Adaptive Beamforming With a Focal-Fed Offset Parabolic · PDF file ·...

Adaptive Beamforming With a Focal-Fed Offset Parabolic Reflect or Antenna

by

Jason Duggan

A thesis submitted to the Department of Elect r i d and Computer Engineering

in conformity with the requirements for the degree of Master of Science (Engineering)

Queen's University Kingston, Ontario, Canada

April 1997

Copyright @ Jason Duggan, 1997

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Abstract

The bandwidth and power restrictions on the communications system between a

geostationary satellite and a mobile terminal are severe. Adaptive antenna array

processing, which exploits the spatial distribution of the mobile users, is seen as one

possible solution to these problems. Adaptive m a y processing allows the satellite to

maximize the signal received from a desired user and null out the contributions of

co-channel interfering users based on their distinct locations.

The adaptive antenna Literature has used a linear or planar array structure on

which to perform the m a y processing. The most common structure referred to

is the uniform linear array (ULA) which is composed of a line of identical antenna

elements spaced an equal distance apazt. Linear and planar array structures are more

generally referred to as direct radiating arrays (DRA). On a geostationary satellite

offset parabolic reflector antennas are the dominant antenna type. This is due to the

reflector's high gain which is critical in a geostationary satellite to Earth Link. This

thesis considers the use of an offset reflector with an m a y feed as a way of combining the high gain of the reflector with the spatial filtering ability of the antenna array.

This type of antenna is referred to as a multiple beam antenna (MBA). The question is

whether or not adaptive algorithms designed with the direct radiating array stmcture

in mind will work on a multiple beam antenna.

A signal model is developed which is general enough to encompass both the DRA structure and the MBA structure. The key quantity is the steering vector which

describes the response of the antenna m a y to a plane wave arriving from a given

angle. While the steering vector for a uniform linear m a y is analytically derivable

with knowledge of the wavelength of the plane wave, and the spacing of the antenna

elements, the steering vector for a MBA must be found numerically.

One of the necessary steps to finding the steering vector of the MBA is the ability

to determine the secondary field of each of the antenna elements. The theory of

reflector antenna analysis is developed utilizing the physical optics approximation

and the Fourier-Bessel method of solution. This theory is embodied in a computer

program which is capable of generating the antema pattern of the reflector antenna

with an array feed. This computer program is used to find the secondary field of each

of the antenna elements. The program is also used to study some of the properties

of offset parabolic reflectors including the effects of tapering of the feed's primary

pattern and lateral displacements of the feed from the focal point.

Using the principie of reciprocity and the secondary field of each of the antenna

elements in the array the steering vector for the MBA is numericdyeduated. With

the steering vector and the general signal model optimum minimum mean-squared

error (MMSE) array processing is investigated demonstrating the ability of the MB A to perform adaptive beamforming. One of the conceptually simplest adaptive dgo-

ri t hms, the Direct Matrix Inversion (DMI) algorithm is described and its performance

on a MBA is demonstrated.

A special class of adaptive algorithms called cyclic beamforming algorithms ase

introduced. These algorithms exploit the inherent cyclostationarity in the desired

user's signal to extract it in the presence of spectrally incoherent interference and noise. One of the highly desireable properties of these algorithms is that they do

not require either a reference signd, or, knowledge of the directions of arrival of the

users of the system. A particular cyclic bedorming algorithm, the Least Squares

Self-coherence Restord (LS-SCORE) algorithm, is demonstrated to work on a MBA and its performance is studied.

The conclusion of this thesis is that adaptive m a y processing can be performed

by using the m a y as a feed to an offset parabolic reflector. Adaptive algorithms will

work on the multiple beam antenna structure without any changes.

Acknowledgements

I would like to sincerely thank Dr. Peter J. McLane for his generous support and

guidance throughout my graduate career. I would also like to thank Eric Amyotte

of SPAR Aerospace for his time and patience in answering my many questions on

my visits down to SPAR. I thank the the Telecommunications Research Institute of

Ontario for their financial support and industry Canada, Communications Research

Centre, Ottawa for their financial support and background material on cyclic beam-

forming. Specifically, I would like to thank Chun Loo of CRC and Mark Rollins,

formerly of CRC for their assistance. Thanks also goes to a l l of the staff in the TRIO office for their assistance,

I would like to thank all the friends I have made during my time as a graduate

student. In partidax I would like to thank Ken Gracie, Dave Young, Oguz Sunay, Dave Parrtnchych, Jean Au, Joubin Kaximi, Alex Seyoum, William Wan, Chris Tan aad Walid Ahmed. I thank them for their friendship, their support and for always

being available to help me out when I needed it.

I would like to thanlc my friends who not only supported me and gave me their

friendship but also gave my a place to stay when I traveled back to Kingston to work

on my thesis. Thank you to Jeniffer Sartor, Jason Pantarotto, Sarah Jones, Me1 Clancy, and Finola Shankar.

To my family, thank you for your love, support and encouragement throughout

my life.

To Deirdre, thank you for your love and your belief in me.

Table of Contents

Abstract

Acknowledgements

Table of Contents

List of Symbols xii

1 Introduction 1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Motivation 1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Literature Survey 4

. . . . . . . . . . . . . . . . . . . . 1.2.1 Adaptive A n t e ~ a Arrays 4

. . . . . . . . . . . . . . . 1.2.2 Adaptive Multiple Beam Antennas 5

. . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Contributions of Thesis 6

. . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Presentation Outline 8

2 Beamforming Theory 10

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction LO

. . . . . . . . . . . . . . . . . . . . . . . . 2.2 The Unlfom Linear Array 11

. . . . . . . . . . . . . . . . . . . . . . . 2.3 The Wideband Signal Model 13

. . . . . . . . . . . . . . . . . . . . . . 2.4 The Narrowband Signal Model 14

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Beamforming 19

. . . . . . . . . . . . . . . . . . . 2.6 Statistically Optimum Beamforming 21

. . . . . . . . . . . . . . . . . . . 2.6.1 The Minimum MSE Solution 21

2.7 An Example: Optimum Combining With A ULA . . . . . . . . . . . 23

. . . . . . . . . . . . . 2.8 Beadorrming With a Multiple Beam Antenna 29

3 Offset Parabolic Reflector Antenna Analysis 30

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.2 Geometry of the Offset Reflector . . . . . . . . . . . . . . . . . . . . . 32

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 CoordinateSystems 35

3.4 Reflector Antenna Analysis . . . . . . . . . . . . . . . . . . . . . . . 36

3.5 Derimtion of the Radiation Integral . . . . . . . . . . . . . . . . . . . 38

3.6 Solution of the Radiation Integral . . . . . . . . . . . . . . . . . . . . 41

3.7 The Physical Optics Approximation . . . . . . . . . . . . . . . . . . . 41

3.8 Evaluation of the Radiation Integral . . . . . . . . . . . . . . . . . . 42

3.9 The Fourier-Bessel Method . . . . . . . . . . . . . . . . . . . . . . . . 46

3.10 S u m m q of Fourier-Bessel Method . . . . . . . . . . . . . . . . . . . 49

3.11 Implementation and Verification . . . . . . . . . . . . . . . . . . . . . 50

3.12 Properties of Offset Reflectors . . . . . . . . . . . . . . . . . . . . . . 57

3.12.1 Edge Taper, Aperture Efficiency and the Effect of the q Pasameter 57

3.12.2 Reflector Antenna Pattern Characteristics of

Off-Focus Feeds . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.13 Extension to an Array Feed . . . . . . . . . . . . . . . . . . . . . . . 66

3.14 Calculation of the Directivity . . . . . . . . . . . . . . . . . . . . . . 69

4 Beamforming With An Offset Parabolic Reflector Antenna 72

. . . . . . . . . . . . . . . . . . . . . . . . 4 . 1 Introduction and Overview 72

. . . . . . . . . . . . . 4.2 Beamforming With a Multiple Beam Antenna 73

4.3 Optimum Combining With an Offset Reflector Antenna . . . . . . . . 76

. . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Adaptive Algorithms 84

. . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Direct Matrix Inversion 85

4.6 Simulation of the Direct Matrix Inversion Algorithm . . . . . . . . . 86

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Discussion

5 Cyclic Beamforming Algorithms on a Multiple Beam Antenna

5.1 Introduction and Overview . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Cydostationary Signal Analysis . . . . . . . . . . . . . . . . . . . . . 5.3 Cyclic B h d Spat i d Filtering Algorithms . . . . . . . . . . . . . . . .

5.3.1 Cyclic B h d Spatial Filtering Algorithms - A Brief Overiew . 5.3.2 LS-SCORE . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.4 Cyclostationarityof BPSK . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Simulation of LS-SCORE . . . . . . . . . . . . . . . . . . . . . . . . .

6 Conclusions

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Conclusions

6.2 Recommendations for Further Study . . . . . . . . . . . . . . . . . .

Bibliography

A Antenna Basics

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Time-Harmonic Fields and MaxweU's Equations . . . . . . . . . . . . The Magnetic Vector Potential . . . . . . . . . . . . . . . . . . . . . . Solution of the Complex Wave Equation . . . . . . . . . . . . . . . . Antenna Near and Far Fields . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plane Waves

Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Far-Field Representation of the Antenna Radiation Field . . . . . . Radiation Intensity and Antenna Pat terns . . . . . . . . . . . . . . .

