Numerical methods for the high speed solution of integral equations in wireless communications

159 

Loading....

Loading....

Loading....

Loading....

Loading....

Full text

(1)

LEABHARLANN CHOLAISTE NA TRIONOIDE, BAILE ATHA CLIATH TRINITY COLLEGE LIBRARY DUBLIN OUscoil Atha Cliath The University of Dublin

Terms and Conditions of Use of Digitised Theses from Trinity College Library Dublin Copyright statement

All material supplied by Trinity College Library is protected by copyright (under the Copyright and Related Rights Act, 2000 as amended) and other relevant Intellectual Property Rights. By accessing and using a Digitised Thesis from Trinity College Library you acknowledge that all Intellectual Property Rights in any Works supplied are the sole and exclusive property of the copyright and/or other I PR holder. Specific copyright holders may not be explicitly identified. Use of materials from other sources within a thesis should not be construed as a claim over them.

A non-exclusive, non-transferable licence is hereby granted to those using or reproducing, in whole or in part, the material for valid purposes, providing the copyright owners are acknowledged using the normal conventions. Where specific permission to use material is required, this is identified and such permission must be sought from the copyright holder or agency cited.

Liability statement

By using a Digitised Thesis, I accept that Trinity College Dublin bears no legal responsibility for the accuracy, legality or comprehensiveness of materials contained within the thesis, and that Trinity College Dublin accepts no liability for indirect, consequential, or incidental, damages or losses arising from use of the thesis for whatever reason. Information located in a thesis may be subject to specific use constraints, details of which may not be explicitly described. It is the responsibility of potential and actual users to be aware of such constraints and to abide by them. By making use of material from a digitised thesis, you accept these copyright and disclaimer provisions. Where it is brought to the attention of Trinity College Library that there may be a breach of copyright or other restraint, it is the policy to withdraw or take down access to a thesis while the issue is being resolved.

Access Agreement

By using a Digitised Thesis from Trinity College Library you are bound by the following Terms & Conditions. Please read them carefully.

(2)

N u m erica l M eth o d s for th e H igh S p eed S o lu tio n o f

In tegral E q uations in W ireless C om m u n ica tio n s

Eamonn Pol O Nuallain

D ep artm en t of Electronic and Electrical Engineering

University of Dublin

Trinity College

S ubm itted for the Degree of

Doctor of Philosophy

(3)
(4)

I, the undersigned, declare th a t this thesis is entirely my own work, except where otherw ise

accredited, and th a t it has no t been su b m itted for a degree in any other university or

in stitu tio n . T his thesis may be borrowed on application to the Dean of G rad u ate Studies,

University of D ublin, Trinity College and w ith the w ritten consent of th e author.

It m ay be borrowed or copied from O ctober 1st 2005 upon request w ith the perm ission of

th e L ibrarian, U niversity of D ublin, Trinity College.

Signed on the 30th of Septem ber 2000.

(5)

DEDICATION

(6)

SUMMARY

This thesis investigates the the appropriateness of integral equations for use in determ ining

electric field coverage over sub-urban terrain which is illum inated by an arb itra rily placed

tran sm itter.

Sub-urban terrain is the term used to describe undulating terrain w ith an homogeneous

or piecewise homogeneous distrib u tio n of clu tter (vegetation, buildings, rocks etc.).

The exact num erical solution for electric field coverage is given in the form of coupled

integral equations.

These are the Coupled Electric Field Integral Equations (C EFIE) or the Com bined Field

Integral Equations (CFIE).

The field coverage is evaluated numerically using these equations over different terrain

profiles consisting of various types of m aterials.

The C E F IE is reduced to a sim pler Electric Field Integral E quation (E FIE ) and its ap­

plication in determ ining field coverage is justified m athem atically and w ith field coverage

results.

Published m ethods to speed up th e calculation of coverage using the E F IE are im ple­

m ented on various profiles a t different frequencies.

These m ethods are the N atural Basis M ethod [57], th e G reen’s Function P ertu rb atio n

M ethod [58], the Fast M ultipole M eth o d /F ast Far Field A pproxim ation [61]/[6] and the

T abulated Interaction M ethod [25].

These m ethods are then com pared in term s of order of com plexity of the algorithm , accu­

racy of results, memory requirem ent and complexity of code.

I introduce my own m ethod, the Field E xtrapolation M ethod (FEM ) [62], and apply it as

w ith the published methods.

(7)

re-quireinents and have the simplest code.

I conclude it is the most appropriate fast integral equation method to calculate field cov­

erage over sub-urban terrain.

A statistical model for clutter is developed and the FEM is applied to this model over var­

ious profiles with varying degrees of clutter. The results confirm th a t the FEM is robust

in its application to this type of terrain and the extent of signal distortion due to surface

roughness mirrors the distortion of the original smooth surface.

Finally some of the better known non-integral equation methods for calculating field cov­

erage are discussed.

These are the Physical Optics approximation [52], the Parabolic Equation Method [22],

the Impedance M atrix Localisation Method [43], the Impedance Boundary Condition [21]

and the Geometric and Uniform Theory of Diffraction [55].

The salient advantages and disadvantages of each of these methods are listed.

This thesis concludes th at integral equation methods are an efficient means of estimating

field coverage and th a t the FEM is the most appropriate of these methods for application

to suburban terrain.

(8)

ACKNOLEDGEM ENTS

I would like to th an k Dr. Peter Cullen for his supervision and Teltec Ireland for funding

this research. I would also like to th an k my colleagues and the staff a t the D ep artm ent of

Electronic Engineering for making my time here an enjoyable and m em orable one.

(9)

GLOSSARY OF SYMBOLS

E

Electric Field Strength

V /m

Electric Field Incident from th e Source

V /m

Scattered Electric Field

V /m

E ^

Total Electric Field

V /m

H

M agnetic Field Strength

A /m

J, t/j

Electric C urrent Density

A /m

M; M ,

M agnetic C urrent Density

A /m

F

Electric Vector Potential

C /

A

M agnetic Vector Potential

T

(

Electric P erm ittiv ity

F /m

fo

Electric P erm ittiv ity of Free Space

F /m

fi-

M agnetic Perm eability

H /m

fio

M agnetic Perm eability of Free Space

H /m

a

Conductance

Mhos

V

Intrinsic Im pedance

Ohms

Vo

Intrinsic Im pedance of Free Space

Ohms

U!

A ngular Frequency

R ad ian s/s

Q

Charge Density

C / m ?

P

Propagation C onstant

P

Surface Position Vector to Source

P'

Surface Position Vector to O bservation Point

P"

Position Vector to Observation Point (above Surface)

i

Scattering Point

(10)

V l l

Gi

Scattering Group

Gi'

Receiving Group

I

Centre Point of Scattering Group

/'

Centre Point of Receiving Group

V

t

Transverse Gradient Operator

i

C

Arbitrary Constant

G

Green’s Function

H

Hankel Function

H

Far-Field Approximation to the Hankel Function

T

Fourier Transform Operator

h

Unit Normal Vector

I

Unit Tangent Vector

O

Order of Complexity

(11)

GLOSSARY OF ACRONYM S

EM

Electrom agnetic

CC

C om putational Cost

LHS

Left H and Side

RHS

Right Hand Side

TM

Transverse Magnetic

PEC

Perfect Electrical C onductor

FT

Fourier Transform

F F T

Fast Fourier Transform

D FT

Discrete Fourier Transform

UHF

U ltra High Frequency

2-D

Two Dimensional

3-D

Three Dimendional

(12)

CONTENTS

D e c la r a tio n

i

D e d ic a tio n

ii

S u m m a r y

iii

A c k n o w le d g e m e n t

v

G lo ssa r y o f S y m b o ls

v i

G lo ssa r y o f A c r o y n m s

v iii

1

I n tr o d u c tio n

1

2

E le c tr o m a g n e tic S c a tte r in g T h eo ry

4

2.1 M axwell’s E q u a t i o n s ...

4

2.2 The C o n stitu tiv e Relations for Linear M atter

...

