tube:

adjacent limiting rays generally do not lie in the same plane, which inevitably

5

leads to a more complicated and therefore time-consuming receiver test .

An approximate solution for two-dimensional (2D) curvature can be found

by forcing the reected limiting rays such that the rays pairwise form a limiting

plane [Laq94, CZW96]. For straight tunnels with an elliptical cross section,

rst the projection of the propagation paths into the plane of the cross section

is considered. The strategy is to reect the limiting rays at the arc of the

ellipse, each of them in its own point of reection. Then the reected central

ray is chosen to pass through the intersection point of the limiting rays. The

divergence in the other dimension, which is along the tunnel axis, is kept to

be spherical such that nally ray tubes with limiting planes are achieved.

Although this approximation seems reasonable, it produces unsatisfactory

results by introducing articial caustics [Laq94, CZW96, SW98]. An approach,

which is not restricted to 2D curvature, and which does not suer from the

above mentioned limitations, is presented in the following section.

malization (RDN)

If reception spheres are used, a normalization can be found, being also valid

in curved geometriescontrary to the hitherto usual approach (cf. section

3.1.2.2). The new concept of ray density normalization (RDN) is as follows.

Instead of trying to avoid the existence of multiple rays, it is a priori assumed

that several multiple (or dependent) rays are present on each physical propaga-

tion path. The number of these rays is determined, and this number is used to

normalize the contribution of each ray to the total eld. As a consequence, the

new method requires multiple representatives of each physical electromagnetic

wave at a time. This is opposed to classical ray tracing, where the one ray

representing a locally plane wave front is searched.

In order to determine the number of rays, which reach the receiver on the same

propagation path, the ray density is used. The ray density is dened as the

number of rays per unit area. In addition to its amplitude, phase, polarization

etc., each ray carries the ray density along its path. If a ray hits a reception

sphere, the theoretical number of multiple rays hitting the same sphere can be

calculated by simply multiplying the ray density with the area of the sphere.

5 For the general case, this test can only be performed on a numerical basis.

3.2. RAY LAUNCHING IN CURVED GEOMETRY 41

and the restriction to plane surfaces, the ray density nd at a distance r from6

N

nd =

4r2 : (3.2)

1=r2 is not longer valid in general. At curved surfaces the rays can be focused

or defocused. It is e.g. possible that after a reection at a parabolic reector

the rays are parallel, i.e. nd becomes independent of r (plane wave). The de-

termination of the ray density nd in curved geometries is performed according

to geometrical optics in analogy to the calculation of the electric eld after re-

ection from a curved surface. However, the ray density nd is not proportional

to the electric eld ~

E but to the radiation density S of a ray. Since ~ j2 ,

S / jE

the ray density nd at a distance s from a curved surface can be written as

r r i

nrd (s) = r 1 2r

(1 + s)(2 + s) nd;

(3.3)

where nid denotes the incident ray density just before the reection, nrd(s) is

the ray density after reection at a distances from the point of reection, and

r1;2 denote the radii of curvature of the wave front after reection at s = 0,

i

which is represented by the reected ray (cf. chapter 2). The ray density nd;1

before the rst reection at a curved surface is given by (3.2). Using (3.2)

and (3.3) it is possible to determine the ray density of all rays along their

propagation path.

The special case of an initially stigmatic ray ( 1 = 2 = r) reecting

exclusively on planar surfaces leads back to (3.2) according to

2 2

rm i rm r12 N N

( )=

nrd;m sm

(rm + sm)2 d;m = (rm + sm)2 (r1 + s1)2 4r12

n = 4r 2;

(3.4)

rj+1

reections and r = rm + sm .

If a ray hits the receiver, the theoretical total number M of rays reaching

the receiver on the same propagation path is now calculated using the visible

area A of the receiver normal to the propagation direction of the ray, i.e.,

M = ndA: (3.5)

42 CHAPTER 3. RAY TRACING IN A TUNNEL ENVIRONMENT

Again, these multiple rays are physically identical, i.e. they arrive from the

same direction, they have the same delay, number of reections etc. Hence,

these rays have to be weighted, such that the ensemble of M multiple rays

leads to the same result as if only (exactly) one ray would reach the receiver.

The weighting thereby depends on the type of analysis, described in section

3.3. The determination of A depends on the type of receiver, treated in section

4.5. For a reception sphere, the area equals A = rR2 , where rR denotes the

radius of the sphere.

Prerequisites of the RDN are a large number of rays and their homogeneous

distribution in space. The large number requirement results from the discret-

ization in the computer: the theoretical number of multiple rays M given by

(3.5) is a real valued number. However, the number of actual registered rays is

an integer. Consequently, in order to keep the error introduced by the weight-

ing as small as possible, the number of multiple rays has to be suciently large.

The second requirement directly results from (3.5): the relation is only valid for

a homogeneous distribution of the rays. This distribution can be achieved by

a stochastic generation of the ray directions (Monte-Carlo method, cf. section

4.2). In order to full the rst requirement with tolerable computational com-

plexity, all ray-tracing calculations, especially all intersection routines, should

be solvable analytically (cf. section 4.1).

In the remainder of this section, only the ratios of received values (index R)

to a reference value (index 0) are examined, whereby absolute values become

obsolete. Directly at the transmitter (r = 0m) the ratio cannot be determined,

since the electric eldand consequently the received powertheoretically

become innite (cf. section 2.2). Without loss of generality, suppose that a

reference distance r0 = 1m is assumed, i.e.,

PR (r0 = 1m)

P0

= VR (r0V= 1m) = 1; (3.6)

0

where the notation V = jV j is used for the magnitude of the induced voltage

at the receiver.

3.3. APPLICATION OF THE RAY DENSITY NORMALIZATION 43

density

As shown in section 2.4, the loss along a propagation path due to reection,

absorption etc. can be represented by the propagation transfer factor TP , which

also includes the inuence of the antennas directivities. The attenuation due

to the divergence of the tube of rays can be represented by the divergence

transfer factor TD . For exclusively planar boundaries, the divergence factor

is given by TD = TD;0 = 1m=r, where r denotes the unfolded path length.

However, at curved boundaries, the radii of curvature of the astigmatic tube

of rays have to be considered, leading to

v

u

r1;m r2;m r1;1 r2;1 1m2

u

TD =

(1;m + sm)(2;m + sm) (1;1 + s1)(2;1 + s1 ) r12 ;

t (3.7)

r r r r

representing the number of reections,

after thej -th reection, sj the distance from reection point j to the next

one, and r1 the distance from the transmitter to the rst point of reection.

TD can also be expressed in terms of nd by taking the ray density of the ideal

isotropic source (3.2) as start density and by cascading (3.3) accordingly. By

comparison to (3.7) the following relation can be established

r

TD =

4 nd1m2: (3.8)

N

Now it is possible to state the normalized induced voltage at the receiver for

a single physical propagation path or ray in dependence of the ray density

r

VR

V0

= TP TD = TP 4N nd1m2: (3.9)

The relative received power results from the square of (3.9), i.e.,

VR 2

PR

= = 4

TP2 nd1m2 : (3.10)

P0 V0 N

In the following sections is shown how the ray density normalization is applied

to two dierent calculation schemes: the so-called eld and power traces.

The classical approach in ray launching/tracing is to assign a certain electric

eld strength to each ray, which is normalized to a reference level, as indicated

44 CHAPTER 3. RAY TRACING IN A TUNNEL ENVIRONMENT

in chapter 2. The rays are traced until they eventually hit a receiver, or until

they surpass a certain maximum attenuation. This approach is referred to as

eld trace.

In order to correctly weight the contribution of each ray to the overall

received power or the induced voltage at the receiver, weighting factors XFc=i

are introduced. This is because the number of rays reaching the receiver on

the same physical propagation path is a priori unknown (cf. section 3.2.2).

Furthermore, one has to dierentiate if the analysis of multipath propagation

is of coherent or incoherent nature. For the coherent analysis, the weighting

takes the form

The following considerations are only valid for a single physical propagation

path, on which several discrete rays with identical amplitude V R reach the re-

ceiver. The number of multiple rays is theoretically given by M . The coherent

summation of all rays belonging to the same propagation path leads to

2

V cR;k VRc 2

= M 2TP2 4N nd1m2XFc 2;

M

PRc X

P0

=

V = M

V

(3.13)

k=1 0 0

XFc = M1 = n 1A : (3.14)

d

The incoherent summation leads to

M V i 2

VRi 2

PRi

= M V0 = MTP2 4N nd1m2XFi 2:

X

P0

= R;k

V0

(3.15)

k=1

Comparing the result to (3.10) gives

p

XFi (3.16)

d

With (3.14) and (3.16) the RDN-based weighting factors are known, by which

the predicted values of the discrete rays have to be normalized at the receiver

in order to obtain valid results.

3.3. APPLICATION OF THE RAY DENSITY NORMALIZATION 45

The idea of the power trace is to look at the power (respectively energy) of each

ray, instead of the habitual electric eld. This method is the standard approach

in ray tracing for computer graphics [Gla89]. The total radiated power of the

transmitter is spread over all rays. Each ray keeps its portion of the power

on its way through the tunnel, attenuation may occur due to propagation

phenomena like reection etc. If a ray hits a receiver, it is assumed that its

remaining power is transferred totally to the receiver. The summation over

all received rays gives the received total power. Focusing eects for instance

may occur, if a huge number of rays hit the receiver. On the other hand, the

eect of free-space attenuation is included implicitly: at increased distance,

less rays and thus less power reach the receiver due to geometrical divergence

(cf. Fig. 3.6). In contrast to the eld trace, where each ray represents a locally

plane wave and therefore can be treated separately, only the ensemble of all

rays reaching the receiver makes sense in physical terms for the power trace.

source

distance less rays reach the area A

the transmitting antenna characterized by GT and C~ T , and the number of

launched rays N , yielding

P G jC~ j2

Pt = T T T : (3.17)

N

If no propagation losses occur, the assumption that a ray delivers its total

46 CHAPTER 3. RAY TRACING IN A TUNNEL ENVIRONMENT

P

SA = t : (3.18)

A

Here A denotes the area of the receiver and SA the radiation density, which the

ray tube (corresponding to the discrete ray) would have, if its area equalled

A. The approximation is valid for

where Art denotes the actual area of the ray tube and AeR the eective area

of the receiving antenna given by (2.47). The contribution of the ray to the

total received power is obtained using AeR , leading to

Pt 2

PR = SA AeR = AeR = 0 PT GT GR TP2 :

A 4AN (3.20)

TP is necessary in the previous equation to account for the dissipation of en-

7

ergy on the propagation path due to the dierent propagation phenomena.

However, no divergence factor is introduced since the power of the ray is re-

garded, not its radiation density. Equation (3.20) points out that only the

ensemble of all received rays has a physical meaning. This can easily be veri-

ed for the free-space case with TP = 1. The area of the receiver A at a

distance r to the source is reached by

M = 4AN

r2

(3.21)

rays, assuming N rays equally distributed over the solid angle 4 (cf. Fig. 3.6).

This results in a total received power of

M

X 0 2

PR;tot = PR;k = MPR =

k=1 4r PT GT GR ; (3.22)

For the general caseincluding reection etc.the same approach as in the

previous section is adopted. First, the power of each single ray is normalized

leading to

PR

P0

= TP2 ; (3.23)

7T

P also includes the inuence of the antenna directivities (cf. (2.52b)).

3.3. APPLICATION OF THE RAY DENSITY NORMALIZATION 47

or equivalently

VR

V0

= TP : (3.24)

The phase of the induced voltage at the receiver is determined by the tra-

versed path length and phase jumps of +90Æ at potential passages through

caustics [Bal89, LCL89]. However, (3.24) only states the contribution of a

single ray. Again, the number of rays arriving at the receiver on the same

physical propagation path is a priori unknown. Therefore, according to the

eld trace case, weighting factors are introduced which are based on the RDN.

Finally, the actual value at the receiver is achieved by weighting and superpos-

ition of all incoming rays. In analogy to section 3.3.2, the coherent summation

is in the form

2

Vc 2

M

PRc X V cR;k

P0

=

V

= M R

V

= M 2TP2 XPc 2; (3.25)

k=1 0 0

yielding

r s r

XPc =

4 nd1m2 = 4 1m2 = 41m2 1

N M2 N nd A2

(3.26)

NA M

after comparison with (3.10). The power sum, on the other hand, is given by

M V i 2

VRi 2

PRi

= M V0 = MTP2 XPi 2;

X

P0

= R;k

V0

(3.27)

k=1

leading to

r

XPi = 4NA

1m2 = X c pndA:

P (3.28)

For both the eld and the power trace, the following relation holds

p

X i = X c nd A; (3.29)

Because of the introduced factors, the actual computation of the coherent

8

results is essentially the same for both the eld and the power trace . Only the

8 One dierence lies in the determination of the phase jumps when crossing caustics. The

change in phase is performed implicitly by (2.13) for the eld trace case. For the power

trace, passages through caustics have to be determined by additional tests.

48 CHAPTER 3. RAY TRACING IN A TUNNEL ENVIRONMENT

way how these methods are derived diers: the conventional straight forward

analytical formalism for the eld trace and the more graphical approach for

the power trace. The dierence between the two methods, however, becomes

signicant for the incoherent analysis. The incoherent weighting factor XPi of

the power trace is constant according to (3.28) and therefore independent of

the ray density. This means, the complicated and time-consuming calculation

of the ray density can be omitted, speeding up the simulation considerably, if

only the power sum is of interest in the analysis. A method, making solely use

of the incoherent analysis, is treated in detail in the next section.

overcome one of the major disadvantages of geometrical optics: its failure at

caustics. Section 2.2 has shown that in a caustic the predicted GO eldand

therefore the received powerapproach innity. For the power trace approach,

however, the received power is determined by the number of rays, which ac-

tually reach the receiver. This number is always nite and smaller than the

number of launched rays N. Consequently, even if all rays reached the re-

9

ceiver, the maximum received power would always be lower than the input

power of the transmitting antenna PT in the incoherent analysis. The only

variable, which reects the GO behaviour, is the ray density nd needed for

the weighting of the coherent analysis. In (3.26), the ray density nd is used

together with the visible area of the receiver A to determine the theoretical

number of multiple received rays M = ndA. In a caustic, this value would

approach innity. Heuristically, it can now be bound by the maximum value

of N , or by a fraction of it.

A fast way to integrally characterize the propagation through a tunnel is given

by the method of power ow. For this method, the total time-averaged for-

ward propagating power Pprop through the tunnel cross section is computed

at dierent locations in a tunnel [LL98]. Mathematically Pprop is given by the

surface integral

Z

Pprop = Sn dA; (3.30)

component) normal to the cross section. In ray tracing this can be achieved

3.4. THE METHOD OF POWER FLOW 49

by the power sum of all impinging rays on the tunnel's cross section

X

Pprop = PR;q ; (3.31)

q

where PR;q are the (remaining) powers of each discrete ray according to the

power-trace approach. Furthermore, if an equal distribution of the power over

the whole cross section is assumed, the radiation density is given by

P

S~ = prop ; (3.32)

Acs

where Acs denotes the total area of the cross section. Using the eective

2

area

incoherent power can be obtained by

2

P~R;tot

i ~ ei = Pprop Aei

= SA = 40 PNA

T GT X

2 :

TP;q (3.33)

Acs cs q

This result can now be compared with the received incoherent power of section

3.3.2. Summing over all incoming rays at the receiver and substituting (3.16)

into (3.15) yields

2 0

2 41m2 = 4N1m

2 X TP;q

i

PR;tot X

P0

= Ml TP;l

NAl

: (3.34)

q0 Aq

0

l

With GR = 1 for the isotropic antenna, the reference power P0 is given by

2

P0 = 0

41m GT PT ; (3.35)

such that

2 X 2

i

PR;tot = 40 PTNGT TAP;qq0 0 : (3.36)

q0

ate the actual received power: the important dierence lies in the considered

area. In (3.36), the actual visible area Aq0 of the receiver is taken for each

ray (cf. section 4.5), resulting in a certain number of discrete rays (index q0 ),

which actually hit the receiver or reception sphere. Conversely on the other

hand in (3.33), the total cross section Acs is taken, where all remaining rays

(index q), i.e. all rays that reach the cross-sectional area, are used for the

50 CHAPTER 3. RAY TRACING IN A TUNNEL ENVIRONMENT

the source, where the main assumption, namely an equal distribution of power

over the cross section, is rarely fullled. Nevertheless, the simulation becomes

very fast for this incoherent analysis, since the calculation of the ray density

is not required, and since only a signicant smaller number of rays is needed

than for the coherent analysis [LL98, CG99]. Therefore, the method of power

ow can be used as a rst and fast estimate of the mean received power and

thus of the propagation behaviour in a tunnel.

3.5 Summary

In this chapter, the question of how the ray trajectories are determined in

a tunnel has been answered. The terms ray tracing and ray launching were

introduced and explained. The dierent existing techniques, which are not

able to cope with curved boundaries, were illuminated. In order to solve

the problem of ray tracing in curved geometries, a novel ray-launching/ray-

tracing technique was developed, termed ray density normalization (RDN).

Additionally, this method leads to nite results in caustics, which is not the

case in conventional GO. Finally, a very fast approximate way to characterize

the propagation behaviour in a tunnel, the method of power ow, together

with its relation to the correct solution was established.

The application of the theoretical considerations presented and the integ-

ration into a functional simulation tool require an adequate representation of

a tunnel in the computer. Therefore, the handling of the geometry of a tun-

nel is presented in the following chapter, together with other features of the

simulation approach.