A.10 Antenna Gains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A . l l The Antenna As A One-Port Device . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.12 Reciprocity

A.13 Feed Approximation by cosq(8) . . . . . . . . . . . . . . . . . . . . . 137

B Coordinate ~ansforrnations 140

B.1 Transformation From One Cartesian Coordinate System to Another . 140

B.2 Transformations Between Spherical, Cylindrical and Cartesian Coor-

dinates.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

B.2.1 Transformations Between the Rectangular and Cylindrical Ce

ordinates. . . . . . . . . . . . . . . . . - . . . . . . . . . . . . 148

B .2.2 Transformations Between the Cylindrical and Spherical Coor-

dinates . . . . - . . . . . . . . . . . . . . . . . . . . . . . . . . 151 8.2.3 Transformations Between the Rectangular and Spherical Coor-

dinates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

C Proofs in the Development of the Fourier-Bessel Method 156

C.1 Proof #1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 C.2 Proof #2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

C.3 Bessel Function Definition . . . . . . . . . . . * . . . . . . . . . * . . 158

Vita 160

List of Figures

. . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Spatial filtering structure 2

. . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Spatial atering structure 11

2.2 Uniform linear array (ULA) . . . . . . . . . . . . . . . . . . . . . . . 12

. . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Beamforming overview 20

. . . . . . . . . . . . . . . 2.4 Optimum combining with the ULA: case 1 26

. . . . . . . . . . . . . . . 2.5 Optimum combining with the UCA: case 2 27

. . . . . . . . . . . . . . . 2.6 Optimum combining with the ULA: case 3 28

. . . . . . . . . . . . . . . . . . . . . . 3.1 Offset reflector geometry: 3D 32

3.2 Offset reflector geometry: 2D . . . . . . . . . . . . . . . . . . . . . . 33

3.3 Coordinate systems used in reflector analysis . . . . . . . . . . . . . . 35

. . . . . . . . . . . . . . . . . . . . . 3.4 ParraIel rays approximation .. 39

. . . . . . . . . . . . . . . . . . . . 3.5 Projected aperture of the reflector 47

. . . . . . . . . . . . . . . . . 3 -6 Reflector analysis: P-series convergence 53

. . . . . . . . . . . . . . 3.7 Reflector analysis: (m. n)-series convergence 55

. . . . . . . . . . . . . . . . . . . . . . 3.8 Reflector analysis: verification 56

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9 Effect of edge taper 59

. . . . . . . . . . . . . . . . . . 3.10 Displacement of feed from focal point 61

3.11 Effect of lateral displacement of feed (F/D, = 0.625) . . . . . . . . . 63

3.12 Effect of lateral displacement of feed ( F / D p = 0.625) . . . . . . . . . 64

3.13 Effect of lateral displacement of the feed ( F / D p = 0.4) . . . . . . . . 65

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.14 The feed plane 70

Antenna transmitting when excited by a unit amplitude source . . . . 74

. . . . . . . . . . . . . . . . . . . . . Antenna receiving a plane wave 75

Hexagonal configuration of the feed array on the feed plane . . . . . . 77

MBA optimum combining: case 1 . . . . . . . . . . . . . . . . . . . . 79




. . . . . . . . . . . . . . . . . . . . . . . . DM1 simulation on a MBA 89

5.1 Simulation of LS-SCORE on a MBA . . . . . . . . . . . . . . . . . . 103

A.1 A current I over an incrementd length At . . . . . . . . . . . . . . . 119

. . . . . . . . . . . . . . . . . . . . . . . A.2 Source and field coordinates 121

. . . . . . . . . . . . . . . . . . A.3 The 3 radiating regions of an antenna 122

. . . . . . . . . . . . . . . . . . . . . . . . . . . . A.4 Linear polarization 125

A.5 Circular polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

. . . . . . . . . . . . . . . . . . . . . . . . . . . A.6 Elliptical polarization 127

A.7 The vectors b, d, and k at the far field point r . . . . . . . . . . . . . 129

. . . . . . . . . . . . . . . . . . . . . . . h.8 An example antenna pattern 131

A.9 An antenna fed through a transmission line or waveguide . . . . . . . 134

h.10 Reciprocity case 1: antenna transmitting . . . . . . . . . . . . . . . . 135

. . . . . . . . . . . . . . . . . . A.11 Reciprocity case 2: antenna, receiving 136

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . A 12 COS~(@) feed model 139

B.1 2 Cartesian coordinate systems oriented arbitrarily with respect to one

another . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

B.2 2-D: 1 coordinate system rotated with respect to another . . . . . . . 144

B.3 Transformation of Cartesian coordinates: step 1 . . . . . . . . . . . . 145



. . . . . . . . . . . . . . . . . . . B.6 The recta.ngu1a.r coordinate system 149

. . . . . . . . . . . . . . . . . . . . 8.7 The cylindrical coordinate system 150

. . . . . . . . . . . . . . . . . . . . . B.8 The spherical coordinate system 151

. . . . . . . . . . . . . . . . . . . . . . . . . . B.9 l i intermsof iand& 152

. . . . . . . . . . . . . . . . . . . . . . . . . . B.10 f in termsof f iand~ 153

. . . . . . . . . . . . . . . . . . . . . . . . . . . B.11iintermsofiande 154 A . . . . . . . . . . . . . . . . . . . . . . . . . . . B.l2$intermsofiandB 155

List of Symbols

the gamma function

power reflection coefficient

carrier offset

carrier offset for desired user

carrier offset for interfering user

incremental length of a current element

incremental volume

temporary scalar variable

projected aperture

the surface of the reflector

an angle

correlation matrix of received signal vector

sample correlation matrix

correlation matrix of desired user's received signal vector

correlation matrix of the it h interferer's received signal vector

correlation matrix of the noise vector

xii

P c u l

angle to angular center of reflector

ange to aperture center

angle to lower edge of reflector

half of the angle subtended by the reflector

angle to upper edge of reflector

cycle frequency

Eularian angie

direction cosines

angular parameter

Eularian angIe

direction cosines

propagation constant

Eularian angle

direct ion cosines

cyclic temporal correlation function

temporal cross-correlation function of y ( t ) and r ( t )

Kronecker delta function

complex permittivity

fkee-space permittivity

error signal

integral used in Fourier-Bessel development

free space impedance

aperture efficiency

spillover efficiency

taper efficiency

efficiency of antenna

distance parameter

unit vector in 0 direction

angle in a spherical coordinate system from the z axis in the associated

cartesian coordinate system

beam scan angle

feed tilt angle

direction of arrival of the desired user (the variable 0 in a spherical coor-

dinat e system)

direction of arrival of the ith user's signal (the variable 0 in a spherical

coordinate sys tern)

angle from z, axis in spherical coordinate system centered at the source

expected main beam posit ion (0 variable)

scalar constant

wavelength

complex permeability

xiv

fie-space permeability

parameter used in the directivity calculation

complex variable

3.14159265359

radial variable in a cylindrical coordinate system

signal-twnoise ratio of the desired user's signal

signal-to-noise ratio of the intedering user i's signal

radial variable in polar coordinate system centered at the aperture center

(integration coordinates)

distance variable in polar coordinate system to define the location of feed

m with respect to feed n

a scalar constant

conductivity

free-space conductivity

variance of the noise

effective area of the antenna aperture

physical area of the antenna aperture

tilt angle of polarization

time lag

represents x, y, z in expression of scalar components of a vector

unit vector in 4 direction

XV

angle in a spherical coordinate system from the s axis in the x - y plane

of the associated cartesian coordinate system

direction of asrival of the desired user (the variable # in a spherical CCF

or dinat e system)

direction of asrival of the ith user's signal (the variable $ in a spherical

coordinate system)

angle fiom xc axis in polar coordinate system centered at the aperture

center (integration coordinates)

angle from x, axis in spherical coordinate system centered at the source

expected main beam location (4 variable)

phase of x component of b

phase of y component of b

angular miable in polar coordinate system to define the location of feed

m with respect to feed n

electric scalar potential

interelement phase shift of desired user's signal in a uniform linear array

phase of the ith sensor to a unit-amplitude plane wave from the direction

of the desired user

polarization parameter

temporaq scalar variable

random phase offset of desired user's signal

random phase offset of the ith interfering user

xvi

phase offset of reference signal

polarization parameter of source

polarization parameter of field pattern

angular frequency

magnetic vector potential

matrix used in finding Pmd

temporary scalar variable

amplitude of desired user's signal

amplitude of interfering user's signal

complex constant

amplitude of reference signal

x component of A

y component of A

z component of A

temporary vector variable

radius of the reflector

complex wave amplitude in a transmission line or waveguide (forward

direct ion)

data sequence

wave amplitude at z = 0 in forward direction in the receiving situation

xvii

polarization parameter of the source

polarization parameter of the field pattern

wave amplitude at z = 0 in the forward direction in the transmitting

situation

B magnetic flux density

B complex magnetic flux density

B coordinate transformation matrix

B square area which encloses the aperture

BL:), Bkt) parameters used in directivity calculation

BDF beam deviation factor

b polarization vector

6.4 complex wave amplitude in a transmission h e or waveguide (reverse

direct ion)

LHCP component of b

RHCP component of b

wave amplitude at z = 0 in the reverse direction in the receiving situation

polarization parameter of the source

polarization parameter of the field pattern

wave ampiitude at z = 0 in the reverse direction in the receiving situation

component of b in x direction

component of b in y direction

cross polarization unit vector

coordinate transformation matrix

plane wave amplitude

parameter used in the directivity calculation

conventional cross-covariance function of u(t) and v ( t )

control vector in LS-SCORE:

electric flux density

complex electric flux density

directivity in all polarizations in direction k

directivity in direction k and in polarization R

coordinate transformation matrix

aperture diameter

parent paraboloid's diameter

parameters used in the directivity cdculation

polarization vector

spacing between antenna elements

offset height to aperture center

electric intensity

complex electric field intensity

secondary field due to feed i in direction k

xix

Ei(k,a) component of the secondary field due to feed i in direction k aod in

polarization R

E, E field in receiving situation

I% complex electric field of the source

E f complex electric field of source i

Et E field in transmitting situation

E coordinate transformation matrix

EzVtl Fourier coefficients

ET edge taper

e(x7 Y) describes the E-field transverse variations in a transmission Line or waveg-

uide

F(k) amplitude vector in direction k

F(k, R) amplitude vector components for feeds i = 1, . . . , NE in direction k and

in the polarization R

Fi(k) amplitude vector of secondary field due to feed i

&(k, R) component of Fi in direction k and polarization R

F ( k , R) component of amplitude vector in the direction k and in polarization R

F focal length of reflector

f effective aperture distribution function

f frequency

fa baud rate

fc carrier frequency

G a vector function

G,, GU, G, x , y, z components of G

Gr, Gyt, Gi sf, y', zf components of G

G(k) gain in direction k

G(k, b) gain in direction k and polarization b

Gnalized (k) realized gain in direction k

Gmnlized (k, b) realized gain in direction k and polarization b

temporary scalar function

periodic version of f

gain of ith sensor to a unit-amplitude plane wave from the direction of

the desired user (Bd, q5d) and at frequency f

magnetic intensity

complex magnetic field intensity

complex secondary magnetic field due to feed i

H field in receiving situation

complex magnetic field of the source

complex magnetic field of source i

H field in transmitting situation

offset distance to lower edge of reflector

h(G Y) describes the H-field transverse variations in a transmission Line or waveg-

uide

h temporary scalar fundion

I vector function used in Fourier-Bessel development

1 current

Imd(k) radiation intensity in direction k

Imd(k, b) radiation intensity in the direction k and in polarization b

I the unit dyad

t index variable

3 electric current density

J complex electric current density

Jz surface Jacobian

Jo Bessel function of 0th order

JP Bessel function of order p

j J=r

k wave vector

k unit vector in direction of k

k0 wave vector in direction of main beam peak

k wavenumber

L the number of incident signals

index variable

number of terms in m-series

index variable

baseband signal of desired user

baseband signal of ith interfering user

normal to reflector

number of terms in n-series

number of antenna elements

number of interfering signals

number of samples used in DM1 window

number of samples

unit normal to reflector surface

index variable

LHCP unit vector

RRCP unit vector

maximum value in the pseries

power of desired user's signal

total power leaving region bounded by surface of integration

complex power leaving a region

power of interfering user's signal

XXiii

noise power

power received at element i from desired signal

total time-averaged power radiated by the antenna

power transmitted

power of a baseband signal x ( t )

index variable

pulse shape

electric charge density

complex electric charge density

feed parameter

q parameter in the E-plane

q parameter in the E-plane of feed i

q parameter in the H-plane

q parameter in the H-plane of feed i

reference polarization unit vector

cyclic autocorrelation matrix of x ( t )

cyclic conjugate correlation matrix of x (t )

cyclic autocorrelation function of the desired signal

conventional cross-correlation function of u(t ) and v(t )

conventional autocorrelation function of x (t )

