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**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

### 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.

**DEDICATION**

**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.

### 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.

**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.

**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

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

**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

**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

**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

**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

**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 **

**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

**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**

**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

**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

**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

**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

**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

**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

**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

**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

**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

**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

**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.

**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.

**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.

**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:

**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.

**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**

**I**

### is th eiden tity operator and

**/i**

### and

**e**

**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

**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 )**

**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.

**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**

**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:

**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 ' \***p-j/3|p-p'l*

**r***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

**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.*

**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 **

**f**

*(p{p)dS =*

**I **

**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

**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

**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 P**2** 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 (J**2**,- ^**2**) in Region 2 produces fields **

**{ E**

**{ E**

*2*

**, H**

**, H**

*2*

**) throughout Region **

**) 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

**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**

**+ 0 ^ r**

**u ■**

**u ■**

** E^ip)**

= **E^ip)**

**u **

---/ ( - n X **u**

**H2{p'))—**

**H2{p'))—**

**r^ds'**

**r^ds'**

**iL o e**

**J g****47t I p — p'**

**r **

g-*/3|p-p'l
**r**

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

**Js**

47T I p - p ' I
**Js**

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).

**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 **

**iLoe**

**J s **

**J s**

**A n \ p - p' \**

**A n \ p - p' \**

**f ****Q-iP\p-p'\**

- V X / ( - £ ( / ) X n ) — - d s '

**Js**

**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

**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.

**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,

**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.

**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

**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**

**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**

**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)

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