Chapter 4

In the previous chapters, techniques to treat electromagnetic wave propagation

in a ray-optical way have been developed theoretically. To simulate wave

propagation in arbitrarily shaped tunnels by applying these methods, a precise

modelling of the tunnel itself is needed. The present chapter deals with the

models for the geometry of a tunnel. Additional aspects of the implementation,

which are inherent to the special geometry of tunnels and to the requirements

of tunnel communication systems, are also presented.

The rst section 4.1 deals with geometries commonly used for tunnels

and their modelling with the computer. This is followed by the concept of

stochastic ray launching in section 4.2, which is mandatory for the RDN of

chapter 3. Furthermore, an approach to allow for transmitter placement out-

side the tunnel is explained (cf. section 4.3). The inclusion of moving obstacles,

like vehicles, is treated in section 4.4. Finally, the dierent types of analysis

are presented (cf. section 4.5), followed by general remarks on the simulation

approach in section 4.6.

4.1 Geometry

Almost all train or street tunnels are either of rectangular or arched shape

[BL74, Kin90, Mar93, Str96]. The tunnel course consists of combinations of

straight and curved sections. Normally the curves have a constant radius of

curvature for security reasons. The geometry of the transition from a straight

section to a curve with constant radius of curvature is depicted in Fig. 4.1.

In roadmaking and tunnelling, the smooth transition from the theoretically

innite radius of curvature of the straight section to a curve with radius rc is

generally performed by a section of a clothoid of length s [Str96, Str95, Mar86].

52 CHAPTER 4. THE SIMULATION APPROACH

rc

curve

straight clothoid

section

s

clothoid

To determine the ray trajectories, the reection points of a ray on the tun-

nels boundaries have to be known. They are found by intersecting the ray with

the respective boundary in three-dimensional space. As already indicated in

section 3.2.2, all ray-tracing calculations, especially the intersection routines,

have to be solvable analytically in order to allow for tracing the required large

number of rays in reasonable computation time. Therefore, the tunnel geo-

metry must be described by only a few geometrical (3D) primitives, staying

as close as possible to reality.

In the simulation approach, the cross section of a tunnel can be either

rectangular or elliptical with a raised oor representing the road or rail level.

Furthermore, an additional ceiling may be inserted in the elliptical tunnel,

which is often the case for arched tunnels of newer kind with ventilation.

For straight rectangular tunnel sections, four planes (rectangles) are needed to

describe the geometry (cf. Fig. 4.2(a)). For straight elliptical tunnel sections,

one elliptical cylinder and one or two planes (rectangles), representing the oor

and an optional ceiling, are sucient to describe the geometry of the tunnel,

as indicated in Fig. 4.2(b). The intersection routines for a ray with a plane

(rectangle) and with an elliptical cylinder are given in appendix B.

For curves with constant radius of curvature, a curved rectangular tunnel

section can be modelled by two planes, representing the oor and the ceiling,

and by the jackets of two circular cylinders, representing the curved side walls

(cf. Fig. 4.3(a)). The curved elliptical tunnel section is modelled by an elliptical

4.1. GEOMETRY 53

torus and one to two planes, representing the oor and an optional ceiling

(cf. Fig. 4.3(b)). The intersection routines for the circular cylinder and the

elliptical torus are again given in appendix B.

Figure 4.3: Curved sections of a tunnel with constant radius of curvature im-

plemented in the simulation approach

In the simulation approach, the clothoid is approximated by an extension of

the adjacent straight section and an intermediate circular arc between this

extension and the actual curved section. The center of the arc and its radius

54 CHAPTER 4. THE SIMULATION APPROACH

ra > rc are chosen so that the transitions straightarc and arccurve have

continuous slopes (cf. Fig. 4.4). This approximation keeps the original ori-

y

αa

rc

ra

curve

y(b,s)

straight clothoid

section

s sa

0 xa x(b,s) x

extension of arc

straight section

and an intermediate circular arc.

entation of the tunnel's course and allows to retain the analytical intersection

routines of appendix B for curves with constant radius of curvature.

A clothoid is a curve, for which the radius of curvature r is inversely pro-

portional to its arc length s

1= s

; (b > 0);

b2

(4.1)

r

12 , which in parameter presentation

with the proportionality factor

+ b is given

by [BS 99]

s

bp Zs

p u2 2

Z

x(b; s) = b cos 2 du = cos 2vb2 dv (4.2a)

0 0

s

bp Zs

p u2 2

Z

y(b; s) = b sin 2 du = sin 2vb2 dv: (4.2b)

0 0

4.1. GEOMETRY 55

For a given clothoid, i.e. s and b are known, the parameters of the intermediate

arc ra (radius of the arc) and sa (arc length), as well as the extension length

xa of the straight

section have to be determined. The slope of the clothoid in

x(b; s); y(b; s) can be written as

dy sin s2 2

tan a = tan cloth. = ds

dx = cos 2b22

s = tan 2sb2 : (4.3)

ds 2b2

Referring to the geometry of Fig. 4.4, the equation of the circle corresponding

to the intermediate circular arc is

xa )2 + (y ra )2 = ra2 :

(x (4.4)

p

y = ra ra2 (x xa )2 ; (4.5)

where the negative sign is chosen to match the area of possible transition points

with the actual curve. The gradient of the arc in (x; y), given by

dy

tan a = dx = pr2 x (xxa x )2 ; (4.6)

a a

is chosen to equal (4.3), leading to

2

tan 2sb2 =! p 2 x xa 2 :

ra (x xa )

(4.7)

With the set of two equations (4.5) and (4.7), the parameters ra and xa can

be obtained, which determine both the intermediate arc (dened by ra , x a )

and the extension of the straight section (dened by xa ), yielding

q

y 1 + 1 + tan2 s2

2b2

xa = x

tan 2sb22 (4.8a)

x xa

ra = y +

tan s2 : 2b2

(4.8b)

The left integrals in (4.2) are the so-called real Fresnel integralsC ( ) and S ( ),

such that these relations can be rewritten as

p

x(b; s) = b C

ps

(4.9a)

b

p

y(b; s) = b S

s

p :

(4.9b)

b

56 CHAPTER 4. THE SIMULATION APPROACH

C ( )

1 + A( ) sin 2 B( ) cos 2

2 2 2 (4.10a)

S ( )

1

2

2

2 A( ) cos 2 B( ) sin 2 ; (4.10b)

A( ) =

1 + 0:926

2 + 1:792 + 3:104 2 (4.10c)

B ( ) =

1

2 + 4:142 + 3:492 2 + 6:67 3 ; (4.10d)

with an absolute error smaller than 2 10 3 for all 0. Finally, the length

of the intermediate arc is given by

s2

sa = ra a = ra 2 :

2b (4.11)

method)

One important prerequisite for the ray density normalization studied in section

3.2.2 is a homogeneous distribution of the launched rays over the solid angle at

the transmitter. De facto it is mathematically impossible to deterministically

distribute equally spaced rays over the whole steradian. Therefore, a Monte-

Carlo based generation of the launch directions is chosen in the simulation

1

approach .

The idea of the stochastic ray launching is to generate random directions

in space, which all occur with equal probability [Sob91]. Thus, the constant

probability density function that a ray is pointing to a specic direction is

p

=

1

4 ; (4.12)

taking into account that the total solid angle equals 4. The dierential solid

angle is given by

d

= sin()dd; (4.13)

spherical coordinates and , the following probability density function (PDF)

1 Additionally, this stochastic sampling technique provides a means of anti-aliasing [Gla89,

chaps. 5,6], resulting in smooth transitions between closely spaced adjacent receiver loca-

tions.

4.3. COUPLING INTO THE TUNNEL 57

is obtained

p;(; ) =

sin() :

4 (4.14)

ates are therefore

Z2

sin()

p () = p;(; )d =

0

2 (4.15a)

Z

p () = p;(; )d =

1;

0

2 (4.15b)

which are mutually independent, because p ()p () = p;(; ). Realizations

of and , which follow (4.15), can be generated via two independent uniformly

distributed random variables 1 in [0; 1] and 2 in [0; 1) by

= 22 : (4.16b)

The previous two equations are derived by the percentile transformation method

[Pap84], i.e. by inversion of the integrals of (4.15) [PT

+ 92].

Monte-Carlo approaches have already been used for ray tracing in other

contexts. In [Sch95, LL98] the directional pattern of the transmitting an-

tenna is used to distribute the ray directions in order to reduce complex-

ity. Furthermore a so-called Monte-Carlo ray tracing has been proposed in

[Mül94a, Mül94b]. In that approach, the positions of the transmitter and im-

age transmitters (for image theory) are varied randomly to simulate clutter,

which is thus somehow comparable to the stochastic scattering approach of

section 2.3.3.3.

The transmitting antennas are installed either inside, ore.g. for maintenance

reasonsoutside a tunnel in the vicinity of the entrance. For an outside loc-

ation directly in front of the tunnel with the antenna pointing towards the

entrance, the total interior of the tunnel up to the rst curve is directly il-

luminated by the transmitting antenna (frontal antenna in Fig. 4.5). For a

lateral location, the transmitting antenna is only partially illuminating the

interior of the tunnel. For this situation, diracted rays at the boundary of

58 CHAPTER 4. THE SIMULATION APPROACH

the tunnel entrance constitute a considerable part of the total power inside the

tunnel [Mar92, MLD94] (lateral antenna in Fig. 4.5). Clearly, the rst case

with frontal location is preferable in order to maximize the coverage.

lateral antenna placement

In the simulation approach, all rays which directly enter the tunnel are

taken into account for exterior transmitter locations, together with the rays

which enter the tunnel via an intermediate ground reection. The transmitter

can be located at any distance over the whole width/cross section of the tunnel,

corresponding to the grey-shaded area in Fig. 4.5. In this case, the additional

consideration of diraction would lead to only marginal improvements of the

simulation results [Mar92, MLD94] and is therefore not considered any further.

A convenient way to allow for this type of transmitter conguration is by

2

restricting the send-range of launched rays to the portion of the steradian,

which corresponds to the area of direct ray entry into the tunnel, and the

area of ray entry via ground reection. The formulas to determine the send-

range for tunnels of rectangular or elliptic (arched) cross sections are given in

appendix C.

The goal of communication systems or broadcasting in tunnels is to provide

the respective services to people (or machinery). Note that in addition to the

empty tunnel scenario, the inclusion of obstacles, especially moving obstacles,

is a major point to provide realistic simulation results. As already mentioned

2 I.e. the directions, in which the dierent rays are launched in space.

4.5. ANALYSIS 59

or transmitters, leads to a Doppler spread, which may aect the overall system

performance.

In the simulation approach, moving obstacles are modelled as oating rect-

angular boxes of any size, with PEC sidewalls, PEC roofs, and ideally absorb-

ing underbodys. In this way, these oating boxes represent simple models of

cars, trucks or train waggons, which are (still) mainly built of metal. Although

a more rened model of e.g. a car could be envisaged, this simple oating

box approach already allows to simulate all important eects, like strong re-

ectivity, generation of Doppler shifts, shadowing eects, and the ability of

electromagnetic waves to propagate between the road and the underbody of

a car [SDW00] via an intermediate reection on the ground. Furthermore, a

more meticulous description of a vehicle may not be constructive, since each

calculation with moving vehicles only represents an instantaneous snap-shot

of the time-variant situation, which almost never exactly maps the real-world

scenario. In order to quantify the time-variant behaviour of the channel, the

automatic generation of time series, as a sequence of successive snap-shots, is

also possible (cf. section 7.3).

4.5 Analysis

The calculation of electromagnetic wave propagation in the simulation ap-

proach is performed according to the principles presented in the chapters 2

and 3. The results obtained by the modelling are the received power level,

the delay spread, the Doppler spread, and optionally the delay-Doppler-spread

function with additional angle-of-arrival information for each point of analysis.

These results include the inuence of the antennas, for which a polarimetric

description by their complex vector directional patterns C~ is used [GW98,

chap. 2], leading to dierent results for dierent states of polarization (e.g. vv,

vh, hv, and hh).

Single points of analysis or sequences of points, as they are used for coaxial

analyses in tunnels, are represented by reception spheres with radius rR (cf.

section 3.1.2.2). The visible area

3

A needed for the ray density normalization

of section 3.2.2 is given by

A = rR2 : (4.17)

In Figs. 5.1 and 5.5, the principal usage of reception spheres for the generation

of line-like analyses is sketched.

3 The visible area is the area of a receiver normal to the direction of propagation of a ray.

60 CHAPTER 4. THE SIMULATION APPROACH

of a tunnel, the usage of reception spheres is disadvantageous: the area would

have to be covered by a large number of reception spheres, resulting in a large

number of objects, which would prolong the simulation time considerably. In

this case, the usage of a reception plane is favourable. A reception plane is

a portion of a plane, generally a rectangle, which is subdivided into a mesh

of small rectangles, each of them representing a point of analysis on the

plane. If a ray hits the reception plane, it contributes to the sub-rectangle it is

crossing, with the advantage that for the whole reception plane only a single

intersection test needs to be performed, no matter how high the resolution

of the sub-rectangles has been chosen. The visible area of an element (sub-

rectangle) of the reception plane is given by

A = Ap jk^ n^ p j; (4.18)

the normal vector of the reception plane, and k^ is the propagation direction of

the ray. As follows from (4.18), the visible area of an element approaches zero

for grazing incidence. Equivalently, the number of rays reaching the element

with grazing incidence approaches zero, too. This reveals the disadvantage

of the usage of reception planes: suppose the majority of rays is incident on

the reception plane under grazing angles. Then a very large number of rays is

needed in the process of ray launching, in order to satisfy the large-number-of-

received-rays requirement of the RDN (cf. section 3.2.2). Fortunately, most

relevant rays impinge under almost normal incidence onto the cross section of

a tunnel so that a cross-sectional analysis can be obtained relatively fast by

the aid of reception planes. However, if a horizontal area is under test, the

simulation time may be considerably slower due to the required large number

of rays (cf. section 6.4). In Fig. 6.9(a), the usage of a horizontal reception

plane is visualized.

In addition to the above mentioned analytical intersection routines and the

possibility to employ reception planes, several other ray-tracing acceleration

techniques are employed in the simulation approach, in order to speed up the

simulation time and thus allow for a large number of rays. The concepts of

spatial subdivision and bounding volumes (both [Gla89, chap. 6]) are utilized

in conjunction with the multiple usage of intersection points [SW98].

4.7. SUMMARY 61

4.7 Summary

In this chapter, the particulars of the simulation approach were introduced.

The possible tunnel geometries and their treatment in the simulation tool were

presented. The stochastic ray launching, which is mandatory for the RDN of

chapter 3, the handling of transmitting antennas at exterior positions in front

of a tunnel entrance, as well as the two principal modes of analysis, i.e. along

trajectories and over entire cross sections, were described.

After the treatment of ray-optical propagation modelling in chapter 2, the

particulars of ray tracing in a tunnel environment in chapter 3, and the sim-

ulation approach addressed here, the next two chapters are devoted to the

validation of the proposed modelling techniques. To do this, a theoretical

verication is performed in the next chapter using simple canonical geomet-

ries, for which analytical reference solutions exist.

Chapter 5

Theoretical verication

In order to validate the dierent ray-optical wave propagation modelling tech-

niques of chapters 2 and 3, the proposed approaches are compared to theoret-

ical reference solutions in canonical geometries. This theoretical verication

will be followed by comparisons with measurements in chapter 6.

The rst canonical problem chosen in section 5.1 is a straight PEC rect-

angular waveguide. In this geometry, a eld-theoretical reference solution is

given by the method of Green's functions. The Green's function of a PEC

rectangular waveguide serves to validate the implementation of the exclusively

existing ray-optical reference solution, namely, image theory. Image theory is

further used to evaluate the performance of the dierent ray-launching tech-

niques, especially the RDN-based eld and power traces. Unfortunately, in

curved geometries no ray-optical reference solution such as image theory ex-

ists (cf. section 3.1.1). Therefore, another strategy for the validation is chosen

in section 5.2. First, an ideal metallic corrugated circular waveguide is utilized

as the second canonical geometry for the purpose of qualitative and quantit-

ative verication of the ray density normalization. The reference solution for

this type of curved geometry is based on fast mode decomposition. Further-

more, the dierent techniques are compared to each other to reveal possible

dierences between them. Finally, the stochastic scattering approach of sec-

tion 2.3.3.3 is compared to the Kirchho rough-surface scattering models in

section 5.3.

5.1. THE RECTANGULAR WAVEGUIDE 63

gular waveguide

5.1.1 Geometry of the rectangular waveguide

The canonical geometry chosen to validate the dierent modelling techniques in

planar geometry is the rectangular waveguide. The geometry of the rectangular

waveguide is depicted in Fig. 5.1. The dimensions of the cross section are

4m

4m

Tx

Rx

1.1m 1.7m

2.1m

1.9m

and receiver positions

from the left wall and 2:1m above ground. The receivers have a variable

distance to the transmitter, 1:9m from the left wall and 1:7m above ground.

The carrier frequency is f = 1GHz for all calculations with this geometry. The

scenario is chosen to avoid any symmetry eects [CJ96a].

tion approach in PEC rectangular waveguide

The ray-optical reference solution available for planar geometries is given by

the image theory [MW74a, MLD94, Kle93, Rem93]. All other techniques have

to match the results provided by the image theory. In order to verify this

reference method, it is rst of all compared to a eld-theoretical solution in

a PEC rectangular waveguide obtained by means of the Green's function.