XXiv

P=(r) cyclic autocorrelation function of x(t )

cyclic conjugate correlation hmction of x ( t )

vector to far-field point

unit vector in radial direction

vector to the field point

vector from source to integration point

vector from focal point to integration point

reference signal

radial variable in a spherical coordinate system (corresponding to the

field cartesian coordinate system)


primed cartesian coordinate system)


source cartesian coordinate system)

Poynting vector

complex Poynthg vector

correlation vector

sample correlation vector

vector from focal point to the source location

signal of the ith user (analytic signal) in the frequency domain

signal of ith user in time domain (analytic)

normalized distance variable

radiation integral

truncation function

0 component of T(6, #)

4 component of T(0,4)

transformation matrix from rectangular to cylindrical coordinates

transformation matrix fkom cylindrical to rectangular coordinates

transformation matrix fkom cylindrical to spherical coordinates

transformat ion matrix fkom spherical to cylindrical coordinates

transformation matrix &om rectangular to spherical coordinates

transformat ion matrix from spherical to rectangular coordinates

x component of T(44)

y component of T(B,(b)

z component of T(O,4)

period of time

symbol period of desired user's signal

period of a symbol

time

steering vector

steering vector of the desired user's signal

steering vector of signal i

E-plane pattern of feed

E-plane pattern of feed i

H-plane pattern of feed

H-plane pattern of feed i

distance parameter

distance parameter

direct ion cosine

x ( t ) shifted in frequency

ith element of steering vector

direction cosine of expected main beam position

vector describing the polarization of a wave

semimajor axis of polarization ellipse

semiminor axis of polarization ellips

distance parameter

distance parameter

direction cosine

x(t) shifted in fiequency


weight vector

statistically optimum weights

direction cosine


weight of element i

received noise vector in frequency domain

the recieved signal vector in the frequency domain

received signal vector (analytic signal)

received signal vector of interferer i (analytic signal)

the received signal vector (analytic signals)

received signal vector of the desired user (analytic signah)

received signal vector of the ith interfering user (analytic signals)

received noise vector (analytic)

received signal of desired user at element i

received signal of lth interfering user at element i

received signal vector (baseband) of the desired user

received signal vector (baseband) of i th interfering user

received signal vector (baseband) of noise

general signal

baseband received noise at element i

unit vector in x direction

z coordinate in unprimed (field) cartesian coordinate system

z coordinate in primed cartesian coordinate system

x coordinate in a cartesian coordinate system

x coordinate in source cartesian coordinate system

x coordinate of feed i on the feed plane

temporary x coordinates

output of beamformer at time t

unit vector in y direction

y coordinate in unprimed (field) caxtesian coordinate system

y coordinate in primed cartesian coordinate system

y coordinate in u cartesian coordinate system

y coordinate in source cartesian coordinate system

y coordinate of feed i on the feed plane

temporary y coordinates

time variable

unit vector in z direction

z coordinate in unprimed (field) cartesian coordinate system

temporary z coordinates

z coordinate in primed cartesikn coordinate system

z coordinate in u cartesian coordinate system

z coordinate in source cartesian coordinate system

Chapter 1

Introduction

1.1 Motivation

Recently there has been considerable research into satellite communications, and in

wireless communications in general, due primarily to a tremendous growth of the

demand for these services. As in any communication system, bandwidth and power

are precious resources, but this is particularly true for wireless and satellite corn-

munication systems. Communications between a geostationary satellite and a small

power-limited mobile terminal places a great deal of the burden on the satellite to

use its very limited power and bandwidth resources to establish and maintain the

communications link. Adaptive antenna army processing is one possible solution or

at least partial solution to some of these problems. Schemes such as FDMA, TDMA

and CDMA all try to increase the capacity of the system. However, one property

that isn't being exploited to its full advantage is the spatial distri6ution of the users

of the system and that is where adaptive antenna arrays step in.

An adaptive antenna array is one implementation of a spatial filter and that is

really what we are trying to do. A spatial filter is analogous to a temporal filter.

Just as a temporal filter discriminates between signals which are in disjoint frequency

bands, a spatial filter discriminates between signals which arrive fmm disjoint spatial

locations. Figure (1.1) shows a narrowband adaptive antenna array which consists

of a linear configuration of NE antennas. The signals received from the antennas are

Figure 1.1: The adaptive antenna army as a spatial filtering structure.

multiplied by the conjugate of a complex weight, w;, . . . , wk, and then summed to

~roduce the output signal y(t).

One interesting thing to note about this structure is its resemblance to a temporal

FIR tapped-delay line filter. Instead of temporally sampling the signals as is done

in the tapped-delay line, the signals are spatially sampled at the discrete locations of

the antenna sensors.

The term beamforming is often used when referring to spatial filtering. Beam-

forming is a term which suggests a slightly different perspective over that of the term

adaptive m a y processing. Beamforming is more of a satellite antenna term rather

than a signal processing term. From an antenna point of view the goal of what we

wish to do is to provide each communication channel with its own dedicated agile

beam. By pointing a narrow beam directly at each mobile terminal, each user gets

the peak gain. This is in contrast to the current state of satellite technology where

regional beams cover a given service area and the user may be at the peak of the

beam or out at the edges of the coverage. Narrow beams pointed directly at the users

prevent the loss incurred by a user at the edge of coverage. In addition, narrow spot-

beams d o w much smaller frequency re-use distance, better power efficiency, and the

flexibility to adjust to variations in the traflic pattern.

Adaptive beamforming or adaptive antenna array processing is a reasonably ma-

ture field of research which hasn't found its way into many real-life applications. The

field is regaining popularity due to the intense need for increased capacity in wireless

systems and advances in digital signal processing which are starting to make real-time

implementations of adaptive algorithms seem possible in the near-hture. Adaptive

algorithms, as their name suggests, adjust the weights after each antenna element

in an adaptive way, utilizing feedback from the output data to adapt to changing

environments.

Since adaptive antenna array processing is a mature field, there axe many adaptive

algorithms in the Literature. One thing that these algorithms have in common is that

they were generally conceived with a linear or planar axray configuration in mind,

such as the one in figure (1.1). In a geostationary satellite to mobile communications

application, the type of antenna that is much more Likely to be used is a laxge offset

reflector antenna with an array of antenna elements near its focal point '%dingn the

reflector. This sort of arrangement has several different names. In this thesis it will

be referred to as a multiple beam antenna, abbreviated as MBA. Other names for this

type of antenna include focal-fed array and hybrid antenna.

The reason that a MBA is Likely to be choosen as the antenna in a geostationary

application is that the gain of the satellite antenna is an absolutely critical value.

There are several reasons why gain is such aa important d u e . The first reason is

that a geostationary satellite is approximately 36,000 lun above the Earth. Any signal

which travels that far is severely attenuated. A second reason that gain is important

is that both the satellite and the mobile terminal are extremely power-limited. The

gain of an antenna is directly related to the physical area of the aperture of the

antenna (a larger antenna gathers in more radiation). Therefore, a reflector antenna

is a very practical solution. In order to achieve the same aperture size, a planar array

of antennas would have to have many more antenna elements than a multiple beam

antenna. This increase in the number of antenna elements leads to a much larger

amount of hardware on the satellite with the associated disadvantages of increased

weight, higher power consumption, higher cost and greater complexity. In addition,

deployment of a large number of antenna elements spread over an expansive area

becomes a problem. Refiector antenna tecbno10gy is mature and it is possible to

design an unfurlable reflector antenna which opens up upon deployment.

There has been very little literature on the use of adaptive dgorithms operating

with multiple beam antennas and the essential question that this thesis sets out to

solve is, can adaptive algorithms work on a MBA, and if not, is there some way of

changing the algorithms so that they will work on a MBA?

Chapter 5 of this thesis considers a special class of adaptive algorithms called cyclic

beamforming algorithms. These algorithms utilize the inherent cyclostationarity in

the desired signal to extract the desired signal from the presence of interference and

noise. These algorithms are attractive because they do not require knowledge of a

training signal or the directions of arrivals of the users in order to extract the signal

of interest. This is highly desirable because supplying a training signal takes up

valuable power and bandwidth resources, and methods based on direction finding are

computationdy very intensive. These types of algorithms are appropriately referred

to as blind adaptive algorithms.

1.2 Literature Survey

One of the interesting aspects of this thesis is its somewhat cross-disciplinary nature.

Adaptive signal-processing with a MBA can be viewed from either a signal processing

or an antenna point of view. This thesis attempts to bridge the gap between the two

views and unite them.

1.2.1 Adaptive Antenna Arrays

Adaptive antenna m a y processing is a very old and mature field. The fundamental

aspects are covered in the textbooks by Compton [49], and Monzingo and Miller [53]

and in an excellent tutorid paper by Van Veen and Buckley [I]. In addition, the reader

can refer to a comprehensive bibliography, compiled by Marr [2], of the work done

up to 1986. One paper that will be mentioned is the 1974 paper by Reed, Mde t t

and Brennan, [3] in which they developed and studied the direct matrix inversion

(DMI) algorithm. The DMI algorithm is a simple algorithm which converges very

quickly but has a high computational complexity. The DM1 algorithm is one of

the algorithms that is studied in this thesis. The other class of adaptive algorithms

studied in this thesis are those based on exploitation of eyclostatiunarity inherent in

the signal. These blind algorithms do not require a reference signal or knowledge of

the directions of arrival of the users. This relatively new class of algorithms, called

the SCORE (Self Coherence Restoral) family of algorithms, was pioneered by William

A. Gardner and his students, primarily Agee and Schell[29, 31, 32, 34, 35, 36, 381.

1.2.2 Adaptive Multiple Beam Antennas

The field of reflector antennas is, of course, very old. Much progress in this field was

made during World War 2 and this progress is documented in the classic text, edited

by Silver [59]. Love was the editor in a compilation of papers on reflector antennas in

1978 [52]. More recently, the chapter on reflector antennas written by Rahmat-Samii

in the "Antenna Handbook", which was edited by Lo and Lee [55], is an excellent

source of information on reflector antennas.