5

2.3 The Vector Wave E q u a tio n ...

6

2.4 G reen’s F u n c tio n s ...

7

2.5 Boundary C o n d itio n s ...

7

2.6 The Electric and Magnetic Vector P o te n tia ls ...

8

2.7 M agnetic C u r r e n t ...

10

(13)

II

2.9 The Electric and M agnetic Field Integral E q u a t io n s ...

15

3 C o v era g e E s tim a tio n w ith In teg ra l E q u a tio n s

16

3.1 Surface Profiles, Terrain Com position and T ran sm itter F req u en cies...

16

3.1.1 Surface P r o f i l e s ...

19

3.2 The M ethod of M oments ( M O M ) ...

21

3.3 The Forward Scattering A pproxim ation ( F S A ) ...

23

3.3.1 R e s u lts ...

24

4 S c a tte r in g from D ie le c tr ic Su rfaces

26

4.1 Coupled Electric Field Integral Equations ( C E F I E ) ...

27

4.1.1 T h e o r y ...

27

4.1.2 Analysis of R e s u l t s ...

30

4.1.3 Tabulated C haracteristic D a t a ...

30

4.1.4 R e s u lts ...

31

4.2 Com bined Field Integral Equations ( C F I E ) ...

33

4.2.1 T h e o r y ...

33

4.2.2 Analysis of R e s u l t s ...

35

4.2.3 Tabulated C haracteristic D a t a ...

35

4.2.4 R e s u lts ...

36

5 T h e P E C M o d e l

38

5.1 I n tr o d u c tio n ...

38

5.2 M athem atical Justification of the PEC Model T e r r a in ...

40

(14)

I l l

5.2.2 Analysis of R e s u l t s ...

43

5.2.3 R e s u lts ...

44

6 S c a tte r in g from P E C S u rfa ces

45

6.1 I n tr o d u c tio n ...

45

6.2 The Electric Field Integral E quation ( E F I E ) ...

45

6.2.1 T h e o r y ...

45

6.2.2 Analysis of R e s u l t s ...

48

6.2.3 T abulated C haracteristic D a t a ...

48

6.2.4 R e s u lts ...

49

7 F ast In teg ra l E q u a tio n M e th o d s

52

7.1 The N atural Basis M ethod ( N B S ) ...

53

7.1.1 T h e o r y ...

53

7.1.2 Analysis of R e s u l t s ...

54

7.1.3 Tabulated C haracteristic D a t a ...

54

7.1.4 R e s u lts ...

55

7.2 The Greens Function P ertu rb atio n M ethod (G FPM )

...

56

7.2.1 T h e o r y ...

56

7.2.2 Analysis of R e s u l t s ...

59

7.2.3 Tabulated C haracteristic D a t a ...

59

7.2.4 R e s u lts ...

60

7.3 Fast M ultipole M eth o d /F ast Far Field A p p ro x im a tio n ...

63

(15)

IV

7.3.2 Theory ( F A F F A ) ...

66

7.3.3 Analysis of R e s u l t s ...

70

7.3.4 Tabulated Characteristic D a t a ...

70

7.3.5 R e su lts...

71

7.4 The Tabulated Interaction Method (TIM) ...

74

7.4.1 T h e o ry ...

74

7.4.2 Analysis of R e s u l t s ...

76

7.4.3 Tabulated Characteristic D a t a ...

77

7.4.4 R e su lts...

78

7.5 The Field Extrapolation Method ( F E M ) ...

82

7.5.1 T h e o ry ...

82

7.5.2 Analysis of R e s u l t s ...

85

7.5.3 Tabulated Characteristic D a t a ...

85

7.5.4 R e su lts...

86

7.6 Comparison of PEC Methods ...

89

7.6.1 In tro d u c tio n ...

89

7.6.2 Tabulated Characteristic D a t a ...

90

7.6.3 R e su lts...

91

8 F E M a p p lie d to R o u g h Surfaces

94

8.1 In tro d u c tio n ...

94

8.2 A Model for Clutter ...

94

(16)

V

8.2.2

R e su lts...

98

9 R e v ie w o f S c a tte r in g M e th o d s

106

9.1 The Physical Optics Approximation (PO)

...106

9.2 The Impedance Boundary Condition (IB C ) ...108

9.3 The Parabohc Equation Method ( P E ) ... 109

9.4 The Impedance Matrix Localisation Method ( I M L ) ...110

9.5 The Geometrical/Uniform Theory of Diffraction ( G T D /U T D ) ...I l l

10 C o n c lu sio n

113

A p p e n d ix A

115

A p p e n d ix B

120

(17)

VI

LIST OF FIGURES

2.1 Tw o regions of space F i an d P2 se p a ra te d by a m a th e m a tic a l surface S. R e­ gion 1 is hom ogeneous w ith (ei,jL(,i) an d R egion 2 co n tain s inh o m o g en eities th a t m ay include p erfectly co n d u ctin g m a terials. A source ( J2, M2) in R e­ gion 2 produces fields ( E2, H2) th ro u g h o u t Region 1. A second source

lo cated in R egion 1 ra d ia te s fields { E i , H i ) th ro u g h o u t R egion 1. . 13

3.1 D an ish (Jerslev) T errain Profile. T h e surface co -ordinates are given a t 5 0 A/ intervals an d are in te rjjo la te d linearly... 19

3.2 D an ish (H jorring) T errain Profile. T h e surface co -ordinates are given a t SOM intervals a n d are in te rp o la te d lin early ... 19

3.3 G erm an T errain Profile. T h e surface co-o rdin ates are given a t lO M in terv als an d are in te rp o la te d lin early ... 20

3.4 E lectric F ield coverage a t 144MHz over th e Jerslev profile... 24

3.5 E lectric Field coverage a t 970MHz over th e Jerslev profile... 24

3.6 E lectric Field coverage a t 144MHz over th e H jorring profile... 25

3.7 E lectric Field Coverage a t 970MHz over th e H jorring profile... 25

(18)

VII

4.2 E lectric F ield C overage a t 970M Hz over G erm an T errain co n sistin g of D ry

C lay (e; = 2.44 - i0 .0 9 8 )... 31

4.3 E lectric F ield C overage a t 970M Hz over G erm an T errain con sisting of D ry S an d (e~r = 2.55 — i0 .0 4 1 )... 31

4.4 E lectric F ield Coverage a t 970M Hz over G erm an T errain con sistin g of D ry Loam (e'r = 2.48 — z0.036)... 32

4.5 C o m p a rativ e p lo t of E lectric F ield C overage a t 970M Hz over G e rm a n T er­ rain consisting of D ry Clay, S and a n d L o am ... 32