In engineering terminology, the Greens's function is equivalent to the spatial

impulse response of a system [Bal89]. Thus, knowledge of the Green's function

64 CHAPTER 5. THEORETICAL VERIFICATION

for a given scenario allows to determine the solution for a variety of driving

sources by means of convolving excitation and Green's function. Consequently,

if the source excitation is a spatial impulse (Dirac function), the response of the

1

system is given by the Green's function itself. The Green's function for a PEC

rectangular waveguide must satisfy a partial dierential equation, the scalar

Helmholtz equation in cartesian coordinates, and the boundary conditions

imposed by the waveguide. The Green's function can be obtained by expansion

into a series of eigenfunctions of the corresponding homogeneous dierential

equation. The derivation is discussed thoroughly in [Bal89, chap. 14], [Tai93,

chap. 5] and is omitted for the sake of simplicity.

For image theory, the scenario shown in Fig. 5.1 is assumed with perfectly

electric conducting boundaries. The transmitting antenna is modelled as a

vertically polarized Hertzian dipole, which is equivalent to a Dirac-impulse

source in vertical direction. If the receiver is also modelled as a vertically

polarized Hertzian dipole, the received voltage will be proportional to the

vertical component of the eld. On the other hand, the vertical eld component

excited by this type of transmitting antenna is given by the Green's function

relating a vertical source to a vertical response

2

(multiplied by j!0 0 and the

weight of the Dirac impulse [Bal89, Tai93]). This allows a direct comparison

of image theory and the Green's function approach.

According to (2.23), the reection coecients for PEC boundaries are

Rk = 1 and R? = 1. In other words, the reference geometry of a PEC

rectangular waveguide is a theoretically lossless environment. As a result,

even rays experiencing a large number of reections still contribute signic-

antly to the total eld at the receiver in the ray-optical imaging approach.

The only limiting factor in this case is the traversed path length and the as-

sociated spreading, leading to a decrease of the eld with increasing distance

(2.14). Consequently, a large number of reections has to be considered in the

simulation in order to achieve sucient convergence. The number Nm of rays

reaching the receiver via m reections in a rectangular waveguide is given by

(

Nm =

1 for m = 0;

4m otherwise.

(5.1)

mX

max

N = Nm = 1 + 2mmax(mmax + 1): (5.2)

m=0

1 In the general case, the Green's function is a dyadic [Tai93].

2 The total eld excited by a vertically polarized Hertzian Dipole would be given by a

vectorial Green's function.

5.1. THE RECTANGULAR WAVEGUIDE 65

rays, but also by the number of intersection/reection routines, which have

to be executed. For a ray with m reections, (m + 1) such tests3 have to be

performed, resulting in a computational complexity of order O(m ) [SDK96].

Figure 5.2 compares results of the Green's function approach and the ray-

optical image theory in the PEC waveguide of Fig. 5.1 for distances between

10m and 20m from the transmitter. Both curves are normalized to their re-

5

received power normalized to max (dB)

-5

-10

-15

-20

-25

Green's function

-30

image theory (up to 275 reflections)

-35

10 12 14 16 18 20

distance to transmitter (m)

Figure 5.2: Comparison of Green's function approach and ray-optical image the-

ory in PEC rectangular waveguide of Fig. 5.1, excited by a vertically polarized

Hertzian dipole, f = 1GHz (here, the speed of light is set to c0 = 3 108 m

s, cf.

[CJ95, CJ96a, CJ96b, CJ97])

resulting in a total number of N = 151801 rays per receiver (cf. (5.2)), still

no perfect convergence and therefore no perfect agreement could be achieved

between image theory and Green's function approach due to the lossless geo-

metry. Furthermore, the innite series solution of the Green's function is also

very sensitive to slight changes in the parameters, e.g. the speed of light c0

[SW98]. Bearing in mind these diculties, the comparison of the two entirely

dierent methods validates the ray-optical image approach quite satisfactor-

ily. Therefore, image theory is used in the following to verify the RDN-based

66 CHAPTER 5. THEORETICAL VERIFICATION

methods of chapter 3.

guide with dielectric boundaries

Fortunately, real tunnels are rarely made of PEC boundaries but of rock, rein-

forced concrete, or other building materials. For this type of (lossy) dielectric

materials, the reectivity is always below unity, resulting in a much smal-

ler number of reections needed for satisfactory convergence. The minimum

number of reections, however, is increasing with increasing distance between

transmitter and receiver: this is because for grazing incidence, which occurs

for large distances, the reectivities approach unity (2.24). To quantify this

-15

received power normalized to P0 (dB)

image theory (up to 30 reflections)

-25 image theory (up to 40 reflections)

image theory (up to 50 reflections)

-30

-35

-40

-45

-50

0 200 400 600 800 1000

distance to transmitter (m)

5.1 made of dielectric boundaries with "r = 5, maximum number of reections

mmax = 10 : : : 50, vertically polarized isotropic transmitter and receivers, f =

1GHz

behaviour, the same rectangular waveguide as in the previous section (cf. Fig.

5.1) is chosen, but this time made of smooth dielectric boundaries with "r = 5,

corresponding to dry concrete. Figure 5.3 depicts the results of image theory

for dierent maximum numbers of reections mmax = 10 : : : 50 at distances

up to 1km from the source. Both the transmitting and the receiving antennas

are modelled with isotropic vertically-polarized patterns. In the vicinity of the

5.1. THE RECTANGULAR WAVEGUIDE 67

distances of about 1km, however, at least mmax = 40 reections should be

considered in order to achieve reliable results.

For the comparison of the dierent ray-optical techniques, again the same

scenario as in Fig. 5.1 is chosen at distances from 10m to 20m to the trans-

mitter. Except for image theory and the method of power ow, which require

much less rays to be traced, 20 million rays are launched equally distributed in

space by the stochastic ray launching and up to 10 reections are included, en-

suring sucient convergence of the results. Figure 5.4(a) depicts the coherent

comparison of the reference solution, the pairwise identication of multiple

rays (IMR), the eld and the power trace with RDN, and the uncorrected

raw ray launching. Obviously, the non-normalized curve does not t the

reference solution. Apart from the very high predicted level, it generally de-

creases too fast with distance, indicating a non-constant oset. At increased

distances, less rays reach the receiver resulting in a decreasing total received

power level. On the other hand, the two RDN-based approaches and the IMR

match the image theory results very well. The required computation times for

the dierent methods on a Hewlett Packard C-series workstation with 240MHz

clock-rate are compared in Table 5.1.

waveguide of Fig. 5.4 on Hewlett Packard C-series workstation with 240MHz

clock-rate

N 201 221 20 106 20 106 1 105

time 44s 1:30 h 2:30h 3:40min

Figure 5.4(b) shows the same comparison but for incoherent ray combina-

tion. In this gure, it is clearly seen that the raw ray launching experiences

not only an oset by a constant value. Additionally to the RDN and IMR ap-

proaches, the method of power ow is also included in the comparison. Even

the power ow method, assuming a constant distribution over the whole cross

section, leads to the same results as the image theory with far less compu-

tational cost than the RDN or IMR approaches. This is mainly due to a

signicantly reduced number of required rays for converging results: gener-

ally convergence is already achieved for less than 105 rays [LL98, CG99] (cf.

chapter 7).

The previous two gures reveal the applicability and power of the proposed

68 CHAPTER 5. THEORETICAL VERIFICATION

40

20

non-normalized data

image theory

field trace (RDN)

0 power trace (RDN)

identification of multiple rays (IMR)

-20

-40

10 12 14 16 18 20

distance to transmitter (m)

15

received power normalized to P0 (dB)

10

0 non-normalized data

image theory

-5 field trace (RDN)

power trace (RDN)

identification of multiple rays (IMR)

-10

power flow

-15

-20

-25

10 12 14 16 18 20

distance to transmitter (m)

waveguide of Fig. 5.1 made of dielectric boundaries with "r = 5, vertically

polarized isotropic transmitter and receivers, mmax = 10, radius of reception

spheres rR = 10cm, 201 receiver points, f = 1GHz

5.2. THE CIRCULAR WAVEGUIDE 69

a circular waveguide is considered in the next section.

lar waveguide

In the previous section it was shown that the proposed RDN-based methods

work well in a rectangular shaped straight tunnel. Unfortunately, in tunnels

of curved cross section no ray-optical reference solution like image theory is

available. This is because an unambiguous image of a point only exists at

planar surfaces (cf. section 3.1.1). Thus, another strategy for validation has

been chosen. First, the RDN-based approaches are theoretically validated in

an ideal metallic corrugated circular waveguide. The reference solution for this

type of curved geometry is based on fast mode decomposition. This is followed

by a comparison of the dierent methods in a tunnel of circular cross section

with dielectric boundaries to reveal any possible dierences between them.

Although approximate modal solutions are available to determine wave propaga-

tion in curved rectangular waveguides/tunnels with large radii of curvature

[MW74b], no exact solution exists for curved waveguidesat least to the best

of the author's knowledge. Therefore, the canonical geometry chosen to the-

oretically validate the dierent modelling techniques is the straight circular

waveguide. The validation in a geometry of real three-dimensional curvature,

i.e. curvature in cross section and course, is performed by measurements in

chapter 6. The geometry of the circular waveguide is depicted in Fig. 5.1. The

radius of the circular cross section is rcs = 2m. The position of the trans-

mitter is either centric or 1m above the tunnel axis. For simulation work, the

receivers are either situated parallel to the tunnel axis, or in the plane of the

cross section. The carrier frequency is f = 1GHz for all calculations with this

geometry.

For the theoretical verication, a ctitious ideal metallic corrugated circular

waveguide is assumed. An ideal corrugated waveguide is a special waveguide

with Fresnel reection coecients Rk = R? = 1 (2.24). This behaviour is

achieved by a special geometry of grooves on the inner surface of the waveguide,

also named corrugation [Kil90]. In practice, corrugated waveguides are e.g.

70 CHAPTER 5. THEORETICAL VERIFICATION

Figure 5.5: Geometry of the circular waveguide with dierent transmitter and

receiver positions

systems. The reference solution utilized in this geometry is based on fast

mode decomposition [MT99]. The eld is decomposed in modes, where only

the propagating modes are considered.

Figure 5.6 shows the comparison of the analytical method and the RDN-

based power trace in the waveguide of Fig. 5.5 at f = 1GHz. An ideal iso-

tropic transmitter is placed in the center of the waveguide, the receivers with

radii rR = 4cm are situated along the line center-to-outer-wall at 10m from

the transmitter (cf. Fig. 5.6). For the RDN-based power trace, two dierent

curves are plotted in Fig. 5.6: one with the weighting factors given by (3.26);

the other is bound to a maximum number of multiple rays Mmax = N=1000

in order to predict correct values in caustics (cf. section 3.3.3). Taking into

account the ideal lossless surface of the corrugated waveguide, even rays with

a high number of reections are still contributing to the overall reception level.

Therefore, sucient convergence of the ray tracing is only possible with con-

siderable computational eort. For the simulation in Fig. 5.6 200 million rays

have been traced with up to 200 reections. The simulation time on the HP

workstation of Table 5.1 was about one day. Despite the critical circumstances,

which represent a numerical worst-case scenario in ray launching, a very good

agreement of the results is observed. This validates the developed new ray

density normalization, as well as the implementation of ray tracing.

5.2. THE CIRCULAR WAVEGUIDE 71

received power normalized to P0 (dB)

RDN (power trace); limited numbers of multiple rays

10

-10

-20

0.0 0.5 1.0 1.5

distance to center (m)

corrugated circular waveguide from center to outer wall, distance to transmitter

10m, radius of the waveguide 2m, rR = 4cm,

radius of the reception spheres 80

receiver points, centric position of isotropic transmitter, f = 1GHz

waveguide with dielectric boundary

After the theoretical validation of the RDN-based methods in the previous

section, the dierent approaches applicable in curved geometry are compared

amongst each other. The same geometry as in Fig. 5.5 is used and, as with the

rectangular waveguide of section 5.1.3, the analysis is performed at distances

from 10m to 20m from the transmitter. Figure 5.7 depicts the coherent and

the incoherent analysis of the raw ray launching, the RDN-based methods,

the IMR, and the method of power ow (incoherent analysis only). The same

simulation parameters as in the rectangular waveguide have been chosen. The

computation times are comparable to the ones given in Table 5.1.

methods and the IMR are equivalent. The non-normalized curves, shown in

the upper right corner of Fig. 5.7 on a dierent scale, are obviously wrong.

The only dierence between the dierent approaches is the behaviour of the

power ow. Although the power ow approaches the RDN and IMR results

72 CHAPTER 5. THEORETICAL VERIFICATION

0

60

received power normalized to P0 (dB)

-5

field trace, coh./incoh.

power trace, coh./incoh. 20

IMR, coh./incoh.

-10 0

power flow, incoh.

-20

-15 10 12 14 16 18 20

-20

-25

-30

10 12 14 16 18 20

distance to transmitter (m)

Figure 5.7: Coherent and incoherent comparison of the dierent approaches ap-

plicable in the circular waveguide of Fig. 5.5 made of dielectric boundaries with

"r = 5, vertically polarized isotropic transmitter and receivers, mmax = 10,

rR = 10cm, 201 receiver points, f = 1GHz

rather closely, no perfect match is achieved for distances > 14m to the trans-

mitter. The reason for this lies in the equi-energy-distribution assumption

for the power ow approach, which is approximately the case in the rectan-

gular waveguide but not in the circular one. The accumulation of energy in

some parts of the cross section becomes even more important in curved tunnels

(known as whispering gallery eect [Wai67]) and is treated in the following two

chapters in more detail.

The IMR method is not used in the remainder of this thesis because of the

3

prohibitive memory requirements for the intermediate ray storage and the

prolonged simulation times due to the post-processing and the storage itself.

Furthermore, the IMR fails in and near caustics [SW98], as already mentioned

in section 3.1.2.3.

3 More than 500 megabyte were necessary only for the simulation presented in Fig. 5.7.

5.3. STOCHASTIC SCATTERING APPROACH 73

In order to validate the stochastic scattering approach described in section

2.3.3.3, it is compared to the two rough-surface scattering Kirchho models

[UMF83, GW98]. The rst is the Kirchho model with scalar approximation

for slightly rough surfaces, which states the mean value of the coherent scat-

tering component. The second Kirchho model is based on a stationary phase

approximation for very rough surfaces, which delivers the mean square value

of the incoherent or diuse scattering component. Unfortunately, the two

Kirchho approximations are only applicable in dierent (opposite) validity

regions and cannot be combined to yield the overall mean scattering intensity

for arbitrary surface roughness, which is dened by the sum of coherent and

incoherent components.

deliver instantaneous realizations of a scattering process, which contain at the

same time the coherent and the incoherent component. For a comparison

with the Kirchho models, mean values of the dierent scattering components

have to be generated, which can be obtained by averaging a suciently large

number of realizations.

For the comparison, the scenario of Fig. 5.8 is chosen: a rough, rectangular

plate of dimension Arp = Lx Ly is illuminated by a point source situated at

i , i and ri . The scattering of the rough plate is calculated by the stochastic

scattering approach and the Kirchho models. For the stochastic scattering,

the plate is hit by N rays, which are equally spaced in the x- and y-directions on

the plate. Each ray is reected according to the stochastic scattering approach

of section 2.3.3.3, i.e. the local tangential planes at each reection point are

varied following (2.41) and a Gaussian height variation. At a distance rs , the

resulting electric eld (coherent analysis) is calculated together with the power

sum (incoherent analysis) with a resolution of and . For the averaging,

the procedure is repeated M times. The results have to be transformed to

yield the polarimetric radar cross section (RCS) matrices per unit area, to

allow a direct comparison with the analytical Kirchho models, which directly

yield the RCS matrices.

74 CHAPTER 5. THEORETICAL VERIFICATION

Tx

i

Eφ

Rx i

Eθ

z ri

s

Eφ s rs

Eθ

θs θi

y

φs

φi

Ly x

Lx

Figure 5.8: Scenario and alignment convention for the comparison of the

stochastic scattering approach and the Kirchho models

Generally, a scattering process is described by the complex polarimetric scat-

tering matrix [S ]

s e jk0 rs

E~ (rs ) =

rs

[S] E~ i ; (5.3)

i s

where E~ and E~ denote the incident plane wave and the scattered spherical

wave in the far eld, respectively, both given in their respective local and

components following the alignment convention depicted in Fig. 5.8. According

to the denition in (5.3), [S ] is independent of the distance rs . The radar

cross section is now used to characterize the magnitude of the scattering. It

is equivalent to the area absorbing the amount of power of the incident plane

wave, which is necessary for an isotropic scatterer to cause the same radiation

intensity at the point of interception as the actual (directive) scattering process

[GW98]. The components of the polarimetric RCS matrix pq are related to

the components of the scattering matrix S pq by

where the polarization indices p, q stand for any combination of the and

components. The dimension of pq is square meter according to (5.3) and

5.3. STOCHASTIC SCATTERING APPROACH 75

(5.4). For area-extensive scatterers of area A, often the RCS matrix per unit

area [0] with

[0 ] = A1 [] (5.5)

+

is used [RB 70, UMF83].

stochastic scattering approach

s

For a given E~ , the scattering coecients can be obtained by the aid of (5.3)

(5.5). Thus, the RCS matrix is determined simply by substituting E ~ s in (5.3)

with the resulting eld of the ray-tracing based stochastic scattering in the

scenario of Fig. 5.8.

As

Art Tx

z

rs

∆θ θs θi

ri

Arp x

θs

Figure 5.9: Divergence of scattered and incident ray tubes linked by the scat-

tering plane Arp , together with the reception areas As given by (5.7)

For the ray tracing, one has to consider that the scattered ray density nsd

is related to the incident ray density nid by (cf. Fig. 5.9)

nsd(rs ) =

cos i ri2 ni ;

cos s (ri + rs )2 d (5.6)

where i and s denote the incident and the scattering angle, respectively.