The analysis of the radiation patterns of reflector antennas is an important part

of this thesis. There are several techniques available as described in the small, but

cleasly written text by Scott [50] and in the review paper by Franceschetti and Mohsen

[XI. Among the many papers describing the theory behind modem reflector antenna

analysis are [12, 13, 17, 19, 20, 24, 251. The method of reflector antenna analysis

used in this thesis is the Fourier-Bessel method based on the work of Mittra, KO, and

Sheshadri in [20] and Hung and Mittra in [12]. Papers [I l] by Rahmat-Samii and Lee

aod [lo] by Lam et. al. outline a method of finding the total power radiated by an

array of feeds which is required to find the directivity of the antenna. Their method

of computation was used in this thesis.

The idea of using multiple feeds to illuminate a reflector is also not a new idea. The

use of cluster ( m y ) feeds has been suggested for the generation of multiple beams [52,

55,46, 13,611, for the generation of contour beams [55,13], compensation of reflector

antenna surface distortion [15], improved scan performit~lce [13], and the optimization

of directivity [lo]. This last paper by Lam et. al. [lo] is an important contribution

as far as this thesis is concerned. This paper aims to optimize the directivity of the

reflector antenna in a given direction using an array of feeds. This is accomplished by

applying a weight vector to the signal received at each of the elements and then adding

up the result. In other words, this paper is using the same beamforming structure

as in this thesis, but with a slightly different ultimate goal. There has been very

little literature on the use of multiple beam antennas for adaptive array processing.

Mayhan [44] has discussed the use of multiple beam antennas for adaptive nulling

in a generic MBA system, not dealing with the physical aspects of the problem.

Other authors have also suggested the use of MBA antennas for adaptive nulling or

adaptive processing [45, 42,431 but their details have either been very sketchy or else

they have neglected to work in depth on the electromagnetics involved. Robert Shore

has written the best account to date on the use of MBAs in an adaptive system [47].

Shore's brief report studied the use of a MBA performing adaptive nulling with the

Generalized Sidelobe Canceller (GSC) algorithm. Essentially what Shore concluded

was that no special adaptive nulling techniques are required for MBAs. Karimi's

Master's thesis [61] suggested the use of adaptive m a y processing with a multiple

beam antenna to improve the uplink performance of a mobile communications system

which utilized a CDMA-based geostationary beamforming satellite.

1.3 Contributions of Thesis

This thesis unifies several methods and theoretical developments in the fields of an-

tennas, and adaptive antenna arrays to produce a digital computer based simulation

tool which provides a vehicle to study statistically optimum beamforming involving a

multiple beam antenna in a multi-user digital communication system. The following

contributions support this general contribution.

A narrowband signal model is developed which provides the framework to study

statistically optimum beaglforming with a multiple beam antenna. The key quantity

is shown to be the steering vector which represents the response of the antenna array

to a unit-amplitude plane wave.

A digitd computer program is developed which efficiently evaluates the radiation

field of an offset parabolic reflector antenna. This program is based on the Fourier-

Bessel method of solution and utilizes the physical optics approximation. The field

due to an array feed is found using secondary field superposition. In secondary field

superposition the secondary pattern of each feed is first computed and then stored.

The stored patterns are then weighted and superimposed to give the field of the entire

antenna. This method of superposition is the most efficient choice since the weights

of the antenna array will be varied fiequently in statistically optimum beamforming

sirnulat ion.

The reflector a n t e ~ a analysis computer program is used to briefly study some of

the properties of offset parabolic reflector antennas. The effect of the edge taper of

the feed and the scanning properties of the offset parabolic reflector axe investigated.

A formulation of the reciprocity theorem is developed to relate the transmitting

properties of the reflector antenna to the recieving properties. This allows the steering

vector for the multiple beam antenna to be computed using the amplitude vector of

each of the feeds which are numerically evaluated with the reflector a n t e ~ a analysis

program. With the steering vector determined statistically optimum beamforming

with a multiple beam antenna is demonstrated. This is followed by the simulation

of the direct matrix inversion algorithm and the LS-SCORE adaptive algorithm.

These simulations of the adaptive algorithms show the convergence properties of the

respective algorithms. For LS-SCORE, baud- and carrier-rate features are exploited

in two separate simulations.

1.4 Presentation Outline

Chapter 2 is the starting point, where a narrowband signal model is developed which

is general enough to accomodate both a direct radiating array and a multiple beam

antenna. This model is the backbone of the theory behind this thesis. In the devel-

opment of this theory it is shown that the fundamental difference between the direct

radiating array and the multiple beam antenna is that they have different steering

vectors. For a uniform linear array, which is a specific form of a direct radiating ar-

ray, this steering vector can be found analytically knowing the direction of arrival of a

signal, the wavelength and the geometry of the may. A simple demonstration of sta-

tistically optimum combining is performed in preparation for a similar demonstration

later in the thesis on a multiple beam antenna.

Chapters 3 and 4 are all about finding the steering vector of the multiple beam

antenna. In order to get the steering vector of the MBA we need to find the antenna

pattern of the reflector antenna (chapter 3) and then use the principle of reciprocity

to find the steering vector (chapter 4). An added benefit of being able to find the

antenna pattern of the offset reflector antenna is an ability to study some of the

properties of these antennas. This will be done in chapter 3.

Chapter 4 applies the principle of reciprocity to the antenna pattern work done

in chapter 3, to come up with the steering vector. Knowing the steering vector we

can study beamforming with multiple beam antennas. The culmination of the signal

model in chapter 2, the antenna, pattern work in chapter 3, and the application of the

reciprocity principle in chapter 4 is a demonstration of optimum combining with a

multiple beam antenna performed in chapter 4. I . . addition, a baseband simulation of

a simple algorithm, called the direct mat* inversion ( D M I ) algorithm, is performed

demonstrating its performance on such an antenna.

Chapter 5 looks at blind spatial filtering algorithms based on exploitation of cy-

clostationarity inherent in the signals. A brief presentation of the theory is followed

up by a baseband simulation of the LS-SCORE algorithm operating on a multiple

beam antenna.

Finally, chapter 6 concludes the thesis by summarizing the results, presenting

some thoughts, and some suggestions for further study.

Chapter 2

Beamforming Theory

2.1 Introduction

As described in the introductory chapter, beamforming is a form of spatial filtering.

Figure (2.1) illustrates the spatid filtering structure introduced. The basis of the

spatial filter is an array of NE senson. The signals received at these NE array elements

are multiplied by the conjugate of a complex weight and summed together to form

the output signal. It is convention to multiply the data by the conjugates of the

weights in order to simplify later notation. Beurnforming is generally concerned with

how we vary the weights to change the response of the antenna array. The response

of the antenna is also controlled by the m a y geometry and the antenna patterns of

the individual antenna elements. Most of the beamforming literature deals with a

direct radiating a m y (DRA) which is simply a number of antenna elements asranged

in some arbitrary configuration. The most common D M considered is the uniform

linear array (ULA) where the antenna elements are identical and the elements are

placed in a Line and spaced an equal distance apart. In this situation the antenna

elements are usually spaced a half wavelength apart. If the elements are spaced M h e r

apart than that, grating nulls can form [49]. If the a n t e ~ a elements are spaced closer

together the spatial discrimination of the antenna array is reduced due to the use of

a smaller spatial aperture. An additional a~umption frequently made in the antenna

array literature is that the antenna elements are isotropic meaning that they radiate

I . . .

C

Figure 2.1: Spatid filtering structure for beamforming.

power equally in all directions.

This thesis is interested in the use of an offset reflector antenna fed by an antenna

array. This antenna is referred to in the antenna literature as a multiplebeam antenna

(MBA). In the next section, we'll take a closer look at the ULA as a step towards a

general model which encompasses both the MBA and the D M .

2.2 The Uniform Linear Array

Figure (2.2) shows a uniform linear array with a spacing of d = X/2 between the NE

elements. This figure also shows an incoming signal arriving from angle Bd. Assume

that the actenna elements are isotropic. The incoming signal is assumed to be a

plane wave and is represented by .the analytic signal G2"ft where f represents the

frequency of the wave. This wave will be received with a phase shift between each of

the elements. The interelement phase shift due to a signal of wavelength X arriving

from angle Bd (assume a 2-dimension$ picture for the moment) is given by, referring

to figure (2.2),

The received signal vector, gd( t ) , can be written as

where,

Incident Signal I

Figure 2.2: A Uniform Linear Array with element spacing of d = X/2.

Sdi ( t ) is the anaiytic sign J received at element i (i = 1, . . . , NE), and

$d is a random phase offset which is assumed to be uniformly distributed in the

range 0 5 5 2n.

The tilde symbol, denoted by -, above a signal signifies the analytic signal. All vectors

and matrices are denoted by bold face symbols. The vector

is called the steel-ing vector for the desired user's signal. It represents the response

of the antenna array to a unit-amplitude plane wave fiom direction Bd and frequency

f. In this case the antenna presents a unit gain to an incoming plane wave and a

phase shifk as described by equation (2.3). The steering vector is also referred to in

the literature as the array response vector, army manifold vector, and the direction

vector.

2.3 The Wideband Signal Model

Now a signal model will be developed which is general enough to encompass both a

direct radiating array and a multiple beam antenna. The model is very similar to

that presented in [31]. Consider the analytic signal g2"ft which corresponds to a red

sine wave having frequency f. Assume that the wavefronts incident on the antenna

axe plane waves which arrive at the antenna fiom angle (Bd, #d). The signal received

by the m a y can be expressed by the vector

where,

f d, ( t ) is the analytic signal received at element i (i = 1, . . . , NE),

NE is the number of antenna elements,

+d is a random phase offset (assumed uniformly distributed in the range

0 52 $d < 2 ~ ) ,

gdi (dd, 4d, f ) is the gain of the ith sensor to a unit-amplitude plane wave horn direc-

tion (Od, 4d) and frequency f,

vd,(Bdr 4d, f ) is the phase of the ith sensor to a unit-amplitude plane wave from

direction (ad, # d ) and frequency f, and

ud(Od, bd, f ) is the steering vector which represents the response of the array to a

unit-amplitude plane wave fiom direction (Bd, &) and frequency f.

The collection of steering vectors for aU sets of angles and frequencies of interest is

referred to as the away manifold.

Now, consider the more general situation where there are multiple sinusoidal sig-

nals impinging on the antenna. The data vector can be modeled (311 using linear

superposition and by decomposing the data in the fiequency domain where we as-

sume that the signals are Fourier-transformable. Therefore for L signals incident on

the antenna, we have in the frequency domain

where,

u (0 , # ) represents the steering vector of the lth (1 = 1, . . . , L) signal (which

arrives from angle (el, 41)) as a function of frequency f ,

sdf I represents the it h ( 1 = 1, . . . , L) signal as a function of frequency f , and

xn (fl represents the noise vector as a function of frequency.

In the next section this wideband model is simplified to produce a narrowband model.