4.6 E lectric F ield Coverage a t 970M Hz over G erm an T errain consistin g of D ry Clay (e; = 2.44 - i0 .0 9 8 )... 36

4.7 E lectric Field Coverage a t 970MHz over G erm an T errain consisting of D ry S and {e~r = 2.55 - z0.041)... 36

4.8 E lectric Field Coverage a t 970MHz over G erm an T errain consisting of D ry Loam (e'r = 2.48 — i0.036)... 37

4.9 C om p arativ e p lo t of E lectric Field Coverage a t 970M Hz over G erm an T er­ rain consisting of D ry Clay, Sand and L o am 37 5.1 C om p arativ e P lo t of E lectric Field Coverage a t 970M Hz over G erm an T er­ rain consisting of D ry Clay 44 5.2 C om p arativ e P lo t of E lectric Field Coverage a t 970M Hz over G erm an T er­ rain consisting of D ry Clay, Sand, Loam an d th e P E C M odel 44 6.1 E lectric F ield coverage a t 144MHz over th e Jerslev profile... 49

6.2 E lectric F ield coverage a t 970MHz over th e Jerslev profile... 49

6.3 E lectric F ield coverage a t 144MHz over th e H jo rring profile... 50

6.4 E lectric Field coverage a t 970M Hz over th e H jorrin g p rofile... 50

(19)

VIII

6.6 E lectric F ield coverage a t 970M Hz over th e G erm an profile... 51

7.1 C o m p a rativ e P lo t of th e E lectric F ield C overage a t 970M Hz over D an ish (Jerslev) T e rra in ... 55

7.2 C o m p a rativ e P lo t of th e E lectric F ield C overage a t 970M Hz over G erm an T e rra in ... 55

7.3 E lectric F ield Coverage a t 144MHz over th e Jerslev profile... 60

7.4 E lectric F ield Coverage a t 970M Hz over th e Jerslev profile... 60

7.5 E lectric F ield Coverage a t 144MHz over th e H jo rring profile... 61

7.6 E lectric F ield Coverage a t 970M Hz over th e H jorring profile... 61

7.7 E lectric Field Coverage a t 144MHz over th e G erm an profile... 62

7.8 E lectric Field Coverage a t 970M Hz over th e G erm an profile... 62

7.9 FAFFA sc a tte rin g geom etry. T he u p p er diag ram shows th e angles 4>u and (f)w su b ten d ed by group G i w ith th e horizontal. T h e lower diag ram shows groups G i a n d Gi< of A/4 d iscretisations of th e surface, th e ir cen trep o in ts I and I ', th e p o sitio n vector p i — pi> connecting th e m and a rb itra ry p o in ts i an d j on th e respective g ro u p s... 66

7.10 E lectric F ield Coverage a t 144MHz over th e Jerslev profile... 71

7.11 C o m p arativ e P lo t of th e E lectric Field Coverage a t 970M Hz over th e Jerslev profile... 71

7.12 C o m p arativ e P lo t of th e E lectric Field Coverage a t 144MHz over th e H jor­ ring profile... 72

i 7.13 C om p arativ e P lo t of th e E lectric Field Coverage a t 970M Hz over th e Hjor-' ring profile... 72

(20)

IX

7.15 C om parative P lo t of the Electric Field Coverage a t 970MHz over th e G er­

m an profile...

73

7.16 TIM scatterin g geometry showing an incident plane wave on a flat segm ent

of surface being considered to scatter a m ultitude of plane waves...

74

7.17 C om parative P lo t of the Electric Field Coverage a t 144MHz over th e Jerslev

profile...

78

7.18 C om parative P lot of the Electric Field Coverage a t 970MHz over th e Jerslev

profile...

78

7.19 C om parative P lo t of the Electric Field Coverage a t 144MHz over the

Hjor-ring profile...

79

7.20 C om parative P lo t of the Electric Field Coverage a t 970MHz over the

Hjor-ring profile...

79

7.21 C om parative P lo t of the Electric Field Coverage a t 144MHz over th e G er­

m an profile...

80

7.22 C om parative P lo t of the Electric Field Coverage a t 970MHz over the G er­

m an profile...

80

7.23 C om parative P lo t of the Electric Field Coverage a t 970MHz over th e G er­

m an profile w ith plate length of lA ...

81

7.24 Electric Field Coverage a t 144MHz over the Jerslev profile...

86

7.25 C om parative P lot of the Electric Field Coverage a t 970MHz over the Jerslev

profile...

86

7.26 C om parative P lo t of the Electric Field Coverage a t 144MHz over th e

Hjor-ring profile...

87

(21)

X

7.28 Comparative Plot of the Electric Field Co\erage a t 144MHz over the Ger­

man profile...

88

7.29 Comparative Plot of the Electric Field Co\erage at 970MHz over the Ger­

man profile...

88

7.30 Comparative Plot of the Electric Field Coveiage at 144MHz over the Jerslev

profile...

91

7.31 Comparative Plot of the Electric Field Coveiage at 970MHz over the Jerslev

profile...

91

7.32 Comparative Plot of the Electric Field Coverage at 144MHz over the

Hjor-ring profile...

92

7.33 Comparative Plot of the Electric Field Coverage at 970MHz over the

Hjor-ring profile...

92

7.34 Comparative Plot of the Electric Field Coverage at 144MHz over the Ger­

man profile...

93

7.35 Comparative Plot of the Electric Field Coverage at 970MHz over the Ger­

man profile...

93

8.1

lOOM of Smooth and Rough German Terrain. Amplitude and frequency of

the

' S i n d

function are l.OM and 1.0 rad /s respectively...

98

8.2

lOOM of Smooth and Rough German Terrain. Amplitude and frequency of

the ’Sine’ function are 5.0M and 1.0 rad /s respectively...

98

8.3

lOOM of Smooth and Rough German Terrain. Amplitude and frequency of

the ’Sine’ function are l.OM and 5.0 rad /s respectively...

99

8.4 Comparative Plot of the Electric Field coverage at 144MHz over the rough

(22)

XI

8.5

C om parative P lo t of th e Electric Field coverage a t 970MHz over th e rough

G erm an profile... 100

8.6 C om parative P lo t of th e Electric Field coverage a t 144MHz over the rough

G erm an profile. A m plitude and frequency of the

'Sine'

function are

I M

and

bra d/s respectively...101

8.7 C om parative P lo t of the Electric Field coverage a t 970MHz over the rough

G erm an profile. A m plitude and frequency of the

'Sine'

function are IM

and

5r a d /s

respectively...101

8.8 C om parative P lo t of th e Electric Field coverage a t 144MHz over the rough

G erm an profile. A m plitude and frequency of the

'Sine' function are 5Af

and

I r a d / s

respectively...102

8.9 C om parative P lo t of th e Electric Field coverage a t 970MHz over th e rough

G erm an profile. A m plitude and frequency of the

'Sine' function are 5M

and

\ r a d / s

respectively...102

8.10 C om parative Plot of th e Field Coverage a t 144MHz over Rough G erm an

Terrain. A m plitude and frequency of the 'Sine'

function are I M

and

I r a d / s

respectively... 103

8.11 C om parative P lo t of the Field Coverage a t 970MHz over Rough G erm an

Terrain. A m plitude and frequency of the

' S i n d

function are IM and

I r a d / s

respectively... 103

8.12 C om parative P lo t of the Field Coverage a t 144MHz over Rough G erm an

Terrain. A m plitude and frequency of the

'Sine' function are IM and bra d/s

respectively...104

8.13 C om parative P lo t of the Field Coverage a t 970MHz over Rough G erm an

(23)

XII

8.14 C om parative P lot of the Field Coverage at 144MHz over R ough G erm an

Terrain. A m plitude and frequency of t h e

' S nc'

function are 5M and

I r a d / s

respectively... 105

8.15 C om parative P lo t of the Field Coverage a; 970MHz over R ough G erm an

Terrain. A m plitude and frequency of th e

'S-,nc'

function are 5M and

I r a d f s

respectively... 105

10.1 Two regions of space Fi and F

2

separated bj a m ath em atical surface S. Re­

gion 1 is homogeneous w ith (ei,yUi) and Region 2 contains inhom ogeneities

th a t may include perfectly conducting materials. A source ( J

2

, M

2

) in Re­

gion 2 produces fields (£'

2

,-^

2

) throughou', Region 1.