Note that for the specular direction given by s = i , (5.6) reduces to (3.3)

76 CHAPTER 5. THEORETICAL VERIFICATION

r1

by the angular spacing and at a large distance rs from the center of

the scattering plane are given by

N

nid = ;

Arp cos i

(5.8)

Ns = nsd(rs )As =

Arp (ri + rs )2 : (5.9)

For a reection with one single locally plane wave (and therefore ray) impinging

on Arp , the correct number, taking into account the divergence of the reected

tube of rays, is

N0 =

(ri + rs ) Arp

2 (ri + rs )2 (5.10)

Since the number of rays is proportional to the power density, the electric eld

at the output of the ray tracing has to be weighted by the square root of the

ratio N0 =Ns given by

r r s

N0 Arp cos s Arp 1:

Ns

= NAs

= N tan s rs

(5.11)

Finally, the results are averaged over M dierent realizations. Using (5.11)

together with (5.3)(5.5), it turns out that the resulting RCS matrix per unit

area is independent of both the distance rs and the area Arp of the scattering

plane.

The Kirchho models with scalar approximation and stationary phase approx-

imation directly yield the RCS matrices. The formulation of the models is e.g.

+

given in [RB 70, chap. 9] or [UMF83, chap. 12].

The scalar approximation for the coherent scattering component implies

that all points of reection lie in the same plane. Under this assumption, all

reected elementary waves of an incident uniform plane wave are in phase and

can therefore be combined coherently in the far eld. The pattern produced

5.3. STOCHASTIC SCATTERING APPROACH 77

surface in the absence of roughness. The eect of a slight roughness is to reduce

the magnitude of this pattern. In addition to the tangential plane assumption

(2.44), the validity of this model is limited by

2h < p1 :

L 2 2 (5.12)

The condition says that the RMS slope of the surface must be small.

The stationary phase approximation of the diuse scattering component is

based on the contention that the averaged scattering from a very rough surface

comes from the few areas, which specularly reect. The additional condition

for the validity of this model is given by

2 h (cos i + cos s) > :

(5.13)

0

The condition requires that the root mean square (RMS) of the phase dierence

between two rays reected at any two points on the surface has to be suciently

large. Thus permitting to gather the scattered power from each specular point

incoherently.

First, a relatively rough surface with a standard deviation of the surface heights

h = 0:70, correlation length L = 40 , "r = 10 j 3 and r = 1 of size Lx =

Ly = 80 is considered. For the ray-optical stochastic scattering approach,

the isotropic source is situated at i = 60 and i = 0 , at a distance of

Æ Æ

ri = 20000 from the plane. The rays impinge separated by 0 =10 in the

x- and y-direction on the surface, resulting in N = 81 81 = 6561 rays

per realization. The resulting eld is determined in the upper hemisphere

at a distance = 20000 from the center of the plane, corresponding to

rs

approximately 8 times the far-eld distance, with a quantization of = =

1Æ. A total of M = 105 realizations is taken for the ensemble averaging. Figure

5.10 depicts the components of the RCS matrix per unit area of the diuse

(or incoherent) scattering component for the Kirchho model with stationary

phase approximation (Fig. 5.10(a)) and for the stochastic scattering approach

(Fig. 5.10(b)). The broken line in Fig. 5.10(a) indicates the limit of the validity

region of the Kirchho model according to (5.13). The absence of colour in the

lower corners in Fig. 5.10(b) indicates that no ray reaches these regions in the

stochastic scattering approach. The absolute value in the specular direction of

e.g.

0 is in both cases 6:3dB.

The coherent scattering component is negligible

for this type of surface (more than 50dB below the incoherent component).

78 CHAPTER 5. THEORETICAL VERIFICATION

0° 0°

σ0θθ σ0θφ

10

radar cross section per unit area (dB)

θs θs

90° φs 90° φs

0° 180° 360° 0° 180° 360°

0° 0°

σ0φθ σ0φφ

θs θs

-40

90° φs 90° φs

0° 180° 360° 0° 180° 360°

0° 0°

σ0θθ σ0θφ

10

radar cross section per unit area (dB)

θs θs

90° φs 90° φs

0° 180° 360° 0° 180° 360°

0° 0°

σ0φθ σ0φφ

θs θs

-40

90° φs 90° φs

0° 180° 360° 0° 180° 360°

Figure 5.10: RCS per unit area of the incoherent scattering component of the

scenario according to Fig. 5.8 (h = 0:70 , L = 40 , Lx = Ly = 80 , i = 60Æ ,

i = 0Æ )

5.3. STOCHASTIC SCATTERING APPROACH 79

0° 0°

σ0θθ σ0θφ

2

radar cross section per unit area (dB)

θs θs

90° φs 90° φs

0° 180° 360° 0° 180° 360°

0° 0°

σ0φθ σ0φφ

θs θs

-98

90° φs 90° φs

0° 180° 360° 0° 180° 360°

0° 0°

σ0θθ σ0θφ

2

radar cross section per unit area (dB)

θs θs

90° φs 90° φs

0° 180° 360° 0° 180° 360°

0° 0°

σ0φθ σ0φφ

θs θs

-98

90° φs 90° φs

0° 180° 360° 0° 180° 360°

Figure 5.11: RCS per unit area of the coherent scattering component of the

scenario according to Fig. 5.8 (h = 0:350 , L = 40 , Lx = Ly = 80 , i = 60Æ ,

i = 0Æ )

80 CHAPTER 5. THEORETICAL VERIFICATION

scattering approach and the Kirchho model with scalar approximation is

somewhat more delicate. The diculty lies in the nature of the Kirchho

model: the scattering intensity in a specic direction, as predicted by this

model, is mainly determined by the diraction pattern of the surface and

not by its roughness. In fact, the inuence of the surface roughness is most

pronounced in the specular direction resulting in a reduced scattering intensity

retaining the characteristic pattern produced by the geometry of the surface.

Figure 5.11(a)) depicts this type of pattern calculated by the Kirchho model

with scalar approximation. The same surface as in the comparison of the

diuse components is used, but with half the roughness, i.e. h = 0:350 . The

typical diraction pattern of a rectangular plate [Bal89] can be distinguished,

the inuence of roughness, as already indicated, is restricted to a diminution of

the intensity. Noting the large dynamic range of these images compared to the

incoherent component of Fig. 5.10, it is obvious that the coherent component

is only signicant in the vicinity of the specular direction. Figure 5.11(b)

displays the equivalent images obtained by the stochastic scattering approach.

In contrast to the Kirchho model, these results solely reect the inuence of

the surface roughness. For a perfectly smooth surface, the stochastic scattering

approach would equal specular reection, i.e. a single point. Also for the

stochastic scattering approach, one can clearly distinguish the interference

pattern introduced by the limited area-extend of the surface. The absolute

value in the specular direction of e.g.

0

is 0:7dB for the Kirchho model

and 1:0dB for the stochastic scattering approach. For the incoherent scattering

component, this value is 12:4dB for both models and thus still much larger than

for the coherent scattering. According to (5.12), the value of h = 0:350 lies

outside the validity range of the Kirchho model with scalar approximation

by a factor of two. A smaller value of the roughness leads to almost the same

images for the Kirchho model, but with a larger maximum value. However,

the actual inuence of the surface roughness on the directional pattern, as

given by the stochastic scattering approach, becomes less visible. Thus for

visualization purposes, the value of h = 0:350 was chosen.

The very good qualitative and quantitative agreement proves the validity

of the proposed stochastic scattering approach, especially when bearing in

mind that the Kirchho models are only approximate solutions. The great

advantage of the stochastic scattering approach, as already mentioned, is its

ability to deliver instantaneous realizations of the scattering process including

both the coherent and the incoherent component.

available: for coverage planning purposes, where the mean values of the re-

ceived power levels are of interest, specular reection together with the mod-

5.4. SUMMARY 81

ied Fresnel reection coecients of section 2.3.3.2 are the best choice. For

system design or evaluation purposes, however, the stochastic scattering ap-

proach should be used instead, with its ability to generate instantaneous (and

therefore varying) realizations of the actual scattering processes.

5.4 Summary

In this chapter, the proposed ray-optical modelling techniques and the stochastic

scattering approach have been veried theoretically with reference solutions

available for simple geometries. In the next chapter, they are further valid-

ated by comparisons with measurements. Special emphasis is given to the

performance in real three-dimensional curvature, which to date could not be

evaluated theoretically due to the lack of reference solutions.

Chapter 6

Experimental verication by

measurements in scaled

model tunnels at 120GHz

In the previous chapter the proposed ray density normalization has been val-

idated theoretically. This was only possible in idealized geometries for which

reference solutions exist. To test the RDN in realistic geometry, measurements

have been carried out at 120GHz in scaled model tunnels built of concrete

and stoneware. The performance of the modelling approach in real three-

dimensional curvature is therefore of primary interest. The choice of scaled

model tunnels rather than real tunnels for the initial verication is primarily

due to availability and to ensure reproducibility. Furthermore, the possibility

of investigating dierent arrangements and scenarios together with the abil-

ity to perform analyses over the entire cross section of the model tunnels are

additional advantages. Comparisons with measurements in real tunnel envir-

onments are performed in chapter 7.

The measurement setup and procedure are described in section 6.1. This is

followed by the comparison of measurements and simulations in various con-

gurations. In section 6.2, a straight concrete tube with and without a vehicle

present is considered. In section 6.3, the modelling approach is validated in a

curved stoneware tube. Finally, an entire model tunnel, including a straight

and a curved stoneware section with a concrete road line is examined in section

6.4.

6.1. MEASUREMENT SETUP AND PROCEDURE 83

The measurement setup is shown in Fig. 6.1, with the model tunnel investig-

ated in section 6.4 as device under test (DUT).

stoneware tubes

waveguide probe

computer controlled

positioner

Figure 6.1: Measurement setup with scaled model tunnel; straight and curved

"r 8, diameter: 20cm, total length: 107cm, concrete road

stoneware tube with

lane with 5, transmitter: D-band horn antenna at tunnel entrance, receiver:

"r

D-band waveguide probe for 2D-scans at tunnel exit (resolution: 2mm 2mm),

f = 120GHz

used as transmitter at f = 120GHz. The input power of the antenna PT

10dBm is generated by a backward wave oscillator (BWO). The receiver is a

D-band rectangular waveguide probe, which can be displaced computer con-

trolled to generate two-dimensional (2D) scans with a resolution of 2mm

2mm. The received power level is measured with a vector network ana-

lyzer (VNWA). The measurement equipment was developed at the Institut

für Hochleistungsimpuls- und Mikrowellentechnik, Forschungszentrum Karls-

ruhe, Germany [Arn97, Sch99]. To avoid noticeable side eects by the edges

of the tubes, the transmitter as well as the waveguide probe are positioned

at least 1cm inside the tubes. The directional patterns of the horn antenna

and the waveguide probe are plotted in appendix D and are considered in the

simulations.

Various sewer tubes of dierent kind and shape are used to build the model

tunnels. The following tubes are utilized in the comparison:

84 CHAPTER 6. EXPERIMENTAL VERIFICATION

2. a bent stoneware tube with angle of curvature of 45Æ, length of 30cm and

diameter of 20cm (DN 200 45 ),

Æ

3. a bent stoneware tube with angle of curvature of 90Æ, length of 47cm and

diameter of 20cm (DN 200 90 ),

Æ

4. a straight stoneware tube with length of 60cm and diameter of 20cm.

The allowable tolerances for this type of sewer tubes are given in [DIN99], e.g.

the curvature may vary up to 5Æ. The material parameters of the tubes have

been determined at 200MHz40GHz via measurements of reection coecients

with a HP 8510 VNWA (assuming that r = 1). The averaged permittivity

values are used in the simulations at 120GHz. They are "r 5 for the con-

+

crete tube [SM 96] and "r 8 for the stoneware tubes. Various simulations

suggested that the inuence of the permittivityeven on the order of 50%

variationis of minor importance compared to the inuence of geometrical

parameters (cf. section 6.3), which coincides with results obtained in the liter-

ature [Mar92, Kle93, MLD94]. The surface roughness of the concrete tube is

h 0:1mm. The roughness of the stained stoneware tubes is h 0:05mm

and thus negligible according to (2.29).

The frequency of f = 120GHz in the scaled geometry is comparable to

a frequency of 1GHz3GHz in real tunnels. However, in contrast to scaled

1

measurements [JM84, YAS89, Kle93, KSB93], i.e. using the values measured

at high frequency in a scaled geometry corresponding to a lower frequency in

the unscaled geometry, the measurements and calculations in this chapter are

both performed at = 120GHz.

f

Despite the tolerances in the geometry of the dierent tubes and the ex-

tremely small wavelengths ( = 2:5mm) a coherent analysis is chosen for com-

parison purposes in the following sections. The measured and simulated power

levels are normalized to their respective maximum values. The agreement of

the absolute values is conrmed by an initial free-space measurement.

To quantify the agreement between simulations and measurements, the

standard deviation M and the 2D correlation coecient M of both results are

calculated. In order to avoid large degradations of these values due to possible

misalignment of the images, the images are shifted by a maximum of 3 pixels

in each direction, performing a maximum search with regard to the correlation

coecient prior to the determination of M and M . This technique is adopted

from image processing [DH73, Jäh97]. Additionally, the results are processed

by a 2D mean lter with rectangular transfer function of size 3 3 pixels

resulting in a second set of smoothed M and M . A sucient correlation

1 For scaled measurements, one has to ensure a correct scaling of the equivalent con-

ductivity of the building materials, otherwise leading to false imaginary parts of "r (2.4)

[Kle93].

6.2. COMPARISONS IN A STRAIGHT CONCRETE TUBE 85

between images is given for M;M > 0:5, although strongly dependent on the

image content.

First, the straight concrete tube of Fig. 6.2 is examined.

transmitter:

horn antenna

20cm

1m

receiver positions

of waveguide probe

Figure 6.2: Measurement setup with a concrete tube; diameter: 20cm, length:

1m, "r 5, transmitter: horn antenna at tunnel entrance, receiver: waveguide

probe for 2D cross-sectional scans at tunnel exit (resolution: 2mm 2mm),

f = 120GHz

Figure 6.3 shows the measured and simulated power level distribution (co-

herent analysis, vv-polarization) at the end of the tube for an eccentric trans-

mitter position 5cm above the center at the tunnel entrance. The measurement

and the simulation results agree very well. Only in the lower regions of the

gures small discrepancies occur, which may be due to a small misalignment of

the antennas. The standard deviation between the measurement and the cal-

culation is = 4:0dB, the correlation between the two images is M = 0:68.

M

The values after mean ltering are M = 2:3dB, M = 0:82.

7

For simulation work 5 10 rays were traced with up to 10 reections. One

single reception plane was used and scattering was taken into account by the

modied reection coecients according to (2.30). The simulation time was

approximately three hours on the HP workstation, described in Table 5.1.

Then, a conguration with a road lane made of PVC ( "r 2:5) and a

matchbox car (London Sightseeing Bus) is used. The thickness of the lane is

86 CHAPTER 6. EXPERIMENTAL VERIFICATION

M

0 0

6 6

4 4

distance to center (cm)

2 −5 −5

2

0 0

−2 −10 −2 −10

−4 −4

−6 −6

−15 −15

−6 −4 −2 0 2 4 6 −6 −4 −2 0 2 4 6

distance to center (cm) distance to center (cm)

Figure 6.3: Results for measurement setup according to Fig. 6.2, transmitter

5cm above center, scanned area: 71 71 points, resolution: 2mm (coherent

analysis, vv -polarization)

width: 3:6cm, height: 6:4cm, distance between lane and underbody of the car:

0:3cm) with various windows, whose dimensions are on the order of several

wavelengths. The matchbox car is positioned inside the tunnel, 5cm from the

tunnel exit, such that a distinct shadow is visible in the analysis. Although

the vehicle is only modelled as a oating PEC rectangular box (cf. section

4.4), the measurement (Fig. 6.4(a)) and the simulation (Fig. 6.4(c)) are sim-

ilar. Additionally, the same scenario has been re-measured with a metallic

rectangular box instead of the matchbox car (Fig. 6.4(b)). The outline of the

measured box is obviously larger than the one of the matchbox car and there-

fore closer to the result obtained by the simulation. In all gures the circular

structures coincide, as well as the horizontal stripes of high (respectively low)

reception levels. Also the boundary condition for grazing incidence is visible

at the surface of the lane, which calls for a minimum of the received power

level.

The standard deviation and correlation between the measurement with the

matchbox car and the simulation is = 11:0dB (M = 9:7dB) and M = 0:7

M

( M = 0:81), respectively. For the measurement with the metallic box, the

corresponding values are M = 10:1dB and M = 0:78 (M = 8:8dB, M =

0:86). If only the right halves of the pictures in Fig. 6.4 are considered, i.e.

if the shadow region of the obstructing object is neglected in the quantitative

6.2. COMPARISONS IN A STRAIGHT CONCRETE TUBE 87

P/PM (dB)

0

4

2

−5

0

−10

−2

−4

−15

−8 −6 −4 −2 0 2 4 6 8

distance to center (cm)

P/PM (dB)

0

4

distance to center (cm)

2

−5

0

−10

−2

−4

−15

−8 −6 −4 −2 0 2 4 6 8

distance to center (cm)

P/PM (dB)

0

4

distance to center (cm)

2

−5

0

−10

−2

−4

−15

−8 −6 −4 −2 0 2 4 6 8

distance to center (cm)

Figure 6.4: Results for measurement setup according to Fig. 6.2 with PVC-oor

("r 2:5) and box-like vehicle or metallic rectangular box, centric transmitter

position, scanned area: 81 46 points, resolution: 2mm (coherent analysis,

vv -polarization)

88 CHAPTER 6. EXPERIMENTAL VERIFICATION

comparison, the values are = 4:1dB (M = 2:7dB), M = 0:58 (M = 0:65)

M

for the matchbox car and = 5:2dB (M = 3:5dB), M = 0:4 (M = 0:47)

M

for the metallic box, respectively. The simulation time was about 5:5 hours on

8

a HP workstation with 1 10 rays.