2.4 The Narrowband Signal Model

Often we are only concerned with a relatively narrow frequency band. Lf this is true

then we may develop a narrowband signal model as a specific case of the wideband

signal model. If the fiequency band of interest is sdciently narrow such that the

steering vector ud(e,q5, f ) is approximately constant with respect to frequency for all

angles (0, #) then we may drop the fiequency dependence [31] and the array data may

be modeled in the time domain by

Essentially what the narrowband model does is treat each of the incident signals as

if they are sinusoids, even though they are not. The narrowband assumption and the

associated signd model wil l be used in the remainder of this thesis.

The problem that we are interested in is one in which we have a single desired user,

and Nr interfering users. Let us modify the notation in the narrowband signal model

to more closely represent this problem. As described above, the received signal is a

linear combination of the signals from the single desired user and the Nr interfering

users. This may be expressed by

where,

j id ( t ) is the received signal of the desired user,

&(t) is the received signd of interferer i (i = 1,. . . , .

kn(t) is the received noise signd.

The nmowband model may then be used to describe the signal of each of the users.

The received signal of the desired user is expressed by

where,

u d is the steering vector of the desired user (i.e. the response of the antenna in

the direct ion of the desired user),

md(t) is the baseband modulating signd for the desired user,

$d is a random phase offset for the desired user, and

f c is the carrier frequency.

Similarly, the received signal of each of the interfering users (i = 1,. . . , Nr) may be

expressed in a more detailed fashion:

In addition to the received signals of each of the users there is noise at each antenna

element which, lor the time being, will simply be expressed by

In this thesis we generally use the sampled complex envelope. Since the complex

envelope is obtained from the analytic signal by a multiplication by exp(-j2a f,t) the

baseband model replaces equations (2.10) and (2.1 1) with the following equations:

One assumption that will be made throughout the thesis is that all data that is

received is zero mean. The noise vector in equation (2.12) is replaced in the baseband

model by

where we assume that the received noise elements, xni(t) (i = 1,. . . , NE) are zero

mean random processes, statistically independent of each other, the desired signal,

and the interfering signal. In other words the expected value of the product of the

noise at element 1 and element i is given by

The power of a baseband signal is given by the expression

Therefore, t732 is the noise power at each element. The power of the desired signal

received at element i is

Similar expressions can be obtained for the interfering signal's power received at the

ith element.

One of the key chaxacteristics of each of the incoming signals is the signal power.

Often the signal to noise ratio is specified. The signal to noise ratio (SNR) of the

desired signal, pd, may be defined by

Similarly, the SNR of interferer i, pi (i = 1, . . . , Nr) , may be defined

The structure used to perform the beamforming is a linear structure as shown by

figure (2.1). Let us define a weight vector by

where the weight on each antenna element is complex in general. The signals re-

ceived at each element are multiplied by the complex conjugate of the weight and

added to give the output signal, denoted y ( t ) . As mentioned in the introduction, the

convention is to use the conjugate of the weight in order to simplify later notation.

Mathematically, the linear combining operation is expressed by

where the symbol is used to denote the Hennitian transpose. Incorporating the

expressions for xd, and into equation (2.23) gives

Equation (2.24) shows that wtud is the gain applied to the desired signal. The

quantity wtu(B,4) can be interpreted as a spatial transfer function which is analogous

to the transfer function of a Linear t ime-in~ant ( LTI) finite-impulse-response (FIR)

temporal filter. The variation of the magnitude squared of wtu(B,+) with the angle

of arrival, ( 8 , #), gives the antenna pattern of this spatial filter.

The output power of the desired signal is

which may be written as

The symbol ad denotes a N' x NE matrix c d e d the correlation matrix of the de-

sired user's signal. The correlation matrix of the desired user's signal, introduced in

equation (2.26), is defined as

Similarly, the output power of interferer i (i = 1,. . . , NI) may be expressed by

where we have defined the correlation matrix for interferer i by

The output noise power may be expressed as

where the noise correlation matrix has been defined by

Due to equation (2.16) the noise correlation matrix may be written

The signal to interference pius noise ratio SINR is a very important quantity.

The SINR is a ratio of the output signal power of the desired user to the sum of the

output signal power of the interferers and the output noise power. Clearly, we wish

to maximize this ratio. The SINR is defined mathematically by

SINR = Pd

cZ1 Pi + P n

Note that the SINR is not changed if the weight vector, w, is scaled by a constant.

Therefore, any weight vector which is within a constant factor of the optimum weights

will give the same SINR. This is an important fact which is used in the simulation of

optimum combining in section 2.7.

2.5 Beamforming

With the signal model established we can set to the task of forming beams. Beam-

forming is the control of the response of the antenna may by wrying the weights

at each antenna element. As described in the tutorial paper [I] by Van Veen and

Buddey and illustrated in figure (2.3), beamforming may be classified as data in-

dependent beamforming or as statistieolly optimum beamfoming. Data independent

I Beamforming I control the response of the antenna by varying the weights

Data-lndepemdent Bearnf orming

determination of the weights based on predetermined locations of maximums and nulls

f

Statistically 0 timum Beamform P ng

usin the statistics of the data to find & wei- which give the sbt is tka l~ optimum response (optimkabon of a cost function)

Statistically Optimum Beamforming I with Constraints - I

a combination of data-independent and statistically I optimum beamforming techniques

Figure 2 -3: 0 verview of beamforming.

beamforming involves the adjustment of the weights to design a desired response.

The weights do not depend on the data input to the array. Statistically optimum

beamforming uses the data received by the array to generate statistics of that data

which are used in the adjustment of the weights to optimize, in some sense, the re-

sponse of the antenna. The optimization is performed with respect to a cost function.

The general goal is to maximize the response in the direction of the desired user and

to minimize the contributions of noise and interferers. Some methods which seek a

statistically optimum solution for the weights use constraints to control the pattern.

This represents a combination of the techniques of statistically optimum and data in-

dependent beamforming. In this thesis we are concerned with statistically optimum

2.6 Statistically Optimum Beamforming

In determining the statistically optimum beamformer we will assume that the input

data is wide-sense stationary (WSS) and that the second-order statistics are know^.

In practice the data may not be WSS and we do not know the second-order statis-

tics. With the assumption that the data is ergodic we can estimate the second-order

statistics. If the data is not WSS then adaptive algorithms are used which, as their

name suggests, adapt the weights to the changing environment.

There axe several different criteria that are used in the literature in determining

the statistically optimum beamformer. Four common cost functions are: (1) mean-

squared error (MSE), (2) signal to noise ratio (SNR), (3) maximum-likelihood (ML),

and (4) minimum miance (MV). It turns out that each solution is within a constant

of the Wiener solution [49,53] and therefore yields the same SINR. In the next section

we will examine the minimum MSE solution.

2.6.1 The Minimum MSE Solution

Assume that a reference signal, r( t ) , is available that is perfectly correlated with the

desired signal and uncorrelated with the interfering signals and the noise. For this

derivation assume that the reference signal is a perfect reference signal in that it

equals the desired signal. The error signal is defined by

The mean-squared error is then given by

This expression may be simplified by introducing the cornlation mat&,

and the correlation uector,

scow = E { r ( t ) ~ * ( t ) ) -

Therefore, we may rewrite the MSE in equation (2.37) as

Minimizing the MSE with respect to the weight vector leads to the following solution

for the optimum weights [49, 53, 571:

Note that we are assuming that O is nonsingular and therefore its inverse exists. This

equation is known as the Wiener-Kopf equation. Since the correlation vector may be

written as

the optimum weight vector may be reexpressed as

The purpose of this expansion of the optimum weight vector is to show the relationship

of the optimum weight vector to the steering vector of the desired signd. If we are

able to determine this steering vector and the correlation matrix of the input data,

then we know the optimum weight vector to within a scalar constant. This is relevant

because a scaled version of the optimum weight vector still attains the optimum SINR.

2.7 An Example: Optimum Combining With A ULA

In this section optimum combining with a uniform linear array will be illustrated.

Optimum combining represents the steady state solution that any adaptive algorithm

at tempts to converge towards.

Let us assume that all of the users are simple continuous wave (CW) signals.

Each of the users is assumed to be uncorrelated with the other users. Also assume

that we have a reference signal that is perfectly correlated with the desired user and

uncorrelated with the interferers and the noise. Let the reference signal take the form

Since the reference signal, r ( t ) , is assumed to be perfectly correlated with the desired

user we set $, = $d. Therefore we have,

The correlation matrix is, from equation (2.38),

which may be rewritten as

The correlation vector is (from equation (2.39)) given by

The optimum solution for the weights is given by equation (2.41)

This example will use a 3-element array with spacing of d = X/2. It is well

known that an array of NE elements has NE - 1 degrees of freedom [49, 541. Each

null created and each maximum created uses up a degree of freedom. Therefore a

3-element m a y can simultaneously place a maximum on a desired user and null one

interferer. A second interferer can't be nulled. This concept of degrees of freedom

will be demonstrated through the simulation.

This simulation is run for three Herent cases. In the first case only a single

desired user exists. In each subsequent case an interfering user is added. A unique

angle of arrival is selected for each user and all of the users have been selected to have

a signal to noise ratio of 8 d B . Looking at equation (2.58) we can find the optimum

weights to within a scdar constant by determining the inverse of the correlation

matrix and the steering vector of the desired user. The steering vector of all of the

users, including the desired user, are determined analytically by using equation (2.2)

and ( 2 4 , fiom section 2.2, with knowledge of the direction of a n i d of each of the

users. The correlation matrix, @, is determined from equation (2.53) to within a

scalar constant with knowledge of the steering vector and SNR for each of the users.

With knowledge of the correlation matrix to within a scala constant, we also know

the inverse of the correlation matrix to within a scalar constant. In summary, we

can determine the inverse of the correlation matrix to within a scalar constant, and

the steering vector of the desired user and with these two components the optimum

weight vector to within a constant.

Knowing the optimum weight vector to within a constant dows us to determine

the optimum output SINR (since the optimum SINR is unchanged if the weight

vector is scaled from the optimum weight vector - see section 2.4) and the normalized

gain of the antenna array. Equation (2.34) was used to find the SINR. The plot

of normalized gain is a normalized plot of the magnitude squared of wtu(8) as a

function of 8. Equations (2.2) and (2.1) were used to find the steering vector, u(B),

as a function of 8.

Figure (2.4) is a plot of the antenna pattern of the 3- element ULA with only a

single desired user. The desired user has a SNR of 8 dB and it's signal arrives fiom

25". Figure (2.4) shows the maximum placed at 25"- The SINR found was 18.93 dB.

Note that the SINR without any interferers can be seen as a useful upper bound to

the SINR we can achieve when interferers are present. The better the beamformer

is able to null out the contribution of interfering users the higher the SMR and the

closer it comes to this upper bound.