A second source

(24)

XIII

LIST OF TABLES

4.]

C om putational Features of th e C E F I E ...

30

4.^

C om putational Features of th e C F I E ...

35

6.1

C om putational Features of the E F I E ...

48

6.2 C om putation tim es for Electric Field Coverage at 144MHz over the Jerslev

(Danish), Hjorring (Danish) and G erm an profiles...

48

6.3 C om putation times for Electric Field Coverage a t 970MHz over the Jerslev

(Danish), Hjorring (Danish) and G erm an profiles...

48

7.1

C om putational Features of th e N B S ...

54

7.2 C om putation times for Electric Field Coverage at 970MHz over the Jerslev

(Danish) and German profiles...

54

7.3

C om putational Features of the G FPM ...

59

7.4 C om putation times for Electric Field Coverage at 144MHz over the Jerslev

(Danish), Hjorring (Danish) and German profiles...

59

7.5 C om putation times for Electric Field Coverage at 970MHz over the Jerslev

(Danish), Hjorring (Danish) and Germ an profiles...

59

7.6

C om putational Featxires of the F A F F A ...

70

(25)

XIV

7.8

C om putation tim es for Electric Field Coverage a t 970MHz over th e Jerslev

(D anish), H jorring (Danish) and G erm an profiles...

70

7.9

C om putational Features of the TIM (Main P rog ram /T ab ulatio n Program )

77

7.10 C om putation tim es for Electric Field Coverage a t 144MHz over the Jerslev

(Danish), Hjorring (Danish) and Germ an profiles...

77

7.11 C om putation tim es for Electric Field Coverage a t 970MHz over the Jerslev

(Danish), Hjorring (Danish) and Germ an profiles...

77

7.12 C om putational Features of the FEM ...

85

7.13 C om putation tim es for Electric Field Coverage at 144MHz over the Jerslev

(Danish), Hjorring (Danish) and G erm an profiles...

85

7.14 C om putation tim es for Electric Field Coverage at 970MHz over the Jerslev

(Danish), Hjorring (Danish) and G erm an profiles...

85

7.15 C om putational Features of the FAFFA, TIM and FE M ...

90

7.16 C om putation times for Electric Field Coverage a t 144/970MHz over the

(26)

X V

LIST OF SYMBOLS

E F IE

Electric Field Integral E quation

M FIE

M agnetic Field Integral E quation

MoM

M ethod of Moments

PE C

Perfect Electric Conductor

C FIE

Combined Field Integral E quation

RWG

Rao-W ilton-Glisson

CG

C onjugate G radient

C G -F F T

C onjugate G radient-Fast Fourier Transform

E F T

Fast Fourier Transform

FAFFA

Fast Far-Field A pproxim ation

TIM

Tabulated Interaction M ethod

UHF

U ltra High Frequency

ANIM

A nalytical Interaction M ethod

NBS

N atural Basis Set

(27)

Chapter 1

1

INTRODUCTION

The need to effectively com m unicate using wireless systems is not easy to satisfy diie to

]»andwidtli lim itations and to the complex behaviour of electrom agnetic rad iatio n as it

propagates, scatters and attenuates.

Scattering and atten u atio n are more pronounced a t higher frequencies posing severe prob­

lems in providing ubiquitous coverage for mobile com m unications providers whose band-

v id th is at the upper end of the U ltra High Frequency (UHF) radio wave spectrum

-(300 - 3000MHz).

Ih e increased dem and for b e tte r d a ta transm ission integrity, which is a current phe-

ronienon in developed countries in the advent of the personal mobile phone and fax ma-

ciines, m eans the provision of adequate field coverage via surface based tra n sm itte rs will

bx:ome an ever more challenging task.

I) is conceivable, if not indeed likely, th a t these mobile devices will ultim ately provide the

services of a PC which will only exacerbate this demand.

Given th e above dem ands, there is a relatively new interest in the use of integral ecjuations

ii. estim ating field coverage because they are a form of the exact m ath em atical solution

fcr this problem - which is to calculate the field coverage given by an a rb itrarily based

transm itter over an arb itrary surface profile.

To be in a position to offer a good mobile service, an effective tra n sm itte r netw ork m ust

be in place. For this network to be effective it m ust be derived from a suitable planning

process (the alternative is an ad-hoc tra n sm itte r placem ent).

T ie purpose of this thesis is to aid this fast and accurate planning process by providing

fast and accurate solutions for field coverage over sub-urban terrain.

The solution to this problem is slow by its very nature.

(28)

2

\ concom itant fast and accurate planning tool can be provided using integral equation

nethods, the development of which is the focus of this thesis.

The propagation and scattering problem itself is expressed exactly as an integral equation

md this provides the ideal startin g point.

Elements of this integral equation can,

a priori,

be elim inated by v irtue of th eir negligible

:ontribution.

t is this feature of the integral equation form ulation which makes it a suitable environm ent

tor finding fast com putational m ethods which do not significantly com prom ise accuracy.

] am concerned w ith the com putation of UHF radio wave propagation in a su b urban en-

nronm ent w ith application to cellular radio systems planning.

By suburban environm ent I mean undulating terrain w ith an homogeneous or piecewise

lomogeneous distribution of clu tter (vegetation, buildings, rocks etc.).

I wish to develop a fast, efficient determ inistic approach to this problem , taking into ac-

o u n t clu tter as a param aterised random (probabilistic) distrib u tio n of scatterers on the

sir face.

I; nuist be pointed out th a t the integral equations describing the problem can be w ritten

ir differential form and so the integral equation m ethods presented here have th eir analogy

ii the difi'erential domain.

Irtegral equations are by their n atu re easier to conceive, being as they are, sim ply sum-

natio n s. They are therefore preferable to use in the search for fast m eans of solving this

problem.

T ie exact num erical solution of the integral equations for the problem would take days to

sclve for a couple of kilometres of terrain, even on a high speed com puter. Clearly this is

the reason th a t until recently integral equation m ethods were not popular.

A i im p o rtan t model is commonly used w ith this problem which speeds up th e algorithm

significantly. This is the PEC model.

Here the surface is assumed to be a PE C which allows use of the much sim pler integral

equation for PECs.

(29)

3

Both give com parable results w ith the exact solution in the case of grazing incidence which

impLes a surface based tra n sm itte r on terrain which is gently undulating.

It should be noted th a t the com puted coverage results presented in this thesis are derived

assuming the atm osphere and terrain have, respectively constant electrical p erm ittiv ities

(eo i- used for th e atm osphere).

Atmospheric effects such as poor weather, hum idity and convection currents, to nam e b u t

a few, will result in greater atten u atio n , fading and scintillation effects respectivley [1], [2].

Wet or snow/ice-covered terrain will yield different coverage results th a n when th e terrain

is dry [21],

Polarization effects are ignored; all scattered radiation is taken to have the same polariza­

tion as th e incident field [5], and the possibility of resonance effects having a significant

effect on coverage is discounted as being unlikley a t the frequencies considered (144 and

970MHz).