The measurements in the previous section show a very good agreement with the

simulations in a straight tube of circular or arched cross section. To validate

the modelling in real 3D curvature, a bent stoneware tube depicted in Fig. 6.5

is used as the device under test in this section.

transmitter:

horn antenna

20cm

receiver positions

of waveguide probe

Figure 6.5: Measurement setup with a bent stoneware tube; angle of curvature:

45Æ , diameter: 20cm, length: 30cm, "r 8, transmitter: horn antenna at tunnel

entrance, receiver: waveguide probe for 2D cross-sectional scans at tunnel exit

(resolution: 2mm 2mm), f = 120GHz

tube. A closer inspection reveals that the tube is actually constituted by two

short straight sections and an intermediate (generally non-circular) bend. For

the modelling, the curvature is approximated by a circular arc, resulting in

the geometry plotted in Fig. 6.6(b).

In order to validate the performance of the modelling, andat the same

timeto examine the inuence of 3D curvature on the propagation properties,

the bent tube of Fig. 6.5 has been measured and compared with three dierent

simulation setups:

1. a straight tube with the same length and diameter as the actual probe,

6.3. COMPARISONS IN A BENT STONEWARE TUBE 89

3

x

c.

se

y

l3

.2

sec

l2

a rc

sec.1

l1

Figure 6.6: (a) Schematic plot of the longitudinal prole of a bent stoneware

tube with corresponding approximative tri-sectional geometry used in the sim-

ulation (b)

2. a tube constituted by one single circular arc, having the same angle of

curvature ( 45Æ), the same length and the same diameter as the actual

probe ( d = 20cm),

3. a bent stoneware tube modelled according to Fig. 6.6(b), with a rst

straight section of length l1 = 7:5cm, a second curved section of length

l2 = 14:85cm, a radius of curvature rc = 18:91cm, and a third straight

section of length l3 = l1 = 7:5cm.

tion 5cm from the center in direction of the bend. Figure 6.7 depicts the meas-

urement and the simulations at the other end of the model tunnel. Although

the actual geometry of the bent stoneware tube is only roughly approximated

by the tri-sectional geometry of Fig. 6.6(b), the measurement in Fig. 6.7(a) and

the simulation in Fig. 6.7(b) suggest a good agreement over a large area. The

eects of the relatively strong bend can be identied by comparing the images

with the results obtained for the pure (and therefore less distinct) bend in Fig.

6.7(c), and for the equivalent straight tube in Fig. 6.7(d). The correlations

and the standard deviations between the measurement and the calculations

are given in Table 6.1. In addition to the comparison of the entire images,

the values are also determined for the portions of the images containing the

most distinct parts of the interference patterns, extending from [ 2cm; 6:5cm]

over the x-axis, and from [ 4cm; 4cm] over the y-axis (the axes' orientations

90 CHAPTER 6. EXPERIMENTAL VERIFICATION

M

0 0

6 6

−5 −5

4 4

distance to center (cm)

−10 −10

2 2

−15 −15

0 0

−20 −20

−2 −2

−25 −25

−4 −4

−30 −30

−6 −6

−35 −35

−6 −4 −2 0 2 4 6 −6 −4 −2 0 2 4 6

distance to center (cm) distance to center (cm)

M M

0 0

6 6

−5 −5

4 4

distance to center (cm)

−10 −10

2 2

−15 −15

0 0

−20 −20

−2 −2

−25 −25

−4 −4

−30 −30

−6 −6

−35 −35

−6 −4 −2 0 2 4 6 −6 −4 −2 0 2 4 6

distance to center (cm) distance to center (cm)

Figure 6.7: Results for measurement setup according to Fig. 6.5, transmitter in

eccentric position 5cm from the center in the direction of the bend, scanned

area: 71 71 points, resolution: 2mm (coherent analysis, vv -polarization)

are depicted in Fig. 6.6(a)). It is apparent from the gures that the actual

shape (i.e. the course) of the simulated tunnels is of major importance. The

dierences between the straight tube, the pure bend, and the tri-sectional geo-

metry are signicant. The comparison between the measurement in the bent

tube and the simulation of the straight section even leads to a negative correla-

tion. Thus, a precise modelling of the tunnel's geometry including curvature is

6.3. COMPARISONS IN A BENT STONEWARE TUBE 91

Table 6.1: Correlations and standard deviations between the measurement and

the simulations of Fig. 6.7 together with computation times of the simulations

entire images:

straight section pure bend tri-sectional bend

(Fig. 6.7(d)) (Fig. 6.7(c)) (Fig. 6.7(b))

M (M ) 11:9dB (11:7dB) 3:3dB (2:5dB) 3:0dB (2:2dB)

parts of the images (x 2 [ 2cm; 6:5cm], y 2 [ 4cm; 4cm]):

straight section pure bend tri-sectional bend

M (M ) 7:7dB (7:4dB) 3:4dB (2:3dB) 2:5dB (1:6dB)

simulation times

straight section pure bend tri-sectional bend

M

0 0

6 6

−5 −5

4 4

distance to center (cm)

−10 −10

2 2

−15 −15

0 0

−20 −20

−2 −2

−25 −25

−4 −4

−30 −30

−6 −6

−35 −35

−6 −4 −2 0 2 4 6 −6 −4 −2 0 2 4 6

distance to center (cm) distance to center (cm)

Figure 6.8: Results for measurement setup according to Fig. 6.5, transmitter

5cm above center, scanned area: 71 71 points, resolution: 2mm (coherent

analysis, vv -polarization)

mandatory to predict the results with sucient accuracy. For all simulations,

5 107 rays have been launched and up to 20 reections have been considered.

92 CHAPTER 6. EXPERIMENTAL VERIFICATION

The results are also very sensitive to the location of the transmitter, in

addition to the geometry of the tube. In Fig. 6.8 simulations and measurements

are compared with a dierent transmitter position being situated 5cm above

the center of the bent tube. Apart from that the same scenario as for the

previous gures is assumed. The measured (Fig. 6.8(a)) and the simulated

(Fig. 6.8(b)) power distribution dier signicantly from the ones obtained in

Figs. 6.7(a) and 6.7(b). The agreement between the measurement and the

simulation is very encouraging, with M = 4:9dB, M = 3:2dB, M = 0:89,

and = 0:96. Again, 5107 rays have been launched, resulting in a simulation

M

time of 1:20h.

straight and curved sections

After the analyses of single pieces of tubes, two stoneware tubes are combined

to build an entire model tunnel. The model tunnel consists of a straight section

of length ls = 60cm and a 90Æ-bend of length lb = 47cm, resulting in a total

length of lt = 107cm. Due to the tolerances in the geometry of the two tubes,

the transition from the straight to the curved tube is not homogeneous, but

leaves a gap of approximately 0:5cm width. Furthermore a concrete road lane

is cemented into the tunnel up to a height of 5:5cm above the lowest point

of the tubes. Figure 6.1 shows the measurement setup with the entire model

tunnel.

The aim of the measurements in the model tunnel is in analogy to the

previous sections to validate the RDN-based modelling approach. Moreover,

the focusing of energy introduced by a concave curvature is investigated. For

this purpose, horizontal scans into the tunnel rather than scans over the cross

section of the tunnel are carried out. The geometry for this type of analysis is

depicted in Fig. 6.9.

The rst horizontal scan is performed before the concrete road lane is put

into the model tunnel. The transmitter is situated in a centric position, the

scan is taken10cm above the lowest point of the (still circular) cross section.

After the inclusion of the road lane, a second horizontal scan is measured

at the same position (i.e. 4:5cm above the road), with the transmitter 9:5cm

above the road at the tunnel entrance. Figure 6.10 shows the two measured

(Figs. 6.10(a), 6.10(b)) together with the corresponding predicted results (Figs.

6.10(c), 6.10(d)).

For the simulations the bent tube is, according to the previous section,

approximated by a tri-sectional geometry, i.e. a rst straight section with l1 =

6.4. COMPARISONS IN A MODEL TUNNEL 93

18cm

12cm

6cm

18cm 12cm

20cm

60cm

5.5cm

Tx

Figure 6.9: Measurement setup for horizontal scan in the scaled model tunnel

of Fig. 6.1, height of concrete road lane: 5:5cm, scanned area 4:5cm above road

lane,18cm 12cm at 6cm from the tunnel exit (resolution: 2mm 2mm),

f = 120GHz

5:5cm, a second curved section with l2 = 36:1cm and rc = 21:8cm, and a third

straight section with l3 = l1 = 5:5cm. The utilized permittivities are "r = 8 for

the stoneware tubes and "r = 5 for the concrete road lane. The roughness of

the surfaces is h < 0:05mm, which is negligible. A horizontal reception plane

is used in the simulation. A signicant part of the launched rays is intersecting

the horizontal reception plane near grazing incidence. Therefore, the number

of rays has to be signicantly large to ensure a sucient convergence compared

to a cross-sectional analysis (cf. section 4.5). For the results shown in Figs.

6.10(c) and 6.10(d), 5 108 rays have been launched and up to 20 reections

were traced. The computation time was about 4 days.

Bearing in mind that the real geometry of the model tunnel cannot be

constructed perfectly in the simulation (especially the gap on the order of two

wavelengths and the geometry of the bend), the measurements and the simu-

lations in Fig. 6.10 agree surprisingly well. The standard deviations and the

correlations between the measurements and the simulations are calculated for

the right halves of the images, containing the most distinct parts of the inter-

94 CHAPTER 6. EXPERIMENTAL VERIFICATION

M M

2 0 2 0

distance to center (cm)

0 −5 0 −5

−10 −10

−2 −2

−15 −15

−4 −4

−20 −20

−6 −6

−25 −25

−8 −30 −8 −30

−10 −35 −10 −35

6 8 10 12 14 16 18 20 22 24 6 8 10 12 14 16 18 20 22 24

distance to exit (cm) distance to exit (cm)

M M

2 0 2 0

distance to center (cm)

0 −5 0 −5

−10 −10

−2 −2

−15 −15

−4 −4

−20 −20

−6 −6

−25 −25

−8 −30 −8 −30

−10 −35 −10 −35

6 8 10 12 14 16 18 20 22 24 6 8 10 12 14 16 18 20 22 24

distance to exit (cm) distance to exit (cm)

Figure 6.10: Results for the scenario of Fig. 6.1, measurement setup according

to Fig. 6.9, scanned horizontal area: 92 60 points, resolution: 2mm (coherent

analysis, vv -polarization)

ference patterns . The values are M = 7:8dB (M = 5:5dB) and M = 0:33

2

( = 0:53) for the tunnel without road lane, and M = 6:7dB (M = 4:9dB)

M

and M = 0:5 (M = 0:63) for the tunnel with road lane. In the upper parts of

the images, the focusing eect of the bend becomes clearly visible by the light

stripes (focal lines) indicating a high level of received power. Thus, for the

rst time, the RDN-based ray-optical modelling approach enables a precise

prediction of electromagnetic wave propagation in curved tunnels including

the focusing of energy, without the problems of traditional GO solutions in

the vicinity of caustics. Figure 6.11 depicts the same simulations but over the

2 Otherwise the strong inuence of the shadow regions, indicated by the dark areas in the

lower left corners of the images, would dominate the determination of the correlation.

6.4. COMPARISONS IN A MODEL TUNNEL 95

20cm

scanned

area

20cm

20cm

scanned

area

20cm

Figure 6.11: Horizontal analysis in the curved section of the model tunnel (dot-

ted rectangle in Fig. 6.9(b)), resolution: 2mm 2mm (coherent analysis, vv -

polarization)

96 CHAPTER 6. EXPERIMENTAL VERIFICATION

whole bent area of the model tunnel. It is seen that the transitions from the

straight to the curved section can be clearly distinguished and the inuence of

the road lane is pronounced.

6.5 Summary

In this chapter, the proposed ray density normalization (RDN) was veried

by measurements in scaled model tunnels at 120GHz. In particular the per-

formance in real 3D curvature was investigated and validated. The various

qualitative and quantitative comparisons identify the RDN as a very powerful

approach. All relevant eects in straight tunnels, curved tunnels, and tun-

nels with vehicles are predicted with sucient accuracy when compared to

measurements.

After the theoretical validation in chapter 5 and the validation by meas-

urements under laboratory-like conditions, the next chapter deals with the

ability of the technique to cope with real life scenarios. For this purpose,

measurements in the Berlin subway have been performed.

Chapter 7

In the previous two chapters, the practicality of the proposed ray density

normalization (RDN) was proven both theoretically and experimentally by

measurements in scaled model tunnels of dierent shapes. In this chapter,

the application of the RDN in real tunnels, which are mostly of non-idealized

geometry, is investigated. For this purpose, a measurement campaign has been

carried out in the Berlin subway. Further to verify the modelling approach,

the behaviour of electromagnetic wave propagation in dierent scenarios and

its sensitivity to parameter changes are investigated. Moreover, the eects

of exterior transmitting antenna locations and the inuence of curves on the

power distribution inside the tunnels are determined, and the ability of the

modelling approach to estimate broadband channel parameters is pointed out.

Section 7.1 is devoted to the measurements and comparisons in the Berlin

subway. First, the measurement setup and the dierent tunnels are described

(section 7.1.1). Second, the measurements are compared with simulations in

section 7.1.2, together with several analyses of the inuence of the geometry

on the prediction accuracy and on the power distribution in curved tunnels.

Scenarios with dierent antenna positions in front of a tunnel's entrance are

examined in section 7.2. Finally, the impact of moving vehicles on the delay

and Doppler spread is shown in section 7.3.

In order to characterize wave propagation in underground railroad tunnels in

the GSM900 and GSM1800 frequency bands, a measurement campaign was

carried out in the Berlin subway. The propagation scenario of subway tunnels

is by far less homogeneous than the idealized geometries of the model tunnels in

98 CHAPTER 7. SCENARIOS AND APPLICATIONS

the previous chapter. Hence, this environment can be seen as worst-case scen-

ario to test the applicability and performance of the modelling approach. Two

tunnels of dierent shape, length, and building materials are investigated. Fur-

thermore, dierent transmitting antenna positions are analysed. The received

power levels at 945MHz and 1853:4MHz are used to evaluate the attenuation

and the fading characteristics of the dierent constellations and environments.

7.1.1.1 Measurement equipment

receiving antennas

(l /4-monopoles)

transmitting antennas

test receivers

power supply

lorry

optical pulse generator

Figure 7.1: Measurement setup with transmitting antennas and receiving equip-

ment mounted on a lorry with an optical pulse generator

test transmitter, a Rhode&Schwarz SME 23 GSM1800 test transmitter, and

two Rhode&Schwarz extended test receivers ESVD for digital mobile radio net-

works. The transmitters generated two harmonic signals at fGSM = 945MHz

and fDCS = 1853:4MHz, respectively. The intermediate frequency (IF) meas-

urement bandwidth was 10kHz. For the GSM900 band, a log-periodic (Lo-

gPer), vertically polarized transmitting antenna with 12dBi gain was used

(Kathrein K73226), whereas for the GSM1800 band, a wide-band Yagi an-

tenna with 17dBi gain (Jaybeam J7360) was employed [Bal97, chapter 10].

Both are standard antennas for the deployment in tunnel environments. As

receiving antennas, two =4-monopoles were chosen due to their omnidirec-

tional antenna patterns. The patterns of the transmitting antennas and the

7.1. THE BERLIN SUBWAY 99

ted on a lorry which was manually pulled through the tunnels at an average

speed of 1:5m=s. The measurements were recorded approximately every 30cm,

where each measured value corresponds to the averaged received signal during

a 10ms time interval. The actual measurement location was retrieved with a

pulse-generator coupled to the wheels of the lorry (cf. Fig. 7.1). Each meas-

urement was run and recorded twice in order to determine the time variance of

the transmission channel. Furthermore, the two corresponding measurements

were compared and aligned to each other to ensure a reasonable precision

in the absolute location of the measured values. This was necessary due to

the imprecise performance of the pulse generator, which provided an impulse

approximately every 17:9cm with a precision of 0:5cm. The positions of

the transmitting antennas were varied for each measurement. The receiving

monopoles were xed on the lorry at a height of hR = 1:47m above the rails.

The measured path loss has been deduced from the ratio of the measured

received power to the input power of the transmitting antennas, including

the antennas characteristics. The attenuation of the connecting cables was

taken into account, and an additional 1:5dB loss was assumed for any kind

of mismatch in both the receiving and transmitting branches. The simulated

path loss obtained by (2.54), normalized to PT , also includes the antenna

characteristics.

Two dierent tunnels are investigated:

of subway U5 between Friedrichsfelde and Tierpark), built in the early

70's,

Nervenklinik and Rathaus Reinickendorf ), built in the late 80's.

4:3m, and its length l 110m.

U5 U5

concrete, the side walls are covered by a variety of cables (e.g. power supply-

and phone-lines) and mountings. The rail sleepers lie on gravel. The roughness

of the walls was estimated to h = 2cm, the roughness of the oor to h = 5cm.