Now let us examine optimum combining with interferers. Figure (2.5) shows a

plot of the antenna pattern of the optimum combiner with a desired user at SNR = 8

dB arriving from 25" and a single interferer with a SNR = 8 dB arriving at 60". The

plot shows a maximum near the incident direction of the desired user and a null near

60°, the direction of the interferer. Clearly the antenna may has enough degrees

of freedom to place a maximum near the desired user and to place a null near one

interferer. The SINR achieved was 15.27 dB.

Now let us add a second interferer with a SNR = 8 dB and arriving from angle 90".

Figure (2.6) shows the antenna pattern of the optimum combiner. Figure (2.6) shows

that the second interferer can't be nulled. There aren't enough degrees of freedom

to do so. The level of the second interferer is reduced but the null previously placed

on interferer #I has been shifted and so the gain of the 1st interferer rises. Overd,

ULA - Antenna Pattern (1 desired user, 0 interferers)

Angle 0

Figure 2.4: Antenna pattern of the belement ULA with a single desired user with SNR = 8 dB and arriving fiom 25'.

ULA - Antenna Pattern (1 desired u w , 1 interfern)

0 30 60 90 120 150 180 210 240 270 300 330

Angle B

Figure 2.5: Antenna pattern of the %element ULA. There is a single desired user with a SNR = 8 dB arriving from 25' and a single interfering user with a SNR = 8 dB arriving from 60'

ULA - Amtenna Pattern (1 d.rimd user, 2 intebnn)

0 30 60 90 120 150 180 210 240 270 300 330 360

Angle 0

Figure 2.6: Antenna pattern of the 3-element ULA: (pd = 8 dB, pl = 8 dB, pz = 8 dB) and (Bd = ZO, O1 = 60°, $ = go0)

it is clear that there are not enough degrees of hedom to simuitaneously null both

of these interferers and place a maximum on the desired user. The SINR acheived in

this third case was 10.47 dB.

These examples have been generated both to demonstrate the b e d o n n i n g theory

derived in this chapter and to get a feel for the ULA and the concept of the degrees

of Freedom of an antenna array. The reader should refer to [l, 49, 53,541 to get more

in depth coverage of beamforming with a ULA.

2.8 Beamforming With a Multiple Beam Antenna

The signal model developed in this chapter is valid for any type of antenna array

including a MBA. The key difference between beamforming with a ULA (or any DRA)

and a MBA is in the steering vector. For a ULA with isotropic elements the steering

vector consists of a vector of phase shifts. For a MBA it isn't that simple. There

will be a gain and phase at each element in response to an incident unit-amplitude

plane wave. It is these gains and phases which we need to determine in order to

find the steering vector. The next chapter is an important step in that direction. In

that chapter the far-field electric field of an offset reflector with an array feed will be

determined. Using this and the theory of reciprocity we can determine the gain and

phase presented to an incident unit-amplitude plane wave at each of the elements in

the feed array. Chapter 4 will expand upon these ideas and demonstrate optimum

combining with a MBA.

Chapter 3

Offset Parabolic Reflector Antenna Analysis

3.1 Introduction

In chapter 2 a signal model was developed for beamforming with antenna arrays. This

model was general enough to include both direct radiating arrays and multiple beam

antennas (a reflector antenna fed by an array feed). It was found in chapter 2 that a

key quantity in beamforming is the steering vector. In chapters 3 and 4 the steering

vector for a MBA will be derived. This chapter finds the far-field electric field of

an offset reflector fed by an array feed. This information dong with the reciprocity

theorem is used in chapter 4 to find the steering vector of the MBA.

Offset parabolic reflector antennas are the dominant antenna type in geostation-

ary satellite communications. Reflectors provide the large antenna size necessary to

acheive sdlicient gain. The gain of an a n t e ~ a is related to the width of the main

beam with a narrow beam implying a large gain. To give some perspective on how

the beam width for a geosynchronous satellite relates to the coverage area on the

Earth surface consider the following example. A circular beam with a lo beam width

has a solid angle of approximately 3.05 x steradians which at approximately,

36,000 km away covers an area on the Earth of approximately 395,000 km2 which

corresponds to roughly a 630 km x 630 km area.

The offset reflector has several advantages and disadvantages when compared to

symmetric reflectors [16,55,56,52]. The most obvious advantage of the offset reflector

is that it prevents aperture blockage fiom the feed. This is particularly important

with an array feed. When the feed blocks the aperture of the reflector, the radiation

is scattered resulting in a loss of gain, higher sidelobes and higher crosspolarized

radiation. Since gain is a critical parameter, particularly when the ground user is a

small mobiie terminal, this loss of gain makes the offset reflector very attractive. A

second significant advantage of the offset reflector is that the reflector imposes much

less of a reaction upon the primary feed than the symmetric reflector. This allows

the primary feed voltage standing wave ratio to be essentially independent of the

reflect or.

The offset reflector has several disadvantages. Linearly polarized feeds have a

cross-polarized component generated by the reflector. Circularly polarized feeds ex-

perience beam squint meaning that the beam peak is shifted fiom its normal direction.

In Appendix A the foundation of electromagnetics and a n t e ~ a theory necessary

for the study of offset reflector antennas is presented. A number of the results and

derivations in this and the following chapters have their origins in that Appendix.

In this chapter we study the offset reflector and the electromagnetic fields produced

by that reflector when illuminated by a feed element (and later by an array of feed

elements). We will first discuss the geometry of the offset parabolic reflector and the

coordinate systems involved. After this we will derive and solve an integral called the

radiation integral which is used to find the secondary electric field due to a single feed.

The method of solution of the radiation integral which is used is called the Fourier-

Bessel method. After describing this some of the details of the implementation of the

method are mentioned and the antenna patterns generated by a computer program

based on the theory are verified against those in the literature. At this point we will

study some of the properties of offset reflectors. We are particularly interested in the

scanning properties or, in other words, the effect on the reflector's antenna pattern as

we move the feed off of the focal point of the reflector. Following this investigation of

the properties of offset reflectors, the theory is extended to the case of an array feed.

Findy, the calculation of the directivity of the reflector antenna with the array feed

is discussed. The directivity is a measure of how an antenna radiates preferentially

in one direction over another and it is a fundion of angle. The directivity in a given

direction is defined as the ratio of the radiation intensity of the antenna in the given

direction to the radiation intensity of an isotropic antenna radiating in that direction.

Appendix A has a more in-depth discussion of both directivity and radiation intensity.

3.2 Geometry of the Offset Reflector

This chapter focuses on finding the electromagnetic fields fiom a reflector antenna.

The reflector antenna is constricted to an offset parabolic reflector. An offset reflector

is formed by intersecting a parent paraboloid with a circular cylinder. The resulting

reflector has a circulaz aperture. Figures (3.1) and (3.2) illustrate the geometry of

the offset reflector.

Offset

Figure 3.1: 3 dimensional illustration of the geometry of the offset reflector

Figure 3.2: 2 dimensional illustration of the geometry of the offset reflector

The defining parameters of the offset parabolic reflector axe:

a the radius of the circular aperture,

F the focal length of the parent pilraboloid, and

d, the offset height &om the focal point to the aperture center.

The symmetric reflector is a special case of the offset reflector where tiofl = 0. The

other geometric parameters of the offset reflector can be found fiom the above three

parameters and the equation for the parabola

where the (sf, y', z') coordinate system has origin at the focal point and the 2' axis

points dong the axis of the parent paraboloid. The other distance parameters as

illustrated in figure (3.2) are

D the aperture diameter,

Dp the paxent paaboloid's diameter, and

H the offset distance to the lower edge of the reflector.

All of the angular parameters (except for Gs) in figure (3.2) are defined from the axis

of the parent paraboloid. They are:

Q L the angle to the lower edge of the reflector,

arr the angle to the upper edge of the reflector,

Qc the angle to the aperture center,

the angular center of the reflector, and

Bs 112 of the angle subtended by the reflector.

The following equations derived from the geometry are used to relate the derived

parameters to the three defining parameters a, F, and do*:

3.3 Coordinate Systems

There are several different coordinate systems used in solving the reflector radia-

tion analysis problem. Figure (3.3) illustrates the different coordinate systems used.

z Primed coordinates (sf, y', 2') are used as integration coordinates over the surfacc

x, K &r

Source

Y. Y'

Figure 3.3: The coordinate ment .

systems used in the reflector antenna analysis develop

the reflector. The origin of the primed coordinate system is at the focal point of the

reflector and the z' axis points out away from the reflector along the axis of the parent

paraboloid. The I' and y' coordinates lie in the plane of the projected aperture of

the reflector which we choose such that it contains the focal point. We let the x'

coordinate be such that the sf-z' plane is a plane of symmetry for the reflector (see

figure (3.3)). The y' coordinate is then defined by the right-hand rule. A second set

of Cartesim coordinates (I,, y,, 2,) is defined with the unit vectors all pointing in

the same direction as the primed coordinate system but with the origin at the center

of the projected aperture. Therefore, I' = xc + doH, y' = y,, and z' = 4. We define

a set of corresponding cylindrical coordinates, (p', 4, z'), with origin at the center of

the project aperture. They are primed to represent the fad that they will be used a s

integration coordinates.

Unprimed coordinates (I, y, z) are used to specify the field coordinates. Usually

the field point at which we evaluate the radiation field is expressed in terms of the

corresponding spherical coordinates (r, 6,d).

Another set of Cartesian coordinates that are used is the source coordinate system,

(xs, ys, z,). This system is necessary since the radiation pattern of the source is

nonndy expressed in terms of a source-centered spherical system. Therefore, the

spherical coordinates (T,, O,, q5& a.re used frequently.

The relations between these coordinate systems is important since frequently we

need to find a point or a vector, previously defined in one Cartesian system, in a

different Cartesian system. The primed and unprimed coordinate systems are equiv-

alent. The (x,, y,, 2,) system is related to the primed system by a translation along

the I' axis. The source system, on the other hand, may be oriented arbitrarily with

respect to the primed system and the origin may be translated by an arbitrary vector

s. To represent the relationship between the primed and the source system we need

to specify the translation vector s and the three Eularian angles (aar, Peur, yeur). The

Eularian angles represent three successive rotations performed on the primed system

to orient it with the source system. More details on the definitions of the Eularian

angles and the transformations between coordinate systems are given in Appendix B.

3.4 Reflector Antenna Analysis

With an understanding of the geometry of the offset reflector and the coordinate

systems used throughout the derivations, we are prepared to tackle the problem at

hand. The problem is defined as follows: given a certain offset reflector geometry

(a, F, do*), and a given source (location, orientation and radiation pattern) what

is the electric field at a point in space far fiom the antenna defined by the field

observation coordinates (r, &4). For the moment we restrict the problem to that of a

single feed with an extension to an array feed coming later. Physically what is going

on, is that the source is generating an electromagnetic field which induces a current on

the surface of the reflector. This current, in turn, generates its own electromagnetic

field. It is this field that is observed far away from the reflector in what is c d e d

the far field. The term primary field is used to refer to the electromagnetic radiation

field of the feed. The secondary field is the electromagnetic field of the reflector when

illuminated by the feed.