However, th is phenomenon would likely become a significant problem as service providers

are forced to move up the UHF band. Here, raindrops and snowflakes would be likley to

form resonant cavities in which case rain/snow fall may cause effective blackouts [3].

This thesis a tte m p ts to provide the reader with an intuitively acceptable m eans w ith

which to und erstand integral equations in electrom agnetics and th e fast m eans used to

solve them.

Research presented in this thesis justifies the PE C model and provides th e fastest and

m ost efficient m ethod to date to calculate the field over sub-urban terrain.

(30)

Chapter 2

4

ELECTROM AGNETIC SCATFERING THEORY

M odern E lectro m ag n etic S catterin g T h eo ry is f o u n d d on th e laws of E lectro m ag n etism , w hich are M axw ell’s E q u atio n s [5] an d th e constitutive rela tio n s for m a tte r.

2.1

M axw ell’s Equations

M axw ell’s E q u atio n s are given here (a tim e depender.ee of 6**^^ is assum ed ) [52]:

V X H{ p) = l u Di p ) + J( p) (2.1)

V X E{p) = - iluB {p) (2.2)

V • Di p ) = q{p) (2.3)

V • B{ p) = 0 (2.4)

- w here uj is th e ra d ia tio n frequency ( r ad/ s ) , q is charge den sity an d p is a p o sitio n vector. T h ey are respectively th e laws of A m pere a n d M axvell, F arad ay a n d Lenz, G au ss an d B iot an d S avart.

A m p e re ’s law was corrected by M axwell to include the D isplacem ent C u rre n t te rm i u D . T h e above po sition-only vectors are com plex qu an tities an d are re la te d to th e o riginal p o sitio n a n d tim e d ep end en t q u an tities by:

(31)

2.2. T h e C o n s titu tiv e R e la tio n s for L inear M a tte r

5

2.2

T h e C o n s t it u t iv e R e la tio n s for L in ear M a tte r

The following consituitive relations apply for linear m atter - Harrington[19]:

^

^

d E

d^E

D - e E + e i — + £ 2 - ^ + ( 2 .6 )

d H

_ d^ H

,

,

5 -

+

/ i i —

+

+ ...

(2.7)

d E

d^ E

,

,

J - a E + a i —

+ a 2 - ^ +

(2,8)

and can be approximated by:

D = f E

(2.9)

D =

fiH

(2.10)

,7 =

d E

(2.11)

- where the tilde superscript denotes a complex quantity and the bar superscript denotes

a tensor.

Empirically:

e = e + (2.12)

to

M atter is termed ’simple’ if the above complex quantities denoted with the tilde superscript

can be replaced by scalars.

(32)

2.3. T h e V c to r W ave E q u a tio n

6

2.3

T h e /e c to r W ave E q u ation

The Vector Wve Equations in

E

and

H

are derived from Maxwell’s equations by taking

the curl of (2.) and using the vector identity:

V

X

V ) 1 / = V V • 1 / - V V

(2.13)

- where

V

is ai arb itrary vector.

The Vector W,ve E quation for the Electric Field is:

V

X

f/r^7

X

E(p) — Lo'^e ■

E{p)

=

iu>J{p)

(2-14)

- where /I and^, the m agnetic perm eability and electric perm ittivity, are rank 2 tensors.

In an homogeiaous isotropic m edium the Vector Wave equation becomes:

V

X

V

xE{p) — €^E{p)

=

iu)fiJ{p)

(2.15)

- which can bevvritten

V ^ E { p ) - e'^E{p) = - i c o p ^ I ■

J{p)

(2-16)

-where

I

is th eiden tity operator and

/i

and

e

are scalars.

By the D ualityPrinciple [5], (2.16) can be w ritten in term s of

H

and

M.

The Vector W a’e Equation is comprised of three coupled scalar wave equations.

The derivation )f (2.16) is to be found in Chew[6].

The integral eqiations th a t describe the electrom agnetic scattering problem we are ab o u t

(33)

24. G r e e n ’s F u n ctio n s

7

2 4

G r e e n ’s F u n c tio n s

A G reen’s Function is a physical system response to a Dirac d elta type pulse.

Tie scalar G reen’s function,

g{ p, p) ,

is the solution to the Scalar Wave E quation where

tie current is the Dirac D elta function and it is:

(2.17)

- vliere /? is the wave number.

Tie Dyadic G reen’s Function, so nam ed because it is a dyad or rank two tensor, is the

anilogous solution to the Vector Wave Equation.

Fo: an homogeneous isotropic m edium it is:

linear, the solution to an electrom agnetic problem can be obtained by superposition.

2.5

B o u n d a r y C o n d itio n s

At the interface between two m aterials having relative perm ittivities e^i,

6 ^ 2

and perme-

abi.ities

Hri,

and where there are no sources, the following conditions can be shown to

ho ll a t the boundary [5]:

a { p , p )

(2.18)

Derivations of (2.17) and (2.18) can be found in Chew[

6

] Because the Wave Equations are

n

X

[El — E

2

)

= 0

(2.19)

n

X

{Hi — H

2

) —

0

(2.2 0)

n

• (e,.i£'i — €r

2

-£'

2

) —

0 (2.2 1)

n

■ { f l r l H i - /ir2^2) = 0

- where

h

is the outw ard norm al unit vector.

(34)

2.6. T h e E le c tr ic an d M a g n e tic V e cto r P o te n tia ls

8

2.6

T h e E le c tr ic an d M a g n e tic V e c to r P o te n t ia ls

C onsider E lectric an d M agnetic Fields an d Ha due to th e E lectric C u rre n t J only.

T h en [16] shows how:

VV • /I + /3M

iLoe

- w here A is th e solutio n to:

(2.23)

\ / ^ A + P^ A = - J (2.24)

w hich gives:

(

2

-

25

)

J v 47T I p - p' I

or

- 4 = / J s { p ' ) j ~ ,--- (2.26) J s 47 T I p - p' I

- d ep en d in g on w h eth er one is solving th e Wave E q u a tio n in a volum e or on a surface.

By exact an alogy w ith th e above it can be shown th a t

Ef = - S / X F (2.27)

- w here F is th e solu tion to:

V ^ F + P ^ F = - M (2.28)

w hich is:

F = My { p') - — j---— dv (2.29)

J v 47T I p - p' I

or

r p-iP\p-p'\

F = / M ,( p ') — ---- d s ' (2.30)

Js

47T I p - p' I

A an d F are th e M agnetic an d E lectric V ector P o te n tia ls respectively.

By S u p erp o sitio n th e to ta l E lectric F ield is then:

(35)

2.6. T h e E le c tic an d M a g n e tic V e cto r P o te n tia ls

9

Hence the general olution for the field rad iated by a surface current is:

w 4-/?2

r

r , - i p \ p - p ' \ r p-j/3|p-p'l

E^{p) =

--- ^

r^ds' - V

X

/

Ms{p' )—

- d s '

(2.32)

ve

J s

47t I p - p' I

J s

A ' n \ p - p' \

Use of the Vector P'tentials lead to interm ediate differential equations which are uncoupled

and simple. T h a t i, each com ponent, of say A, depends on th e corresponding com ponent

of J only.

Use of

A or F

dos not decouple the original vector wave equation as can the use of

M agnetic C urrent, vhich will be discussed in detail later.

Hence the only advmtage in the use of A and F is in avoiding the use of complex operators

(36)

2.7. M a g n e tic C urren t

10

2 .7

M a g n e tic C u rren t

T h e co n ce p t of M agnetic C u rre n t is to be used sh o rtly in th e Surface E quivalence P rin cip le

an d th e C oupled an d C om bined F ield In teg ral E qu atio ns.