The cross section of the arched tunnel is constituted by a circular shape

of radius rcs 2:9m with an elevated oor 1:2m above the lowest point of

the circle (cf. Fig. 7.3). A schematic plot of the tunnel's course is shown in

Fig. 7.2. It consists of nine dierent sections and can roughly be described

by a rst straight part, followed by a left bend with large radius of curvature

and a right bend with smaller radius of curvature. For simulation work, the

100 CHAPTER 7. SCENARIOS AND APPLICATIONS

cur. clot.

clot. 8 9

str. 7

clot. 6

5

cur.

clot. str. clot.

4 fire exit

Tx 1 2 3

d

Figure 7.2: Schematic course plot of the arched shaped tunnel (U8), with a total

length of lU8 1079m

Figure 7.3: View into the arched shaped tunnel (U8) of Fig. 7.2 from Karl-

Bonhoeer-Nervenklinik (d = 0m)

nine sections are modelled according to section 4.1. The total length of the

tunnel is l 1079m, the maximum measured distance is d 1000m. At

d > 420m from the transmitter, the receiver and the transmitter

U8

distances

have no longer a direct line-of-sight (LOS). At distance d = 620m from the exit

transmitter, an open connection (re exit) exists for security reasons between

the actual and the second tube, which runs in parallel, extending over an area

of approximately 25m2 (cf. Fig. 7.2). Figure 7.3 depicts the view into the

tunnel from the station Karl-Bonhoeer-Nervenklinik, with the transmitters

situated at the beginning of the tube.

7.1. THE BERLIN SUBWAY 101

It is apparent from Fig. 7.3, that the walls are not smooth but that they

have a periodic structure due to the special construction by screwed prefabric-

ated elements. Consequently, the occurring (large scale) height variations of

the tunnel walls are not resulting from a statistically rough surface in a strict

1

sense, but rather from a periodically rough surface . Nevertheless, the concept

of rough surface scattering can be applied in an approximate way by adapting

the mean roughness of the walls with growing distances: at small distances

from the transmitter, where most rays impinge under oblique incidence (i.e.

with small incident angles) onto the walls, a mean roughness of h = 2cm is

assumed. At larger distances, where most rays impinge under near grazing

incidence onto the walls, the mean roughnessand thus the attenuationis

increased (up to h = 15cm), to reect the shadowing behaviour of the special

wall structure.

For both tunnels, the parameters of the building materials in the simulation

correspond to dry concrete ( "r = 5 j 0:1). The roughness of the walls is taken

into account by the modied Fresnel reection coecients.

7.1.2 Results

In both tunnels, several transmitting antenna constellations were measured at

the two frequencies. First, the short straight tunnel section (U5) is considered.

In this rectangular geometry, image theory, as a ray-optical reference solution,

can be used to determine the achievable accuracy by simulations. Thereafter,

the measurements in the curved tunnel with arched cross section (U8) are used

in exemplary analyses.

Figure 7.4 depicts the measured and predicted path loss in the straight rect-

angular tunnel (U5) for two dierent constellations. For Fig. 7.4(a), the trans-

mitting GSM900 antenna was situated at a height of = 2:82m, 0:88m to

hT

the left of the tunnel center. For the scenario in Fig. 7.4(b), the transmitting

GSM1800 antenna was situated at a height of hT = 2:2m, 0:04m to the right

of the center. The predicted path loss was calculated by image theory at 200

receiver locations for up to 10 reections per ray. The simulation time was

a few minutes on a standard HP workstation. To quantify the agreement of

predictions and measurements, mean values M and standard deviations M

of the dierence (in dB) between the measured and the predicted losses are

determined. Like in the previous chapter, the values are either obtained by

1 A periodically rough surface generally results in scattering patterns with specic pref-

erential directions.

102 CHAPTER 7. SCENARIOS AND APPLICATIONS

10

measurement

20 measurement (RMS)

prediction

path loss (dB)

30

40

50

60 µM = 2.0dB, σM = 3.4dB

µM = 2.1dB, σM = 3.3dB

70

0 20 40 60 80 100

distance to transmitter (m)

(a) GSM900

20

measurement

30 measurement (RMS)

prediction

path loss (dB)

40

50

60

70 µM = 1.5dB, σM = 4.6dB

µM = 1.9dB, σM = 4.4dB

80

0 20 40 60 80 100

distance to transmitter (m)

(b) GSM1800

rectangular-shaped tunnel section (U5) with dierent transmitter positions at

fGSM = 945MHz and fDCS = 1853:4MHz, running RMS window length for the

measurement: 1m (whereas for the prediction, no RMS generation is performed

due to a reception-sphere spacing of 0:5m)

7.1. THE BERLIN SUBWAY 103

running root mean square (RMS) generation (cf. section 7.1.2.5), indicated by

the subscript M leading to M and M . The imperfect match of the measure-

ments and predictions conrms the already indicated worst-case character of

the environment of subway tunnels for propagation modelling. This mismatch

is due to the irregular structure of the tunnel, modelled as being straight and

rectangular. The mean values and standard deviations obtained in this grossly

simplied scenario serve as benchmarks for the following comparisons. In the

remainder of this section, the curved tunnel of Figs. 7.2 and 7.3 is examined.

In the curved arched-shaped tunnel (U8), several transmitter locations were

deployed. Figures 7.5 and 7.6 depict the comparisons for the conguration

shown in Fig. 7.3.

20

30 measurement

measurement (RMS)

40 prediction

prediction (RMS)

path loss (dB)

50

60

70

80

µM = 2.3dB, σM = 4.9dB

90 µM = 1.8dB, σM = 3.6dB

100

0 100 200 300 400 500 600 700 800

distance to transmitter (m)

shaped tunnel (U8) at fGSM = 945MHz, running RMS window length: 400

(right transmitting antenna in Fig. 7.3)

the right of the tunnel's center (right transmitting antenna in Fig. 7.3). The

GSM1800 antenna was positioned at hT = 2:45m, 0:93m to the left of the

center (left transmitting antenna in Fig. 7.3). The receiving antennas were

aligned accordingly with the GSM900 monopole on the right side and the

GSM1800 monopole on the left side of the lorry. The path loss was simulated

104 CHAPTER 7. SCENARIOS AND APPLICATIONS

30

measurement

40

measurement (RMS)

50 prediction

prediction (RMS)

path loss (dB)

60

70

80

90

µM = 1.2dB, σM = 5.4dB

100 µM = 1.3dB, σM = 4.5dB

110

0 100 200 300 400 500 600 700 800

distance to transmitter (m)

shaped tunnel (U8) at fDCS = 1853:4MHz , running RMS window length: 600

(left transmitting antenna in Fig. 7.3)

30

measurement

40 measurement (RMS)

prediction (straight)

50

prediction (RMS)

path loss (dB)

60

70

80

90

µM = -5.8dB, σM = 7.8dB

100 µM = -5.4dB, σM = 7.3dB

110

0 100 200 300 400 500 600 700 800

distance to transmitter (m)

Figure 7.7: Comparison of measurement and simulation of Fig. 7.6, but assum-

ing a ctitious straight tunnel course for the simulation

7.1. THE BERLIN SUBWAY 105

by the RDN-based power trace method. 150 million rays were traced with up

to 40 reections. The calculation time for the 1600 receivers was about 40h.

The good agreement of the measured and the predicted path loss validates

the RDN modelling approach. The small mean errors ( M = 1:8dB in Fig.

7.5 and = 1:3dB in Fig. 7.6) and standard deviations (M = 3:6dB in

M

= 4:5dB in Fig. 7.6) emphasize the good performance of the

Fig. 7.5 and M

model, especially bearing in mind the imprecise assignment of the absolute

measurement location (cf. section 7.1.1.1).

In order to determine the inuence of curves on the propagation behaviour,

the same measurements as depicted in Fig. 7.6 (U8 scenario at fDCS =

1853:4MHz) are compared with simulations, where the bend of the tunnel

is approximated by a straight line. Compared to the simulation of the ac-

tual curved course in Fig. 7.6, one can clearly distinguish the deviation of the

prediction from the measurement in Fig. 7.7. For distances d> 350m, the

deviation becomes noticeable, which is the region where the left bend of the

tunnel starts. Although the radius of curvature of the left bend is as large as

rcs = 850m, the deviation is rather signicant. The predicted mean level is

increased by 7dB and the standard deviation is almost doubled. This example

shows the importance of an adequate modelling of a tunnel's curvature, being

possible by the proposed novel RDN-based techniques.

curacy

It is commonly assumed that the actual shape of the cross section is of minor

inuence on the propagation behaviour in a tunnel, as long as its actual cross-

+

sectional area is preserved [YA 85, ZH98a]. Figures 7.8, 7.9 and 7.10 show

comparisons of the GSM1800 scenario in Fig. 7.3 on the rst 100m with dier-

ent cross-sectional shapes used for the simulations. In addition to the arched

cross section (Fig. 7.8), a pure circular cross section with radius rcs = 2:675m

(Fig. 7.9), and a rectangular cross section of width w = 5:25m and height

h = 4:28m (Fig. 7.10) are applied, all covering the same area. The mean er-

rors and standard deviations are calculated on the rst 100m. Again, it turns

out that the correct modelling of the tunnel's cross section aects the accuracy

of the modelling results signicantly.

106 CHAPTER 7. SCENARIOS AND APPLICATIONS

30

measurement

measurement (RMS)

40 prediction

path loss (dB)

50

60

µM = -0.3dB, σM = 3.3dB

70 µM = 0.0dB, σM = 3.2dB

0 20 40 60 80 100

distance to transmitter (m)

Figure 7.8: Comparison of measurement and simulation with arched cross sec-

tion, parameters according to Fig. 7.6, fDCS = 1853:4MHz , running RMS win-

dow length: 1m (re-plot of Fig. 7.6 on the rst 100m)

30

measurement

measurement (RMS)

40 prediction

path loss (dB)

50

60

µM = 1.2dB, σM = 6.6dB

70 µM = 1.5dB, σM = 6.3dB

0 20 40 60 80 100

distance to transmitter (m)

Figure 7.9: Comparison of measurement and simulation with circular cross sec-

tion, parameters according to Fig. 7.6, fDCS = 1853:4MHz , running RMS win-

dow length: 1m

7.1. THE BERLIN SUBWAY 107

30

measurement

measurement (RMS)

40 prediction

path loss (dB)

50

60

µM = 1.4dB, σM = 7.1dB

70 µM = 1.8dB, σM = 6.8dB

0 20 40 60 80 100

distance to transmitter (m)

section, parameters according to Fig. 7.6, fDCS = 1853:4MHz, running RMS

window length: 1m

An interesting question is, whether the fast fading in a tunnel, i.e. the uctu-

ation of the received signal on a small-scale basis, can be characterized by a

standard probability density function (PDF). The fast fading envelope is ob-

tained by normalizing the received signal to its local root mean square (RMS)

value. Generally, a window length of at least 40 wavelengths is chosen for

the RMS determination [Lee82, Lee89, Ste92]. The resulting fading envelope

is compared to the following classical distribution functions: Gaussian (nor-

mal), lognormal, Nakagami, Rayleigh, Rician and Weibull [ITU1057, Lor79,

Pap84, Pro89]. In order to determine the parameters of the respective dis-

tributions, a least-mean-square (LMS) based parameter tting can be used

[KE95]. The LMS optimization is based on a simplex method [NM65]. As an

example, Figs. 7.11 and 7.12 show the results of the LMS optimization for the

2

GSM1800 measurement of Fig. 7.6.

The best ts for all measurements are achieved using Rician distributions

at both frequencies, which is congruent with the literature [LD98]. Neverthe-

less, except for the Rayleigh distribution, all other densities lead to similar

2 Generally, a sampling of at least 0 =2 is required for a non-ambiguous fast fading char-

acterization. This requirement is clearly violated by the performed measurements. Never-

theless, for all GSM900 and GSM1800 measurements the obtained fading characteristics led

to similar results.

108 CHAPTER 7. SCENARIOS AND APPLICATIONS

4

measurement

Gauss

probability density p(x)

3 lognormal

Nakagami

Rayleigh

Rice

2 Weibull

ray tracing

0

0.0 0.5 1.0 1.5 2.0

VR normalized to local RMS

(PDF) for the measurement of Fig. 7.6, RMS window length: 600

1.0

cumulative distribution F(x)

measurement

Gauss

0.5 lognormal

Nakagami

Rayleigh

Rice

Weibull

ray tracing

0.0

0.0 0.5 1.0 1.5 2.0

VR normalized to local RMS

(CDF) for the measurement of Fig. 7.6, RMS window length: 600

7.1. THE BERLIN SUBWAY 109

4

measurement

Gauss

probability density p(x)

3 lognormal

PDF Nakagami

Rayleigh

Rice

2 Weibull

ray tracing

0

0.0 0.5 1.0 1.5 2.0

VR normalized to local RMS

Figure 7.13: Chi-square best t probability density functions (PDF) for the

measurement of Fig. 7.6, RMS window length: 600

1.0

cumulative distribution F(x)

measurement

Gauss

0.5 lognormal

Nakagami

Rayleigh

Rice

Weibull

ray tracing

0.0

0.0 0.5 1.0 1.5 2.0

VR normalized to local RMS

(CDF) for the measurement of Fig. 7.6, RMS window length: 600

110 CHAPTER 7. SCENARIOS AND APPLICATIONS

results, as indicated by Figs. 7.11 and 7.12. The interesting area of the curves,

however, is in the lower VR -range corresponding to very low received signal

levels, the so-called deep-fades. In this region, none of the curves obtained by

the LMS optimization leads to satisfactory results. Another way to determine

the parameters of the various densities is given by a recursive application of

the chi-square and Kolmogorov-Smirnov goodness-of-t tests [Pap84, Ste92].

The chi-square test is particularly suited for a tting in the area of low received

values due to its sensitivity in regions of low probability. Figure 7.13 shows the

results for the PDF's obtained by a simplex optimization with the chi-square

criterion. The tted curves now approximate the measurement for low received

values more closely for the chi-square tting compared to the LMS tting (Fig.

7.13 compared to Fig. 7.11). However, an overall match could not be achieved

by neither of the analytical densities. Figure 7.14 depicts the cumulative dis-

tribution functions (CDF) obtained with the Kolmogorov-Smirnov criterion.

Again, no signicant improvement can be achieved compared to the LMS t-

ting in Fig. 7.12. Consequently, a complete characterization of the fast fading

characteristics in a tunnel by standard analytical density functions appears to

be impossible. In contrast, the PDF and CDF extracted from the prediction

in Fig. 7.6 approach the measurement more closely over the entire range of

VR . The corresponding predicted curves are drawn in all gures, marked by

black diamonds (and ray tracing).

In the previous chapter, focusing of energy due to bends has been observed

in the model tunnel by measurements and simulations (cf. section 6.4). In

the curved subway tunnel, the opposite eect is noticeable: a rapid decrease

of the received power level at the inner side of a curve. The measurement

in the GSM900 band plotted in Fig. 7.5 was performed over the total length

of the subway tunnel (U8). As already indicated, the transmitting antenna

and the receiving monopole were both situated on the right side of the tunnel

axis referring to Fig. 7.3. Following the tunnel's course depicted in Fig. 7.2,

the receiver was on the outer side in the left curve and, consequently, on the

inner side in the following right curve at the end of the tunnel. The averaged

850m to 950m from the transmitter,

measured path loss is plotted Fig. 7.15. At

the curve is dropping by almost 8dB. This is the area of the right bend, with

the receiver situated at the inner side of the curve.

In the same gure, the path loss predicted by the method of power ow

is indicated by the dotted line. To allow a direct comparison, the gain of the

receiving monopole in the horizontal direction ( = 90Æ) was considered in the

simulations. Although the overall propagation slope is predicted very well,

the defocusing eect can obviously not be predicted by the integral method

7.1. THE BERLIN SUBWAY 111

20

60

80

measurement (RMS)

power flow (RMS)

100 power flow (RMS), left tunnel half

power flow (RMS), right tunnel half

120

0 200 400 600 800 1000

distance to transmitter (m)

Figure 7.15: Total propagating power through the curved tunnel (U8) calculated

by the method of power ow, compared to the averaged measurement of Fig.

7.5 at fGSM = 945MHz (RMS window length: 2000 )

of power ow. In order to enable the power ow to detect such a shift of

energy, the area of analysis is split. Instead of working on the total cross

section of the tunnel, the method of power ow is now applied separately on

the left and the right halves of the cross section. The power ow in the right

half, shown by the curve with the black diamonds in Fig. 7.15, follows the

measurement quite closely. Furthermore, the power ow in the left half is

drawn in the gure. 350m, where the tunnel is approximately

On the rst

straight, the energy is equally distributed on both sides of the tunnel. In the

left curve (between 350m and 800m), the energy is focused on the right half

of the tunnel. In the following right bend, the energy is shifted from the inner

(right) side of the curve to the outer (left) side of the curve. The break-even

point is at about 900m from the transmitter, after which most of the energy

is gathered in the left tunnel half. This result again indicates the eect of

curves on the propagation behaviour, and the requirement for an appropriate

modelling approach. For the power ow, a total of1 105 rays were launched

with up to 40 reections. The spacing between the reception planes was 5m.

The simulation time for the power ow was about 3min, which translates to a

factor of 800 times below the computation time of the power trace.

112 CHAPTER 7. SCENARIOS AND APPLICATIONS

the propagation behaviour

After the investigations in the Berlin subway in the previous section, the re-

mainder of this chapter is dealing with the inuence of exterior transmitter

positions and the impact of moving vehicles on wave propagation. The dier-

ent scenarios are examined by simulations.