In Appendix A the foundation of electromagnet ics necessary to solve this problem

has been given. There it is found that we can find the electric field produced by a

current distribution by first finding the magnetic vector potential A and then using

equation (A.37) reproduced below:

where,

E is the complex electric field,

A is the magnetic vector potential,

w is the frequency in radians,

c is the permittivity, and

P is the permeability.

As developed in Appendix A, the magnetic vector potential, A, may be found

by superposition of the magnetic vector potential of a large number of infinitesimal

current elements. This was expressed in equation (A.47) of Appendix A and is repro-

duced below for the case where we have surface currents:

where,

37

r is the vector &om the origin to the field point,

r' is the vector from the origin to a point on the surface we are integrating over,

3 is the complex surface current, and

k is the wave number.

The integration is over the surface of the reflector denoted by C. We place the origin of

the integration coordinate system (the primed system) at the focal point. Therefore,

r' represents the vector fiom the focal point to the integration point on the surface of

the reflector as shown in figure (3.3). The vector r represents the fat-field observation

point represented in spherical coordinates by (r, 8,4) (dso shown in figure (3.3)). The

source generates an electromagnetic field which induces the surface current J(r') on

the reflector surface. Equation (3.11) is solved for A at the far-feld point r and then

equation (3.10) is used to find the electric intensity. The &st step is to rearrange

these equations and introduce some simplifications to reach an equation referred to

in the antenna literature as the radiation integral.

3.5 Derivation of the Radiation Integral

Equations (3.10) and (3.11) may be expressed in a single equation,

where r ) is the freespace impedance and I is referred to as the unit dyad. The unit

dyad has the property that I a = a.

We are interested in the electric field edua ted at a large distance fiom the reflec-

tor (evaluated in the far-field). At this point the far-field parallel rays approximation

is made (see figure (3.4)) giving

where r = lrl. Therefore,

Figure 3.4: The pardel rays approximat ion.

and the integral (3.12) becomes

Applying the divergence operator to the integrand in equation (3.15) gives

where the vector identity V - ($a) = a VJ1 was used which applies when the vector

a is constant with respect to the V operator. We can use this identity because the

vector J(r') is not a function of the observation coordinates. In spherical coordinates

the gradient operator is

Throughout the rest of this derivation of the radiation integral we are only going to

retain terms that are of order l / r . The reasoning is that in the far-field r is large

and therefore terms of order l/r2 and I/? fall off quickly. Note that the e and 4 components of the gradient will vasy with respect to r as e-jk/r2. Therefore we PrriII

not keep these terms. The radial component of (3.16) remains as

Keeping only the l / r term the radial component becomes

Now we will apply the second V operator to the remaining integrand. To do so,

we recognize that the integrand can be split into two scalar functions of r, 8, and q5

denoted by g(r, 9, #) and h(r, 0 ,d ) :

e - j b J(rr) - (- jk) + - @'-'

4nr

When the V operator is applied we may use the identity

The fist part of the identity may be evaluated as

where once again d l/r2 terms have been removed.

Since g = J(r') - Z is only a function of 8 and 4 and not r, the application of

the gradient operator in spherical coordinates causes these terms to vary as l/r2.

Therefore we set the hVg term to 0.

Making all of the above changes we arrive at the radiation integral

e-jh. E(B,q5) = -jkq-(I - ii) + 11 ~ ( r ' ) g ~ ~ . ' d ~ .

4nr

This equation is the starting point in modern reflector antenna analysis [50].

3.6 Solution of the Radiation Integral

In the previous section the radiation integral (3.23) was derived which gives the

electric field at the point in space defined by the vector r = (r, 8,4). The derivation

assumed that the point r was in the far field of the antenna. In the far field the

electric field is a spherical wave which has components only in the plane perpendicular

to the direction of propagation of the wave. In other words, the electric field will

have components in the and 4 direction but not in the radial direction. The dot

~roduct of (I - H) with the double integral represents a subtraction of the radial

component. The radiation integral can be rewritten using two equations where the

radial component is explicity taken out to give

where To and TQ are respectively the and 6 components of

which will from now on be referred to as the radiation integral. Since J(r') is given by

it's Caxtesian components, T(O,+) will be expressed in terms of it's Cartesiaa corn-

ponents? Tz(O, +), T,(0, d ) , and Tz(B, +). To find Te and T4 we use the transformation

(see Appendix B)

cos 0 cos 4 cos O sin 4 - sin 0 = ( -sin, cos 4 0 ){$}-

Once we solve equation (3.25) for Tz, T, and Tz, (3.26) is used to get To and T4

which are substituted into (3.24) to get the final result. The objective is to evaluate

equation (3.25) efficiently and accurately.

3.7 The Physical Optics Approximat ion

The source current induced on the reflector surface, J(r'), is often approximated by

what is called the physical optics (PO) approximation which is expressed mathemat-

i cdy by

J(rf ) = 2ii(rf) x H, (r')

where, ii(r') is the unit normal to the rdector surface, and H,(r') is the incident

magnetic field of the source at r'. The PO approximation implies that at each point

on the illuminated side, the scattering takes place as if there were an infinite tangential

plane at that point. The PO approximation is known to give accurate results near

the main beam but the accuracy degrades at the far out sidelobes [55]. There are

methods of improving the accuracy at wide angles, one of which is to include an extra

fringe current at the edges of the reflector (since at wide angles, radiation is mostly

due to current at the edges of the reflector). Since we are primasily interested in the

main beam region and efficiency of computation is important, the PO approximation

will suffice.

The unit normal is given by * N

where, for a paxobolic reflector

and

These equations are used to evaluate equation (3.27).

3.8 Evaluation of the Radiation Integral

Let us consider the radiation integral (3.25) solved using the PO approximation (3.27).

It is possible to numerically integrate this integral. However, the integrand oscillates

quickly and requires a large number of points [21]. For large reflector surfaces the

time taken to evaluate this integral can be quite large. In the antenna literature much

effort has been put into solving this integral by more efficient means [12, 13, 17, 20,

21,24,25,50]. The method used in this thesis is cded the Fourier-Bessel method [12,

20,501. It is one escient method of evaluating the integral in equation (3.25). Before

delving into the details of the Fourier-Bessel method we will expand and rearrange the

radiation integral. First, t r d o r m the integration in (3.25) over the reflector surface

C, into an integration over the projected aperture which is symbolically denoted A.

The projected aperture is a circular region since the reflector is formed by intersecting

the parabolic surface with a circular cylinder (see figure (3.1)). Now,

dC = Jc dx, dy, = Jz p' Q' dq5' (3.31)

where a transformation to cylindrical coordinates (4,4') has been made, and Jx is

the surface Jaco bian. The radiation integral becomes

Let us expand the exponential term. First express P by

where the direction cosines have been defined as

u = sin8cost$,

v = sine sin#,

and w = cose.

Next, we express r' by

r' = x'f + y'ji + z r i

where (x', y', 2') may be expressed in terms of (p', #):

1 and zf = -(zR + - F

4F

Now we can eduate the dot product r' - P as

+ 2dO8.pf cos 47 +

At this point the angular spherical coordinates of the expected main beam position

are defined as (go, 40) with the corresponding direction cosines defined as

uo = sin eo sin

md wo = COSflo.

Introducing these variables into equation (3.42) gives

k] cos qS + (V - vo) sin 4' rt r = {[(. - u.) + (W - w0) 2F

Substituting (3.46) into (3.32) and letting p' = as gives

where,

where,

Let us take a closer look at equation (3.48) which is the integral we now set out

to solve. f (s, q5') is a function which is independent of observation coordinates (0, q5),

instead solely depending on the integration coordinates (s, 4'). f (s, 4') is c d e d the

effective aperture distribution. The third term is a Fourier kernel involving the far field

coordinates (5, G). It is the second factor which makes solving this integral difficult.

That term will be dealt with momentarily. Before proceeding, note that equations

(3.47) - (3.49) may be solved on a component-by-component basis. In other words,

we solve for the I, y and z components separately. The scalar x , y, z components of

T(B, 4) will be denoted by Tv where v represents any of s, y, and z. The same will

be true for the vectors I, f , and J. Each vector equation becomes 3 scalar equations.

Now it is time to deal with the second term in equation (3.48). This term may be

expanded in a Taylor series in the complex miable

The variable 5 will be sm& in the direction of the main beam as w approaches wo.

Away from the main beam the antenna pattern is governed primarily by diffraction

at the edges of the reflector. At the rim of the reflector the variable s approaches 1.

The result of this factorization is that the variable will be small over a wide range

of far field angles.

The Taylor series expansion is given by

For the purposes of computation, the series is truncated to P terms. The series

converges rapidly and only a few terms are required (as will be shown in section 3.11).

Applying the truncated Taylor series expansion described above and substituting

x, = as cos #' and y, = as sin # equation (3.48) becomes

3.9 The Fourier-Bessel Method

Now it is time to introduce the Fourier-Bessel method to efficiently solve the integral

in equation (3.54). Let us express f,(s, 4') in terms of two other functions

where T is the truncation function defined by

1 , inside the aperture area A (i.e. s 5 1) 0 , outside the aperture areaA (i.e.s > 1)

and g(s, #'), which has scalar components g,(s, 4) (u = x , y, z ) , is a periodic vector

function with period axea B, which is the D x D square in figure (3.5). In other

words, g,(s, 4') has period D in both the z and y directions. Clearly by equation

(3.55)

f,,(s, 4') = gV(s, 4') inside the aperture area A. (3.57)

For the moment, we will leave g,(s, 4') undefined in the shaded region in figure

(3.5) between the perimeters of the regions A and B. With f, = g, inside A we can

rewrite equation (3.54) as

The motivation behind defining g,(s, 4') in equation (3.57) is that since g, is

periodic it may now be expressed in terms of a 2-D Fourier series giving

Figure 3.5: The projected aperture A and the D x D square it inscribes, B.

where E;,, are the Fourier coefficients given by

It is this expansion into a Fourier series which is the key idea behind the Fourier-

Bessel method. The point is that the resulting integral can be analytically evaluated

as will be shown shortly. Now from equatiou (3.58),

where,

Further, using I, = as cos 4' and y, = as sin4', and letting

we can write equation (3.62) as

Equations (3.65) and (3.66) imply that,

The integrals in equation (3.67) are of the general form

Performing the integration of the Q variable gives (see Appendix C)

where J,(x) is the zeroth order Bessel function. The s integration of equation (3.71)

gives (see Appendix C)

where, Jp+l (I) is the Bessel function of order p+ 1 as defined in Appendix C. Applying

the above results to equation (3.67) gives the final expression for I@, 4) as

3.10 Slrmmary of Fourier-Bessel Method

It is time to summarize the results to this point and to make a few remarks about

the foregoing equations (3.24), (3.26), (3.47), (3.73), (3.60), (3.5?), (3.49), and (3.27).