M ag n etic c u rren t does n o t exist in th e sense of electric cu rren t. I will devote som e space

here to ex plain w h at it is an d w here an d why it should be used.

F irs t it shoiild be n o ted th a t m ag n etic cu rren t can n o t be iso lated because m ag n etic charges

do n o t exist.

Secondly, m ag n etic cu rren t is used only as a m a th e m a tic a l convenience. I t is an a lte rn a tiv e

re p re se n ta tio n for electric cu rren t.

In sh o rt, M agnetic C u rren t (M ) is an a lte rn a tiv e rep resen tatio n in F a ra d a y ’s Law for the

E lectric C u rre n t ( J ) w hich ap p ears in A m p ere’s Law. T h ere is th u s a tra n sfo rm th a t

rela tes b o th . Use of eith er or b o th is a q uestion of convenience since th e so lu tio n for th e

field is unaffected.

F rom V an B ladel [17] volum e electric a n d m agn etic cu rren ts are re la te d by:

A U p) = - ( — )V X J, {p) (2.33)

lUJt

a n d

U p) = ( — )V X M „(p) (2.34)

liUli

A good exam ple of th e usefulness of M agnetic C u rre n t is in m ag n etic m a te ria ls w here

ro ta tin g electric cu rren ts exist.

A p p ly in g th e C o n tin u ity E q u atio n to these electric cu rren ts yields:

V • Jt, = iuiq = 0 (2.35)

- since th e re is no net inflow /outflow of charge.

Since any vector field is specified by its curl an d divergence free com po nen ts, th is ty p e of

electric c u rre n t can be com pletely described by a m ag n etic c u rren t M .

If th e o rig in al electric cu rren t is a fu n ctio n of two o rth o g o n al vectors - say x a n d y - th e n

th e equ ivalent m ag netic cu rren t will b e a function of th e z vector only.

(37)

2 .7 . M a g n e t i c C u r r e n t 11

E q u a tio n for My an d converting th e resu lt to th e equ ivalent Jy using (2.33).

T h e a lte rn a tiv e to th is would be having to solve th e E lectric V ector W ave E q u a tio n for

Jy, w hich m eans solving two coupled scalar wave eq uatio ns.

H ence, w here Jy is ro ta tin g , does n o t diverge an d is in v a rian t in one d irectio n , use of

m a g n e tic c u rre n t allows one avoid th e use of dyadic an aly sis in solving for Jy.

It sh o u ld also be p o in ted o u t th a t if J varies in all d irectio n s x, y a n d z, conversion

to m a g n e tic c u rren t will resu lt in having to solve two coupled scalar (m ag n etic) wave

eq u a tio n s as opposed to th ree (electric).

In su m m ary , a p p ro p ria te use of m ag n etic c u rren t (i.e. w here V ■ Jy = 0) reduces th e

d im en sio n ality of solving th e V ector W ave E q u atio n by one.

T h e re la tio n s for surface electric an d m ag n etic cu rren ts Jg an d Mg are as follows [17]:

Ms = ( — ) n X J8s (2.36)

iuj€

an d

Js = - i — ) n X Mbs (2.37)

iue

w here 6^ is defined by th e functional:

< 6 s , ( p > =

f

(p{p)dS =

I

8s4>{p)dV (2.38)

J s J v

T h e n o tio n of 6s stem s from th e D irac D elta F un ctio n, from whose definition th e conversion

of a line, surface or volum e integral to a p o in t value of a fun ction is possible. T h is gives

one th e m ean s to describe a d istrib u tio n on a half-line or plane.

O n th e o th e r h an d , th e definition of bg enables one to convert a volum e in teg ral to a

surface in teg ral. T his gives one a to o l to describe a d is trib u tio n on a surface w hich is n o t

necessarily p la n ar.

T h e usefulness of surface m ag netic c u rren t follows from its rela tio n to surface electric

cu rre n t. T h a t is, th a t a surface electric c u rren t w hich is a fu n ctio n of tw o o rth o g o n al

vectors m ay be described l)y a m ag n etic cu rren t w hich is a function of th e th ir d only.

It is im m ed iately clear in th is circum stan ce th a t if th e V ector W ave E q u a tio n is to be

solved on a surface, ap p ro p ria te use of surface m ag n etic c u rren t gives th e sam e ad v an tag es

(38)

2 .7 . M a g n e t ic C virrent 12

The key word here is ’ap p ro p riate’.

Above it was the use of vohime m agnetic current to describe a ro ta tin g electric current

w ith zero divergence.

Here its use is in describing a surface electric current, which is a function of two orthogonal

vectors, in term s of a surface m agnetic current which is a function of the th ird orthogonal

(39)

2.8. T h e Su rface E qviivalence P r in c ip le

13

2.8

T h e Surface E q u ivalen ce P rin cip le

Consider the situation depicted in the figure below. Here we have two regions of space Fi

and F

2

separated by a m athem atical surface

S.

One of the regions is unbounded.

Region 1 is homogeneous with electric and magnetic permeabilities ei and /ii where Region

2 contains inhomogeneities th a t may include perfectly conducting materials.

OO

Figure 2.1: Two regions of space Fi and P2 separated by a m athem atical surface S. Region 1 is homogeneous with and Region 2 contains inhomogeneities that may include perfectly conducting materials. A source (J2,- ^2) in Region 2 produces fields

{ E

2

, H

2

) throughout Region

1. A second source located in Region 1 radiates fields throughout Region 1.

A source electric and magnetic current

{J

2

, M

2

)

is located in Region 2 and radiating in the

presence of the inhomogeneities produces fields

E

2

and

H

2

throughout Region 1.

We postulate also a second source (Ji,

Mi)

located in Region 1 but radiating fields

Ei

and

Hi

in an homogeneous space having constitutive parameters Ci and

ni.

The fields of both sources satisfy the Sommerfeld radiation condition [4] on the boundary

(40)

2.8. T h e S u rface E q u iv a len ce P r in c ip le

14

T he Surface Equivalence P rin cip le can be w ritte n as follows - P ete rso n [16]:

V V •

+ 0 ^ r

u ■

E^ip)

=

u

---/ ( - n X

H2{p'))—

r^ds'

iL o e J g 47t I p — p'

r

g-*/3|p-p'l

- u • V X { - E2{ p ' ) x f i ) -— j---— ds' (2.39)

Js

47T I p - p ' I

T h is eq u a tio n is a sta te m e n t th a t th e field p ro d u ced by {J2, M2) a t som e lo c a tio n o u tsid e

of R egion 2 can be expressed in th e form of an in te g ra tio n over th e ta n g e n tia l fields on th e

surface of Region 2.

T h e eq u atio n is of th e form of (2.32) w hich is th e general so lu tion for th e field r a d ia te d by

a volum e or a surface cu rren t.

For th is reason we identify th e ta n g e n tia l co m po nent of th e m a g n etic field a t th e surface as

a surface electric c u rren t a n d th e ta n g e n tia l co m po nent of th e electric field a t th e sm'face

as a surface m ag n etic cu rren t.

T h e Surface Equivalence P rin cip le m akes it reasonable to p o s tu la te t h a t th e field sc a tte re d

from a surface can be com pletely specified according to an eq u atio n of th e form of (2.39).