-10

received power normalized to P0 (dB)

-20 external transmitter position

-30

-40

-50

-60

-70

100 200 300 400 500 600 700 800 900

distance to transmitter (m)

level compared to a transmitter situated inside a straight rectangular tunnel

compared to a transmitter situated inside a tunnel is investigated by two initial

simulations in a straight rectangular tunnel. The cross section of the tunnel

is 10m wide and 5m high. For the rst simulation, an omnidirectional trans-

mitter is situated inside at the beginning of the tunnel at a height of 2:5m

in a centric position. Secondly, the same transmitter is positioned outside,

30m in front of the tunnel entrance at the same height. Figure 7.16 shows

the predicted received power levels as a function of the distance to the two

respective transmitters. The power levels are computed with image theory

and normalized to the reference power P0 (3.35), according to chapter 5.

At small distances, the power level for the interior antenna relative to the

exterior antenna is signicantly higher. Also, the uctuations of the power level

7.2. EXTERIOR ANTENNA PLACEMENT 113

are more pronounced. At larger distances, however, the two curves converge.

The reason for this is quite evident. In the vicinity of the interior antenna,

many rays impinge almost homogeneously spread over a solid angle of 2, res-

ulting in a high reception level and a highly uctuating interference pattern.

Consider an increasing distance from the transmitter. The rays reaching the

receivers via multiple reections under large angles with respect to the tunnel

axis, are highly attenuated. These rays can be interpreted as belonging to

highly attenuated propagation modes. The remaining rays almost exclusively

hit the receivers from the frontal direction, reecting under grazing incidence,

which corresponds to the less attenuated modes. For the exterior antenna pos-

ition, only rays of the latter type enter the tunnel, resulting in the convergence

of the two curves at large distances, where for the interior antenna position

also the less-attenuated rays dominate.

7.2.2.1 Description of the scenario

b=5.5m 1000m

0m

60

a=6m 5m

800m

Tx

2.5m positions

Figure 7.17: Cross section and course of the tunnel used for the simulations with

exterior transmitting antenna locations

tenna positions, the method of power ow is applied in the following. For

all simulations, the tunnel illustrated in Fig. 7.17 is used. The basic form of

the cross section is an ellipse with a horizontal half axis of a = 6m, and a

vertical half axis of b = 5:5m. 2:5m above the lowest

The oor is situated

point of the ellipse, the total height of the tunnel is 5m. The course consists

of two sections: a right bend of length l1 = 600m with radius of curvature

rc = 800m, followed by a straight section of length l2 = 1000m. The trans-

mitting antenna, a vertically polarized dipole, is located at dierent positions

in front of the tunnel entrance. The mean received power level is calculated

by the power ow approach over the entire cross section for each meter inside

114 CHAPTER 7. SCENARIOS AND APPLICATIONS

the tunnel. The walls of the tunnel are smooth with a relative permittivity of

"r = 5, the frequency in all simulations is f = 1GHz.

7.2.2.2 Antenna height

First, the transmitting antenna is situated at a distance d = 20m in front of

the tunnel entrance at three dierent heights: hT;1 = 0:5m, hT;2 = 2:5m, and

hT;3 = 4:5m. The predicted power level for each position is plotted in Fig.

7.18. Not surprisingly, the centric position ( hT;2 = 2:5m) leads to the lowest

path loss. The dierence between the curves for hT;1 and hT;3 is partially due

to the ground reected rays in front of the tunnel entrance and the asymmetric

shape of the cross-section.

-20

received power normalized to P0 (dB)

-40

-45

-50

-55

-60

0 200 400 600 800 1000 1200 1400 1600

RX distance to tunnel entrance (m)

Figure 7.18: Comparison of the mean received power level for exterior antenna

positions with dierent heights

To determine the inuence of the distance between the transmitter and the

tunnel entrance, several simulations with a spacing of1m, 20m, 50m, 80m and

120m between the transmitting antenna and the tunnel entrance are carried

out. The height of the transmitters is always hT = 2:5m. Figure 7.19 shows the

results for various distances. The larger the distance from the transmitter to

the tunnel entrance, the lower is the coupled power into the tunnel. Deep inside

the tunnel, however, the curves are converging in analogy to the observations of

7.2. EXTERIOR ANTENNA PLACEMENT 115

section 7.2.1. For a Tx-spacing of 80m, there exists a break-even point at about

800m from the tunnel entrance. Such a break-even point is only possible in

curved geometries. Its occurrence strongly depends on the actual geometry of

the tunnel and the (lateral) transmitter location, which will become apparent

in the following section.

-10

received power normalized to P0 (dB)

Tx distance to entrance: 1m

Tx distance to entrance: 50m

-30 Tx distance to entrance: 80m

Tx distance to entrance: 150m

-40

-50

-60

0 200 400 600 800 1000 1200 1400 1600

RX distance to tunnel entrance (m)

Figure 7.19: Comparison of the mean received power level for exterior antenna

positions with varying distance to the tunnel entrance

In the preceding cases, no lateral displacement of the transmitting antennas

was examined. For the scenario of Fig. 7.17, the impact of such a lateral shift

is investigated. = 20m and d = 80m from the tunnel

At two distances d

entrance, three dierent transmitting antenna locations are used, respectively.

The rst is the centric position, the second position is shifted by 5:5m to the

left, whereas the third position is shifted accordingly by the same amount to the

right. The Tx height at all positions equals hT = 2:5m. As one would expect,

the transmitting antenna at the left position results in the highest received

power level, due to the largest area of direct tunnel illumination compared to

the two other locations. Additionally, the dierence in received power between

the left and right Tx position is smaller for the larger spacing ( d = 80m).

Furthermore, no crossing of the curves is observed for the left Tx position,

like in the case of a purely straight tunnel. It can therefore be concluded that

the inuence of a curve is less pronounced for a transmitter position moved

116 CHAPTER 7. SCENARIOS AND APPLICATIONS

-20

left Tx position, (d=20m)

-30 centric Tx position, (d=20m)

right Tx position, (d=20m)

-40

-50

centric Tx position, (d=80m)

-70 right Tx position, (d=80m)

0 200 400 600 800 1000 1200 1400 1600

RX distance to tunnel entrance (m)

Figure 7.20: Comparison of the mean received power level for exterior antennas

at dierent lateral positions

To summarize, the simulations reveal a signicant inuence of the exterior

transmitting antenna position on the electromagnetic wave propagation.

In this section, the inuence of moving vehicles on the propagation channel

is considered. Three moving vehicles are situated in a straight rectangular

tunnel. The tunnel is 9:9m wide and has a height of 4:8m. A truck with

a speed of = 15m=s and two cars with v2 = v3 = 20m=s are driving

v1

through the tunnel. An omnidirectional transmitter is positioned at the tunnel

entrance 1m below the ceiling. The frequency is f = 1GHz. This scenario has

been calculated with a time interval of 1s at three consecutive time instants.

Figure 7.21 depicts the results on the rst 44m of the tunnel at a height of

1m above the oor obtained by the RDN-based power trace method. The

resolution of the employed horizontal reception plane is 0:1m 0:1m, resulting

in 440 99 = 43560 discrete points. The vehicles are modelled as oating

rectangular boxes according to section 4.4. Due to the horizontal orientation

of the reception plane, a large number of 200 million rays has been traced

to ensure sucient convergence (cf. sections 4.5 and 6.4). Because of the

PEC boundaries of the vehicles, up to 200 reections were considered in the

7.3. MOVING VEHICLES AND BROADBAND ANALYSIS 117

-55 -5

4

2 v3

0

2 v1 v2

4 t=t1

0 5 10 15 20 25 30 35 40

distance to center (m)

4

2 v3

0

2 v1 v2

4 t=t2

0 5 10 15 20 25 30 35 40

4

2 v3

0

2 v1

4 t=t3

0 5 10 15 20 25 30 35 40

distance to entrance (m)

vehicles at three consecutive time instants at f = 1GHz, v1 = 15m=s, v2 = v3 =

20m=s, (black dot at time instant t = t2 indicates Rx position for power delay

prole in Fig. 7.23)

ed resulting from reections at the vehicles' boundaries. Delay and Doppler

spreads at the second time instant t = t2 are shown in Fig. 7.22. The delay

spread is very small and thus negligible. The Doppler spread, however, is

signicant, exceeding 130Hz.

Finally, the power delay prole (PDP) at the receiver position indicated

by a black dot in Fig. 7.21 is plotted in Fig. 7.23. The height of the receiver

is 2m above the oor. The PDP is normalized to the input power PT of

the transmitting antenna, and thus obtained by the magnitude squared of

(2.59). The rst contribution corresponds to the direct path, which reaches the

118 CHAPTER 7. SCENARIOS AND APPLICATIONS

ns

4 35

dis. to cent. (m)

2 v3

0

2 v1 v2

4 t=t2

0

0 5 10 15 20 25 30 35 40

distance to entrance (m)

Hz

4 120

dis. to cent. (m)

2 v3

0

2 v1 v2

4 t=t2

0

0 5 10 15 20 25 30 35 40

distance to entrance (m)

Figure 7.22: Delay and Doppler spreads at the time instant t = t2 of Fig. 7.21

receiver with a delay of = 55:8ns. The impulses with a black square indicate

contributions by single reected rays. In total, ve rays reach the receiver via

a single reection. The rst four are either reected on the ground, the ceiling,

or at the two side walls of the tunnel. The last single reected contribution

belongs to a backward reected ray at the rear of the truck. It has the lowest

attenuation of all reected rays because of the lossless PEC boundary of the

vehicle. Due to the relatively short delay lengths for this scenario, a system

bandwidth of at least 100MHz (i.e. 10%-bandwidth) would be necessary to

resolve the important multipath components.

In general, the Doppler spread in tunnels, especially road tunnels, can reach

considerably large values and thus become important from a system point of

view. The delay spread is less critical due to the absence of signicant distant

scatterers, unlike the scenarios for urban or rural environments [Lee82].

7.4. SUMMARY 119

-60

-70

-80

-90

-100

-110

-120

-130

delay (ns)

Figure 7.23: Power delay prole (PDP) at the receiver position indicated in Fig.

7.21 at the time instant t = t2 at f = 1GHz, single reected rays marked by

black squares

7.4 Summary

In this chapter, the eectiveness of the proposed new modelling techniques

has been proven. The approaches were applied to a variety of dierent scen-

arios and were able to deliver valuable prediction results. By comparison to

measurements in the Berlin subway, it was shown that the geometry of tun-

nels, especially the cross-sectional shape and curves, have a major impact

on the propagation behaviour. In order to obtain suciently accurate path

loss predictions, it is mandatory to describe the special geometry of tunnels

in propagation modelling adequately. It was also shown that the fast fading

in tunnels cannot be characterized by standard analytical probability density

functions over the entire range of values. The densities derived from propaga-

tion modelling provide a superior t. Furthermore, the impact of exterior

transmitting antenna locations was examined. Finally, the ability of the ray-

based modelling approach to deliver broadband channel parameters has been

pointed out.

Chapter 8

Conclusions

The research conducted in this thesis deals with the prediction of electro-

magnetic wave propagation in arbitrarily shaped tunnels on a ray-optical

basis. The main motivation for this work was the inadequate knowledge of

the propagation channel in such tunnels, and the lack of a technique for its

sound description and analysis.

i.e. geometrical optics (GO) and ray-tracing techniques, were presented. It

has been shown that the available methods were not able to accurately de-

termine wave propagation in curved geometries. In order to overcome this

shortcoming a new approach has been derived. The proposed ray density

normalization (RDN) proves to be a powerful tool, able to predict electro-

magnetic wave propagation in arbitrarily shaped tunnels at frequencies in and

above the UHF frequency range ( > 300MHz) with an adequate precision. Ad-

ditionally, the failure at caustics, which is inherent to GO, can be avoided by

this new RDN approach. Furthermore, a fast, approximate way to determine

the power ux through a tunnel has been presented. The inclusion of directive

rough surface scattering in the deterministic propagation modelling, retaining

the stochastic nature of the scattering process, has been developed for the use

in system simulations. All techniques were veried theoretically using canon-

ical examples. Experimental verications in scaled model tunnels at 120GHz

proved the validity of the RDN in real three-dimensional curvature in an im-

pressive way. Finally, the applicability and performance of the approaches has

been tested in real tunnel environments. For this purpose, measurements in

the Berlin subway were conducted. It has been shown that the geometry of a

tunnel, especially the cross-sectional shape and the course, is of major impact

on the propagation behaviour and thus on the accuracy of the modelling. In

121

order to obtain reliable prediction results, the joint use of an adequate cal-

culation technique, such as the presented RDN-based methods, and a correct

modelling of the tunnel's geometry is required.

arbitrarily curved geometries with sucient accuracy on a ray-optical

basis,

is overcome,

inistic wave propagation model.

led to novel insights regarding the propagation environment of a tunnel. In

particular, the following:

and eciently by ray techniques,

tunnel geometry model.

As a result, the concepts derived in this thesis oer new perspectives for

both system analysis and coverage planning in arbitrarily shaped tunnels. For

the former realistic time series, broadband parameters, and statistical analyses

are obtained by a ray approach, including the eect of rough surface scattering.

For the latter, suciently accurate path-loss predictions may now be obtained,

including for the rst time the actual geometry of a tunnel, the inuence of

the antennas, and polarization eects.

As a eld for further research, the inclusion of leaky feeders in the modelling

can be envisaged, and again, ray techniques seem to be a promising candidate

[Mor99].

122 CHAPTER 8. CONCLUSIONS

Appendix A

of reected astigmatic tube

of rays

The principal radii of curvature r1;2 and i1;2 of the reected and incident

wavefront, respectively, are related in the following way [KP74, Bal89] (cf.

Fig. 2.4)

1 = 1 1 + 1 + 1 :

r1;2 2 i1 i2

(A.1)

f1;2

In general, f1 and f2 can be obtained by

f1;2 jj2

(

R1 R2

2 2 2 212 112

12 1i 1i + 1i 1i 4 cos i 22 12

+

1 2 1 2 jj2 R1 R2

2 2 4jj2 1=2

" #)

2 2 2 2

+ 4 cos i 22 + 12 + 21 + 11

jj4 R1 R2

; (A.2)

R1 R2

where the plus sign is used for f1 and the minus sign for f2 . R1;2 are the radii of

curvature of the reecting surface; for a convex surface R1;2 > 0, for a concave

surface R1;2 < 0. j;k and jj are determined by the principal directions of the

radii of curvature of the reecting surface ( u^1;2 ) and of the incident tube of

124 APPENDIX A. RADII OF CURVATURE AFTER REFLECTION

rays ( X^1i;2 )

i

^ ^i

[] = 1121 1222 = XX^1i uu^^1 XX^1i uu^^2 (A.3a)

2 1 2 2

jj = det[] = X^1 u^1 X^2 u^2 X^2i u^1 X^1i u^2 :

i i

(A.3b)

In addition to the radii of curvature of the tube of rays, their principal direc-

tions are modied at the reection. They are determined with the aid of the

symmetric curvature matrix for the reected wave front [Qr ]

r

[Qr ] = Q 11 Qr12 ;

Q12 Qr22

r (A.4)

2 2

1 2 cos i 22 21

11 = i + jj2 R1 + R2

Qr (A.5a)

1

Q12 =

r 2 cos i 22 12 11 21

+ R

jj2 R1

2

2

(A.5b)

1 2 cos 2

12 + 11 :

Qr = +

22

i

i2 jj2 R1 R2

(A.5c)

The principal directions X^1r;2 of the reected wave front can now be written

as

1 r

^X1r = r r1 e^1 Q12e^2

Qr22 r r

(A.6a)

2

Qr22 1r1 + (Qr12 )2

X^ 2r = k^r X^1r ; (A.6b)

with

The initial principal directions of the wavefront curvature can be chosen

1

arbitrarily , because for a point source, the radii of curvature are equal

and therefore have no specic direction.

The radii of curvature of the reecting curved surfaces, i.e. the tunnel's

walls, and their directions are given in appendix B.

1 Arbitrarily in this context still means perpendicular to each other and to the direction

of propagation.

Appendix B

Intersection algorithms

Let a ray be dened by the equation

where ~br is the base and d^r is the direction of the ray. In the following in-

tersection algorithms, the parameter is determined, such that the actual

intersection points of the ray and the respective objects can be obtained by

inserting into (B.1).

The algorithms for the plane, the rectangle, the sphere and the circular

cylinder are all based on geometrical considerations and are taken from the

literature [Gla89, Gla95, Web96]. The algorithms for the elliptical cylinder

and the torus are derived applying analytical geometry [SW98].

For the cylinders and the torus, in addition to the intersection algorithms,

the normal vectors in the intersection points, as well as the principal radii of

curvature and their directions are given, since they are needed for the calcu-

lation of the reected tube of rays (cf. section 2.3.2 and appendix A).

The orientation of the respective objects in space is chosen in accordance

to their usage in the simulation approach.

B.1 Plane

Let a plane be dened by its normal form

where ~bp is the base and n^ is the normal of the plane. The parameter , which

denes the intersection point of the plane and the ray of (B.1), is obtained by

126 APPENDIX B. INTERSECTION ALGORITHMS

(~b ~b ) n^

= p^ r :

dr n^

(B.3)

B.2 Rectangle

Rectangles are used in the simulation approach for any planar boundary (i.e.

walls, oors, ceilings, reception planes, the boundary of vehicles etc.). Let

a rectangle be dened by a base point ~b2, which is one of the corners of

the rectangle, and the two orthogonal vectors d~2;1=2 with length jd~2;1=2 j and

direction d^2;1=2.