(41)

2.9. T h e E le c tr ic an d M a g n e tic F ie ld In te g r a l E q u a tio n s

15

2.9

T h e E lectric and M a g n etic F ield In tegral E q u a tio n s

The derivation of Electric Field Integral Equation (E FIE ) is based on the following pos­

tu lates - [5], [16];

E ‘ ' ( f ) = E ' ( p ) + E^ ( p )

(2.40)

T h at is, th e to ta l observed field a t a point equals th e sum of the field incident from the

source plus the field reradiated or scattered by the surface

-and

V V . _ l/5 2 r - i ! 3 \ p - p ' \

E^ { p) =

/ ( _ n X

- d s '

iLoe

J s

A n \ p - p' \

f Q-iP\p-p'\

- V X / ( - £ ( / ) X n ) — - d s '

Js

“I J r I P - P I

(2.41)

- which m eans the scattered field can be expressed in term s of th e tangential com ponents

of the to ta l electric and m agnetic fields a t the boundary, which is the Surface Equivalence

Principle.

Identifying the surface integrals as Electric and M agnetic Vector Potentials, and substi­

tu tin g (2.32) into (2.40) yields:

- A + P^A

E ' i p ) = E ^ { p ) -

- V

X

f

iLoe

(2.42)

Taking the tangential com ponents of both sides yields

f V V • A +

n X E (p) = —Ms { p) — n X

lioe

(2.43)

- which is the EFIE.

The M agnetic Field Integral Equation (M FIE) is derived in a sim ilar fashion yielding:

r v v - A +

n X H \ p ) = J s { p ) — n X V X F

t u t

(2.44)

For a P E C the E F IE and M FIE are simplified by noting the tangential com ponent of the

(42)

Chapter 3

16

COVERAGE ESTIM ATION W ITH INTEG RAL EQ UATIO NS

As outlined in the introduction, fast Integral E quation m ethods are th e focus of th is thesis,

hi this chapter, a formalised approach to solving integral equations exactly is exam ined

along w ith the results this m ethod gives w ith the Forward S cattering A pproxim ation.

These results are com pared w ith the m easured results in superim posed plots,

la C h ap ter 7, fast Integral E quation m ethods applied to the terrain profiles given here are

exam ined chronologically, giving the reader an understanding of the evolution of the latest

m ethods. The coverage results these fast m ethods give, will be com pared w ith the results

given here.

3.1

Surface Profiles, Terrain Com position and Transm itter Fre­

quencies

The sxirface profiles used in this thesis are:

1) llK m of gently undulating Danish (Hjorring) terrain. Profiles and m easurem ents sup­

plied by Prof. Anderson of Alborg University.

2) 6Km of gently undulating Danish (Jerslev) terrain. Profiles and m easurem ents sup­

plied by Prof. Anderson of Alborg University.

3) 3.8Km of m ountainous G erm an terrain provided by Deutsche Telekom AG (no m ea­

surem ents available).

The D anish profiles will be used to illustrate:

1) The Forward Scattering Model.

(43)

3.1.

S u rfa ce P r o files, T erra in C o m p o s itio n and T r a n sm itte r F r e q u e n c ie s

17

The German profiles will be used to illustrate:

1) The CFIE and CEFIE methods.

2) The EFIE (PEC) methods.

3) The effect of surface roughness on field coverage.

Dielectric terrain compositions considered are:

1) Dry clay with relative complex electrical permittivity 2.44-z0.098 at 970MHz.

2) Dry sand with relative complex electrical permittivity 2.55-z0.041 at 970MHz.

3) Dry loamy soil with relative complex electrical permittivity 2.48-^0.036 at 970MHz.

These values were obtained from [19].

The transm itter frequencies used in this thesis are 144MHz and 970MHz.

1) A 144MHz transm itter frequency will be used to illustrate all EFIE (PEC) methods.

2) A 970MHz transm itter frequency will be used to illustrate all methods.

The transm itters will be placed 10.4M above the starting point of the Danish profiles

and 52M above the German profile.

fn all cases the surface will be irradiated with

T

radiation emanating from the source,

an infinite lA carrying strip transverse to the 2-D surface profile.

The discretisation length used for the numerical evaluation of the integral equations is A/4

and A/15 (A is the wavelength of the radiation emanating from the source) for PEC and

dielectric surfaces respectively.

The resultant field will be observed 2.4M above the terrain profiles.

In the FAFFA, TIM and FEM group sizes of 100.0 and 3.0 times the radiation wavelength

are used for the Danish and German profiles respectively unless otherwise stated.

In the TIM the tabulation is performed at intervals of

tt

/SOO.

All computations are coded in

and run on an IBM RS6000 computer. Com puta­

tion times are given in seconds for all methods. This information is in itself immaterial

since com putation times will vary depending on coding language, coding methodology,

(44)

com-3 .1 . S u r fa c e P r o file s , T e r r a in C o m p o s it io n a id T r a n s m it t e r F r e q u e n c ie s 18

parisoii.

The im p o rtan t feature of each m ethod is the order rf com plexity of the solution and the

m em ory requirem ents. Coding complexity is also as;essed. This is a relative assessment

and som ew hat subjective. The relative availability oi library code such as th e F F T is not

given consideration.

(45)

3.1. S u rfa ce P r o file s, T errain C o m p o s itio n and T r a n sm itte r F r e q u e n c ie s

19

3 .1 .1

S u rfa ce P ro files

(U

•4—4 0)

35

30

25

20

15

10

V U ndulatm ^ D anislt^^^it^rn

..y

■ A

1000

2000

0

3000

4000

5000

6000

M eters

F iguie 3.1: D anish (Jerslev) T errain Profile. T he surface co-ordinates are given at SOM intervals and are in te rp o la te d linearly.

40

35

30

^

25

S

20

15

10

0

2000

4000

6000

8000

10000

12000

M eters

G e n t l y ] j n d u k i t i n | , D a n i s h T e r r a i n

(46)

M

e

te

r

s

3.1 . S u rfa ce P r o file s, T errain C o m p o s itio n an d T r a n sm itte r F r e q u e n c ie s

20

400

350

300

250

200

150

M ountainous G e n n a n F-^rofile

1000

1500

2000

2500

3000

3500

4000

500

Meters

(47)

5.2. T h e M e t h o d o f M o m e n t s (M O M )

21

3.2

T h e M e t h o d o f M o m e n t s ( M O M )

This is a general m e th o d for reducing fu nction al eq u atio n s defined in a lin ear space to

n a t r i x eq u atio n s - H a rrin g to n [18].

The E F IE is such a fu n ctio n al equ ation.

Consider th e form of th e E F IE over a surface S w here source a n d o bserv atio n p o in ts are

f a n d p' resp ectiv ely (I have ignored th e co n stan t facto r /??//4 in th e E F IE for sim p licity):

D iscretising a 2 — D surface into N segm ents allows us express th e E F IE as a su m m a tio n :

N

= (3-2)

n = l

is th e M O M , we express J( pn) as a p ro d u c t ang{Pn), w here a„ is a co n sta n t over th e

dom ain. g{p'n) is referred to as a basis function.

We fu rth e r enhance c o m p u ta tio n a l freedom by ta k in g th e inn er p ro d u c t over each d o m ain

w ith fu n ctio n s called w eighting functions.

T h a t is, we allow ourselves app ly w eights a t will should th is be helpful to us in speeding

up th e su m m atio n s:

J { p' ) G{ p, p' ) ds ' (3.1)

To have th e freedom to ev alu ate th is su m m a tio n a n d arrive a t a general a lg o rith m , w hich

/ w{p) ■ E \ p ) d s = f w{p) ■ ^ J { p ' ^ ) G { p , p ' ^ ) A s ' ^ d s

Js

Js

1

(3.3)

n = l

(3.4)

(3.5)

If we assu m e a„ is co n sta n t over th e interval th e n a„ can be ta k en o u tsid e th e in te g ra tio n

to give:

N

E {Pn^dSji ^ • g { p ' J G { p , p ' J A s y s n (3.6)

(48)

3.2. T h e M e t h o d o f M o m e n t s ( M O M ) 22

There are

N

such equations, describing the incident field on the surface intervals, which

can be expressed in m atrix form:

[1/| =

[Z\\J\

(3,7)

- where

[V],

the supply vector, contains the fields from the source incident on each surface

interval.

[J],

the current vector contains the surface currents on these intervals and

[Z],

known as

the impedance m atrix, relates the two.

In 2 — D the rank of

[Z]

is

0{N' ^)

and in 3 —

it is

0{ N^) .

Should we choose the same weighting and testing functions, we have Galerkin’s method.

This often leads to simpler and symmetric impedance matrices [52].

If we choose the weighting functions to be Dirac Delta functions, we have what is known

Figure

Figure 2.1: Two regions of space Fi and P2conducting materials. A source (J separated by a mathematical surface S
Figure 2 1 Two regions of space Fi and P2conducting materials A source J separated by a mathematical surface S. View in document p.39
Figure 3.3: German Terrain Profile. The surface co-ordinates are given at lOM intervals and are
Figure 3 3 German Terrain Profile The surface co ordinates are given at lOM intervals and are . View in document p.46
Figure 3.5: Electric Field coverage at 970MHz over the Jerslev profile.
Figure 3 5 Electric Field coverage at 970MHz over the Jerslev profile . View in document p.50
Figure 3.6: Electric Field coverage at 144MHz over the Hjorring profile.
Figure 3 6 Electric Field coverage at 144MHz over the Hjorring profile . View in document p.51
Figure 4.1: Terrain scattering geometry. The source, an infinite lA  carrying strip in the z direction, is placed above the starting point
Figure 4 1 Terrain scattering geometry The source an infinite lA carrying strip in the z direction is placed above the starting point. View in document p.52
Figure 4.4: Electric Field Coverage at 970MHz over German Terrain consisting of Dry Loam
Figure 4 4 Electric Field Coverage at 970MHz over German Terrain consisting of Dry Loam . View in document p.58
Figure 5.1: Com parative Plot of Electric Field Coverage at 970MHz over Germ an Terrain con­
Figure 5 1 Com parative Plot of Electric Field Coverage at 970MHz over Germ an Terrain con . View in document p.70
Figure 7.5: Electric Field Coverage at 144MHz over the Hjorring profile.
Figure 7 5 Electric Field Coverage at 144MHz over the Hjorring profile . View in document p.87
Figure 7.7: Electric Field Coverage at 144MHz over the German profile.
Figure 7 7 Electric Field Coverage at 144MHz over the German profile . View in document p.88
Figure 7.9: FAFFA scattering geometry. The upper diagram shows the angles 0^/ and 4>u’ sub­tended by group G; with the horizontal
Figure 7 9 FAFFA scattering geometry The upper diagram shows the angles 0 and 4 u sub tended by group G with the horizontal. View in document p.92
Figure 7.10: Electric Field Coverage at 144MHz over the Jerslev profile.
Figure 7 10 Electric Field Coverage at 144MHz over the Jerslev profile . View in document p.97
Figure 7.12: Comparative Plot of the Electric Field Coverage at 144MHz over the Hjorring profile.
Figure 7 12 Comparative Plot of the Electric Field Coverage at 144MHz over the Hjorring profile . View in document p.98
Figure 7.14: Comparative Plot of the Electric Field Coverage at 144MHz over the German profile.
Figure 7 14 Comparative Plot of the Electric Field Coverage at 144MHz over the German profile . View in document p.99
Figure 7.16: TIM scattering geometry showing an incident plane wave on a flat segment of surface [jeing considered to scatter a multitude of plane waves.
Figure 7 16 TIM scattering geometry showing an incident plane wave on a flat segment of surface jeing considered to scatter a multitude of plane waves . View in document p.100
Table 7.9: Com putational Features of the TIM (Main Program /Tabulation Program )
Table 7 9 Com putational Features of the TIM Main Program Tabulation Program . View in document p.103
Figure 7.17: Comparative Plot of the Electric Field Coverage at 144MHz over the Jerslev profile.
Figure 7 17 Comparative Plot of the Electric Field Coverage at 144MHz over the Jerslev profile . View in document p.104
Figure 7.19: Comparative Plot of the Electric Field Coverage at 144MHz over the Hjorring profile.
Figure 7 19 Comparative Plot of the Electric Field Coverage at 144MHz over the Hjorring profile . View in document p.105
Figure 7.21: Comparative Plot of the Electric Field Coverage at 144MHz over the German profile.
Figure 7 21 Comparative Plot of the Electric Field Coverage at 144MHz over the German profile . View in document p.106
Figure 7.23: Comparative Plot of the Electric Field Coverage at 970MHz over the Germ an profile
Figure 7 23 Comparative Plot of the Electric Field Coverage at 970MHz over the Germ an profile . View in document p.107
Table 7.12; Computational Features of the FEM
Table 7 12 Computational Features of the FEM. View in document p.111
Figure 7.24: Electric Field Coverage at 144MHz over the Jerslev profile.
Figure 7 24 Electric Field Coverage at 144MHz over the Jerslev profile . View in document p.112
Figure 7.26: Comparative Plot of the Electric Field Coverage at 144MHz over the Hjorring profile.
Figure 7 26 Comparative Plot of the Electric Field Coverage at 144MHz over the Hjorring profile . View in document p.113
Figure 7.28: Comparative Plot of the Electric Field Coverage at 144MHz over the German profile.
Figure 7 28 Comparative Plot of the Electric Field Coverage at 144MHz over the German profile . View in document p.114
Table 7.15: Com putational Features of the FAFFA, TIM and FEM.
Table 7 15 Com putational Features of the FAFFA TIM and FEM . View in document p.116
Figure 7.30: Comparative Plot of the Electric Field Coverage at 144MHz over the Jerslev profile.
Figure 7 30 Comparative Plot of the Electric Field Coverage at 144MHz over the Jerslev profile . View in document p.117
Figure 7.32: Comparative Plot of the Electric Field Coverage at 144MHz over the Hjorring profile.
Figure 7 32 Comparative Plot of the Electric Field Coverage at 144MHz over the Hjorring profile . View in document p.118
Figure 7.34: Comparative Plot of the Electric Field Coverage at 144MHz over the German profile.
Figure 7 34 Comparative Plot of the Electric Field Coverage at 144MHz over the German profile . View in document p.119
Figure 8.10: Comparative Plot of the Field Coverage at 144MHz over Rough German Terrain
Figure 8 10 Comparative Plot of the Field Coverage at 144MHz over Rough German Terrain. View in document p.129
Figure 8.12: Comparative Plot of the Field Coverage at 144MHz over Rough German Terrain
Figure 8 12 Comparative Plot of the Field Coverage at 144MHz over Rough German Terrain. View in document p.130
Figure 8.14: Comparative Plot of the Field Coverage at 144MHz over Rough German Terrain
Figure 8 14 Comparative Plot of the Field Coverage at 144MHz over Rough German Terrain. View in document p.131

References

Updating...