Beforehand, the intersection point Q~ of the ray and the plane, which con-

tains the rectangle, has to be determined (cf. appendix B.1). The intersection

point ~ lies in the rectangle, if [Gla89, chap. 2]

Q

0 (Q~ ~b2) d^2;1=2 jd~2;1=2 j: (B.4)

B.3 Sphere

Spheres are used in the simulation approach for the reception spheres and

bounding boxes of vehicles. Let the sphere be dened by its central point ~bÆ

and its radius rÆ . The parameters , which dene the two possible intersection

points of the sphere and the ray of (B.1), are given by [Gla89, chap. 2], [Gla95,

vol. I, chap. 7]

= a b; (B.5)

with

q

b = rÆ2 (j~bÆ ~br j2 a2 ) ; (B.6b)

where a is the distance between the base point of the ray and the central

point of the sphere projected in the direction of the ray, b is half the distance

between the two intersection points. These two solutions only exist, if the

radical in (B.6b) is non-negative. For b = 0, there is obviously only one

possible intersection point.

B.4. CIRCULAR CYLINDER 127

Circular cylinders are used in the simulation approach for the side walls of

curved rectangular tunnel sections. Let a circular cylinder be dened by its

radius rcc and its axis, which is identical to the z -axis of a cartesian coordinate

system with direction e^z . The minimum distance from the ray to the axis of

the cylinder (subscript: rta) is given by

^

d^r e^z :

drta = ~

br

jdr e^z j

(B.7)

The distance from the base of the ray ~br to the point of minimum distance

(subscript: btp) is given by

dbtp = :

jd^r e^z j

(B.8)

the ray with the cylinder (subscript: pti) is given by

p

2

rcc d2rta

dpti =

^ h^ (B.9a)

dr

d^r e^z

h^ = jd^ e^ j e^z :

r z

d^r e^z

(B.9b)

jd^ e^ j e^z

r z

The parameters , which dene the intersection points of the ray dened by

(B.1) with the circular cylinder are therefore given by [Gla95, vol. IV, chap. 5]

In the simulation approach of section 4.1, where the circular cylinder is used

to model the side walls of a curved tunnel with rectangular cross section, only

the smaller corresponding to the minus-signis required.

Let the intersection point Q~ be given in cartesian coordinates by Q ~ =

(xq ; yq ; zq )T . Then, a vector in the direction of the normal to the circular

cylinder in Q ~ is determined by

~n = (xq ; yq ; 0)T ; (B.11)

where the plus-sign is used for the inner side of the curve and the minus-

sign for the outer one. The rst principal radius of curvature in ~ is given

Q

128 APPENDIX B. INTERSECTION ALGORITHMS

R = rcc; (B.12)

~n e^z

u^ =

j~n e^z j : (B.13)

Since a cylinder only has a 2D-curvature, the second radius of curvature in the

direction e^z is innite.

Elliptical cylinders are used in the simulation approach for straight circular,

elliptical, or arched tunnel sections. Let an elliptical cylinder be dened by

x2 2

a2

+ zb2 = 1; (B.14)

where a and b denote the half-axes of the elliptical cross section of the cylinder.

The axis of the cylinder is identical to the y-axis of a cartesian coordinate

^y . The ray equation in cartesian coordinates is given

system with direction e

by

where the index dr is used for the direction of the ray. Inserting (B.15) into

(B.14) leads to the quadric equation [SW98]

a2 2

A = x2dr + 2 zdr (B.16b)

b

a2

B = xr0 xdr + 2 zr0zdr (B.16c)

b

a 2

C = x2r0 + 2 zr20 a2 : (B.16d)

b

The parameters , which dene the intersection points of the ray dened by

(B.15) with the elliptical cylinder, are therefore given by

p

B B2 AC

= : (B.17)

A

B.6. ELLIPTICAL TORUS 129

In the simulation approach of section 4.1, where the elliptical cylinder is used

to partially model the boundary of a straight arched tunnel, only the positive

corresponding to the plus-signis needed. The second (negative) root

of (B.16a) is not of interest, because it leads to an intersection point in the

reverse propagation direction of the ray.

Let the intersection point Q~ be given in cartesian coordinates by Q ~ =

T

(xq ; yq ; zq ) . Then, a vector in the direction of the normal to the elliptical

cylinder in Q ~ (pointing to the inner side of the cylinder) is determined by

~n = ( b2 xq ; 0; a2 zq )T : (B.18)

Q

2 2 23

x

R = a2 b2 4q

a

+ zbq4 ; (B.19)

~n e^y

u^ =

j~n e^y j : (B.20)

the direction e^y is obviously innite.

Notice, that a general algorithm to intersect a ray with a quadric surface

has been proposed [Gla95, vol. III, chap. 6], which can be used alternatively

for the elliptical cylinder.

Elliptical tori are used in the simulation approach for curved circular, elliptical,

or arched tunnel sections. Let an elliptical torus be dened by

p 2

x2 + y2 rt 2

a2

+ zb2 = 1; (B.21)

where a, b denote the half-axes of the elliptical cross section and rt the radius of

the torus. Inserting the ray equation (B.15) into (B.21) leads to the 4th-order

equation (quartic)

130 APPENDIX B. INTERSECTION ALGORITHMS

with [SW98]

2 + a2 zdr 2 2

A = x2dr + ydr

b2

(B.23a)

B = 4 x2dr + ydr 2 dr 2 (B.23b)

b b

2 + a2 z 2 x2 + y2 + a2 z 2 r2 a2

C = 2 x2dr + ydr 0 0 0

b2 dr r r b2 r t

2 2 2 (B.23c)

a2 a2

D = 4 x2r0 + yr20 + 2 zr20 rt2 a2 xr0 xdr + yr0 ydr + 2 zr0 zdr

b b

2 a 2 (B.23d)

+ 8rt b2 zr0zdr

a2 a2

E = x2r0 + yr20 + 2 zr20 x2r0 + yr20 + 2 zr20 2 rt2 + a2

b b

2 a 2 2 2 2 2

(B.23e)

+ 4rt b2 zr0 + rt a :

Equation (B.22) is transformed to its reduced form by substituting =

B

+ 4A .

A reduced quartic can be solved via its resolvent cubic [BS 99], which is a third

order equation. A further substitution of =
%, with

% = 2 AC

1 2 1 2

A 3 4B ; (B.24)

3 + p + q = 0; (B.25)

with

1

p = 2 BD 4AE

1 C2

A 3 (B.26a)

q=

1

2 2 2C 3

(3A) 3 72ACE 27 AD + B E + 9BCD : (B.26b)

+

formulas [BS 99]

1 = + (B.27a)

+ p

2 = + 2 j 3

2 (B.27b)

+ p

3 = j 3;

2 2 (B.27c)

B.6. ELLIPTICAL TORUS 131

with

r

q p p p 3 q 2

=

2+ Æ; = ; Æ=

3 + 2 :

3

3 (B.28)

The parameters in (B.22) can now be obtained from the solutions of the

cubic resolvent given by

leading to

1 =

1 p

1 + 2

p p

3

B

2 4A (B.30a)

2 =

1 p

1

p

2 + 3

p B

2 4A (B.30b)

3 =

1 p p

1 + 2 + 3

p B

2 4A (B.30c)

4 =

1 p

1

p

2

p

3

B

2 4A : (B.30d)

1

Additionally to the sign convention of (B.30), the values of the radicals have

to be chosen to satisfy

D BC 3

+ 8BA3 :

p p p

1 2 3 =

A 2A2 (B.31)

In the simulation approach of section 4.1, the elliptical torus is used to partially

model the boundary of a curved arched tunnel. Therefore only the smallest

positive non-complex leads to the desired intersection point of the ray and

the elliptical torus.

Q~ = (xq ; yq ; zq )T . Then, a vector in the direction of the normal to the elliptical

torus in Q~ (pointing to the inner side of the torus) is determined by [SW98]

a2 q 2 2 T

q q

~n = xq rt x2q + yq2 ; yq rt xq + yq ; 2 zq xq + yq :

2 2 (B.32)

b

The previous equation is obtained from (B.18) by a variable transformation.

132 APPENDIX B. INTERSECTION ALGORITHMS

00 q 12 1 32

rt x2q + yq2 2

zq C

R1 = a2b2 B

@@ A +

a2 b2 A

(B.33a)

q

x2q + yq2 j~nj

R2 = = ;

n^ R^Q

q (B.33b)

rt x2q + yq2

with the corresponding principal directions

~n u^2

u^1 =

j~n u^2 j (B.34a)

Q~ e^

u^2 = ~ z :

jQ e^z j

(B.34b)

R^Q = q

1 (xq ; yq ; 0)T ; (B.35)

x2q + yq2

n^ the direction, and j~nj the length of ~n givenqby (B.32). (B.33a) follows directly

from (B.19) by substituting xq with rt x2q + yq2 , (B.33b) follows from the

theorem of Meusnier [BS 99].

+

Notice, that a general quartic roots solver has been proposed by [Gla95,

vol. I, chap. 8], which can be used alternatively for intersecting a ray with an

elliptical torus [Gla95, vol. II, chap. 5].

Appendix C

Determination of the

send-range for exterior

antenna positions

The send-range determines, in which directions the rays have to be launched for

an exterior antenna placement, in order to enter the tunnel directly or via an

intermediate ground reection (cf. section 4.3). The send-range is expressed in

spherical coordinates by the two limiting anglesmin and max in azimuth, and

by the two angles min () and max() min max .

in elevation, with

For = 0 the rays are launched in the direction of e ^z , for = 2 and = 0

they are launched in the direction of e^y , which is along the tunnel axis. e^x,

1

e^y and e^z are dened according to Figs. C.1 and C.2.

The scenario of an exterior transmitting antenna placed in front of a rectangu-

lar tunnel is depicted in Fig. C.1. Let the width of the tunnel be given by w2

and its height by h2 . Let the distance between transmitter tunnel entrance

be dT , its height hT , and its eccentricity, i.e. its distance from the middle of

the road, be T , where a positive T indicates a shift in the direction of e ^x

(to the right lane, whilst looking towards the entrance), whereas a negative

T indicates a shift in the opposite direction. The send-range in azimuth

2

is

1 Not e^x !

2 is taken in the mathematical positive sense.

134 APPENDIX C. DETERMINATION OF SEND-RANGE

P(xP,zP)

θmin h

^z

e

θmax ^y

hT e

^x

e

|∆T| dT

placement for the determination of the send-range

therefore given by

min = 2 arctan

w2

2 T (C.1a)

dT

max = arctan

w2

2 + T

: (C.1b)

dT

The send-range in elevation is rst stated in dependence of the x-component

xP of the point P in Fig. C.1, which is

p !

(T xP )2 + d2T

min (xP ) = arctan (C.2a)

h2 hT

!

h2 + hT

max (xP ) =

2 + arctan p

(T xP )2 + d2T : (C.2b)

xP = T dT tan ; (C.3)

dT

min () = arctan

(h2 hT ) cos (C.4a)

max () = + arctan

(h2 + hT ) cos :

2 dT

(C.4b)

C.2. ELLIPTICAL (ARCHED) CROSS SECTION 135

The scenario of an exterior transmitting antenna placed in front of an elliptical

(arched) tunnel is depicted in Fig. C.2. Let the horizontal half-axis of the

ellipse be given by a, the vertical half-axis by b, and the height of the road/rail

level be given by hÆ . The transmitting antenna position is given by T , dT ,

and hT , like in the previous section.

b

P(xP,zP)

^z

e

θmin ^y

e

θmax ^x

e

hT

h

|∆T| dT

antenna placement for the determination of the send-range

If the road/rail level is lower than the vertical half-axis of the ellipse, i.e.

hÆ b, the send-range in azimuth is determined by

if

a T

min = 2 arctan (C.5a)

dT

a + T

max = arctan : (C.5b)

dT

Otherwise, for hÆ > b, it is given by

0 q 2 1

a 1 hbÆ 1 T A

min;Æ = 2 arctan @ (C.6a)

dT

0 q 1

a 1 hÆ 12 + T A

max;Æ = arctan @ b : (C.6b)

dT

The send-range in elevation is rst stated in dependence of the x- and z -

components xP , zP of the point P in Fig. C.2. Furthermore, the send-range

136 APPENDIX C. DETERMINATION OF SEND-RANGE

has to be split for the direct ray entry and the ray entry via intermediate

ground reection. For the direct rays, the elevation is limited by

p !

2 2

d

min;max (

xP ; zP ) = arctan z(T+ b xPh) + hdT : (C.7)

P T Æ

For the reected rays it becomes

!

r

min;max (

xP ; zP

zP + b + hT hÆ :

) = 2 + arctan p (T xP )2 + d2T (C.8)

Using the equation of the ellipse (B.14) and (C.3), one nally obtains

0 1

B dT C

d

min;max () = arctan B

@ q C

A

b 1 (T dT2tan )2 +b hT hÆ cos

a

(C.9)

and

0 q 1

b 1 (T dT tan )2 + b + hT hÆ cos C

B a2

r

min;max () = 2 + arctan @

B

dT

C:

A

(C.10)

For min;Æ < 2 and 0 max;Æ , which is always the case for

hÆ b, the two send ranges join and can thus be bound by min = min

d and

max = max .

r

If a ceiling is present in the tunnel, an additional test has to be performed:

if the height of the ceiling

3

is larger than the actual distance from point P ()

to the street/rail level, the send range in elevation is calculated according to

(C.9) and (C.10), otherwise it is calculated according to (C.4).

3 The height of the ceiling is measured from the road/rail level to the ceiling.

Appendix D

In the following, the directional patterns of the antennas used in chapters 6

and 7 are given.

probe

The directional patterns of the D-band ( 110GHz170GHz) standard gain horn

and the waveguide probe have been measured and calculated at f = 120GHz.

The calculation was done according to the usual procedure for pyramidal horns

based on the equivalence principle techniques [Bal97, chap. 13].

For the standard gain horn, the measured and calculated patterns coincide

very well (cf. Fig. D.1). Therefore the measured patterns are taken in the

simulations of the scaled model tunnels of chapter 6. The measured patterns

of the waveguide probe, however, are heavily oscillating, although following

the same trend as the calculations (cf. Fig. D.2). The reason for this might be

an induced current in the aperture edges having a nite thickness, which leads

to additional eld components neglected by the calculation method. Since the

measured patterns itself could not be reproduced, the calculated patterns are

used in the simulations of chapter 6.

138 APPENDIX D. DIRECTIONAL ANTENNA PATTERNS

-5

-10

relative power (dB)

-15

-20

-25

-30

measured E-plane pattern

-35

0 30 60 90 120 150 180

θ (deg)

-10

relative power (dB)

-20

-30

-40

calculated H-plane pattern

measured H-plane pattern

-50

-90 -60 -30 0 30 60 90

φ (deg)

Figure D.1: Measured and calculated directional E- and H-plane patterns of the

D-band standard gain horn, f = 120GHz, gain: G = 22dBi

D.1. D-BAND HORN AND WAVEGUIDE PROBE 139

0

relative power (dB)

-2

-4

-6

-8

calculated E-plane pattern

-10 measured E-plane pattern

measured E-plane pattern +3dB

-12

0 30 60 90 120 150 180

θ (deg)

-2

relative power (dB)

-4

-6

-8

-10

-12

measured H-plane pattern

measured H-plane pattern +3dB

-14

φ (deg)

Figure D.2: Measured and calculated directional E- and H-plane patterns of the

D-band waveguide probe, f = 120GHz, gain: G = 3:6dBi

140 APPENDIX D. DIRECTIONAL ANTENNA PATTERNS

0 θ (deg)

0

30 30

-10

-20 60

-30

-40 90

120 120

150 150

180

elevation plane (vertical) amplitude pattern

0 φ (deg)

relative power (dB)

0

330 30

-10

-20 60

-30

-40 90

240 120

210 150

180

azimuth plane (horizontal) amplitude pattern

f = 947:5MHz, gain: G = 12:3dBi, source: Kathrein-Werke KG, Rosenheim,

Germany

D.3. JAYBEAM J7360 (YAGI) 141

0 θ (deg)

0 30 30

-5

-10

-15 60

-20

-25

-30 90

120 120

150 150

180

elevation plane (vertical) amplitude pattern

0 φ (deg)

relative power (dB)

0 330 30

-5

-10

-15 60

-20

-25

-30 90

240 120

210 150

180

azimuth plane (horizontal) amplitude pattern

1830MHz, gain: G = 17dBi, source: Jaybeam Ltd, Northampton, England

142 APPENDIX D. DIRECTIONAL ANTENNA PATTERNS

0 θ (deg)

0 30 30

-5

-10

60

-15

-20

-25

-30 90

120 120

150 150

180

elevation plane (vertical) amplitude pattern

0 θ (deg)

relative power (dB)

0 30 30

-5

-10

-15 60

-20

-25

-30 90

120 120

150 150

180

elevation plane (vertical) amplitude pattern

Figure D.5: Directional elevation plane (vertical) amplitude patterns of the two

=4-monopoles with circular ground plane, calculated by the method of mo-

ments (MoM) with FEKO, EMSS, Stellenbosch, South Africa

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Curriculum Vitae

Persönliche Daten:

Name: Dirk Leonidas Didascalou

Geburtsort: Hamburg

Schulbildung:

1989 Philipp-Matthäus-Hahn Gymnasium Leinfelden-

Echterdingen: Allgemeine Hochschulreife

1989 - 1991 Studium der Elektrotechnik an der Universität Stuttgart

Vordiplom

Gasthörer in Elektrotechnik und Neugriechisch

Europäisches Gemeinschaftsstudium für Elektroingenieure

Vertiefungsrichtung: Nachrichtensysteme

Electronique (ESIEE), Paris

Diplomarbeit: The potential of turbo codes in trellis coding

schemes

quenztechnik und Elektronik der Universität Karlsruhe (TH)