Bringing these equations together we have,

cos 0 cos 4 cos 0 sin 4 { : I = ( -sin4 cos 4

f x u y ) , inside the aperture area A gu(x"9yu) = { as yet undefined , in the shaded region of figure (U),

(3.79)

jk ( g ) (3 -*)rue jkas [(no+$wo) cos #+vo sin 4' f&, 47 = JJs, 4') JZ e e

The fist rema.rk to be made is that g,(x,, Y,,) has yet to be defined in the region

outside A but within B in figure (3.5) and as shown shaded in that figure. One thing

we can do is to let g,(xc, y,) = 0 in this region. The problem with this is that we axe

introducing a discontinuity at the circular boundary of A. A better solution is to use

the value that f,(xc, y,) would be at these points if f, were dowed to extend beyond

A. This choice of definition of g, gives a quicker convergence of the m, n series [12].

A few remarks on the advantages of the above method should be made. The first

point is the fact that all of the coefficients EV,, are independent of the observation

coordinates (8,4). Therefore we only need to calculate the coefficients once and then

these coefficients are used for aIl observation points. The second important point

is that the coefficients in equation (3.78) can be cdculated using the Fast Fourier

Transform (FFT) algorithm [48]. This is one of the key advantages of the Fourier-

Bessel method.

3.1 1 Implement at ion and Verification

The previous section summarized a set of equations which may be used to find the

far-field electric field of a reflector antenna. A computer program based on these

equations has been developed and it will be used to study the properties of offset

parabolic reflectors and to investigate adaptive bedorming with a MBA. This corn-

puter program doesn't actually evaluate the electric field. Instead, it evaluates the

amplitude vector (see Appendix A) which is related to the electric and magnetic field

by the following equations

where F(k) is the amplitude vector. The amplitude vector isn't a function of r, the

distance to the field point. That dependence is given by the $ term in equation

(3.82). The amplitude vector contains the angular dependence of the radiation term

which is what we are most often interested in. As described in Appendix A we may

define the radiation intensity by

which has units of watts/steradiaa. A normalized plot of the radiation intensity is

often called the antenna pattern of the antenna.

The amplitude vector (and the electric field) may be decomposed into a component

in the reference polarization and a component in the cross polarization. We do this by

defining the unitary vectors R and C. These vectors are unitary and ate orthogonal

to each other and to the direction of propagation. Therefore,

F(k) = R F(k, R) + c F(k, C) (3.84)

where

Our computer program allows a symmetric or an offset configuration for the reflec-

tor, the symmetric reflector being a special case of the offset reflector where the offset

height do8 = 0. Note that the program does not take into consideration blockage

by the feed. Therefore the reduction in the gain, and increase in the sidelobe levels

due to blockage won't show up in the plots of the antenna patterns of symmetric

reflectors.

The feed model used is the cosq(0) model described in Appendix A. The cosq(0)

feed model is a common choice in the reflector antenna literature since it is a reason-

able analytic approximation to real feeds, particularly in the main beam region. The

feed patterns are 0 for (n/2 < 0, 5 s) and in the angular region (0 5 0. 5 n/2) the

E and H fields for a linearly x-polarized feed are defined by [55]

where

UE(~.) = (COS QqE = Eplane pattern

UH(&) = (cos 8s)qa = H-plane pattern

51

and r, is the vector from the source origin to the field point (see figure (3.3)). The in-

dices (qE, q H ) control the shape of the pattern. A linearly y-polarized feed is described

by [=I

A circularly polarized feed element has the pattern given by [55]

where the x parameter is used to select between right-handed circular polarization

(RHCP) and left-handed circular polarization (LHCP)

-1 RHCP x = { + l LAW.

The circularly polarized feed pattern is a superposition of the x- and y-polarized

feed patterns with a rt90° phase shift between them.

The computer program allows the feeds to be placed anywhere and oriented at

any angle with respect to the reflector. The assumption is made that the reflector is

in the far-field of the feed so that we may use the far-field pattern of the feed.

The summations in equation (3.73) must be truncated to compute them. The p

sum has a lower limit of 0 and an upper limit of P. We give the (m, n) sums a lower

limit of (F - 1, - 1) and an upper limit of (F, 5) giving a total of (M, N) terms

in the series. The value of M and N are chosen as a power of 2 because the FFT is

used to compute the E;,, coefficients (481.

Figure (3.6) is a graph illustrating the convergence of the pseries for one reflector

configuration. The geometry used is the same as in paper [17] by Y. Rahmat-Samii

and V. Galindo-Israel. The reflector used is an offset reflector with F = 120A, a =

50X, and dofl = 70A. The feed is a y-polarized feed with the cosV pattern where

52

0 I 2 3 4 5

0 (in O) (+ = 90°)

Figure 3.6: gseries convergence ( M = N = 64) for an offset reflector (F = 120)1, a = 50X, dOf = ?OX, y-polarized feed with q~ = q~ = 15.514)

q~ = q~ = 15.514 giving an edge taper of -10.0 dB. The edge taper (ET) is a value

which describes the feed radiation intensity at the edge of the reflector relative to the

center. The edge taper is the value often quoted when refering to the feed pattern.

The ET is related to the q d u e of the feed by the expression [24]

ET fin dB) q =

\ I

20 log,, cos (*";'r) - It can be seen £?om figure (3.6) that the pseries converges very quickly. In fact,

only the first term is needed for this reflector. It is fortunate that the pseries converges

so quickly because higher order p terms take much more time to compute. This is

due to the need to compute the Bessel function of order p + 1. Higher order Bessel

functions are much more costly to compute.

Figure (3.7) illustrates the convergence of the (m, n) series. The convergence of the

(m, n) series is not as quick as for the pseries but it does converge quickly enough

to make the Fourier-Bessel method an efficient met hod of evaluating the radiation

integral.

As a method of verifying the reflector antenna analysis theory, and the computer

program, it should be noted that the converged solution in figures (3.6) and (3.7)

match the plots from [17] except for the fact that the plots in [17] are plots of the ref-

erence and cross polarization radiation intensity and are normalized to 0 dB. Figures

(3.6) and (3.7) are plots of the reference and cross polarization directivity in dB and

as such are shifted up by the value of the maximum directivity. Recall, fiom section

3.1 that t the directivity in a given direction is defined as the radiation intensity of the

antenna in the given direction to that of an isotropic radiator.

As a second verification of the theory figure (3.8) is another plot of the directivity,

this time, only plotting the reference polarization directivity. The configuration used

is from [lo] and this time the feed is displaced fiom the focal point. This reflector has

F = 94.87X, a = 54.075A and dd = 70.94X. The feed is a x-polarized feed displaced

from the focal point and placed at (I. = 0, y. = -5.8X, z, = 0) where the source

9 (in O ) ($ = Ma)

Figure 3.7: (m, n)-series convergence (P = 2) for an offset reflector (F = 120X, a = 50X, do# = ?OX, y-polarized feed with q~ = q~ = 15.514)

Antenna Pattern of an Offset Reflector

0 1 2 3 4 5 6

9 (in ") (4 = 270°)

Figure 3.8: Plot of the reference polarization directivity for an offset reflector (F = 94.87X, a = 54.08X, doff = 70.941, x-polarized feed with q~ = 3.6 and q~ = 2.8)

coordinates aze such that the z, axis points towards the center of the reflector and

the x, axis is in the plane of symmetry of the rdector. The resulting antenna pattern

matches the one in [lo].

3.12 Properties of Offset Reflectors

In this section we use the reflector antenna analysis program to study some of the

properties of ofiet parabolic reflector antennas with a single feed. We will study the

effect of the q parameter of the feed on the radiation pattern of the reflector. We will

$SO examine what happens as the feed is laterally moved off of the focal point (i.e.

we examine the scanning characteristics of the reflector).

3.12.1 Edge 'Paper, Aperture Efficiency and the Effect of the q Parameter

In this section the effect of the primary pattern (the feed antenna pattern) on the

secondary pattern (the pattern of the reflector) is investigated. We will do this for a

symmetric reflector. The essential ideas and results carry over to the offset reflector. A

s ymrnet ric reflect or acheives its maximum directivity when the aperture is udormly

illuminated [50]. However, if a feed were to illuminate the reflector in this fashion a

large amount of radiation from the feed would spill past the reflector. This represents

lost power in the main beam. In addition a uniformly illuminated reflector will have

fairly high sidelobes. To reduce sidelobe levels and to reduce the amount of spillover

the illumination is tapered across the reflector. The parameter which defines the

degree to which the iUumination is tapered is the edge taper (ET), introduced in

section 3.11. The ET was defined as the ratio of the power density of the feed

radiation at the edge of the reflector to that at the center of the reflector. When

expressed in dB the ET is usually a negative value. As the ET is increased and

more of the feed's power is directed towards the center of the reflector rather than

the edges, the directivity of the main beam is reduced, the width of the main beam

increases and the sidelobe levels are reduced. Clearly there is a tradeoff involved. We

wish sidelobe levels to be decreased but we do not want the directivity of the main

beam to be reduced or the width of the main beam to increase.

Figure (3.9) demonstrates the effect of the edge taper on the antenna pattern of

the reflector. The reference polarization directivity is found for two feeds. The first

has an ET of -5dB (qE = q~ = 4.599). The second feed has an ET of -20 dB

(qE = q~ = 18.3973). Figure (3.9) clearly shows that as the edge taper increases, the

width of the main beam increases but the sidelobes are reduced substantially. Often

there is also a drop in the maximum directivity as the edge taper increases but that

isn't evident from figure (3.9).

Another performance measure for the antenna is the aperture eficiency. The

aperture efficiency relates the effective area of the antenna to the actual physical

radiating area of the ast ema, expressed mat hemat i c d y by

Clearly, the better the aperture efficiency the better use is being made of the physical

axea the antenna covers. The aperture efficiency is broken down into the product of

several efficiency factors, the most important of which are the spillover eficiency and

the taper eficiency,

Vap = f)spiltVtap* (3.98)

The spillover efficiency indicates the degree to which radiation is spilling past the

reflector surface. The taper efficiency indicates how closely the illumination of the

reflector approaches the uniform illumination. Nonuniformity of the illumination

is caused by tapering of the feed radiation pattern and by unequal path loss to

different parts of the reflector. The maximization of spillover efficiency and of the

taper efficiency are contradictory goals. There is a tradeoff involved in order to

increase the overall aperture efficiency

Effect of Edge Taper on Antenna Pattern

Figure 3.9: Effect of the edge taper. The antenna pattern of a symmetric reflector with an x-polarized feed with edge taper of (a) ET = -5 dB and (b) ET = -20 dB (F = 100X, a = 50X)

3.12.2 Reflector Antenna Pattern Characteristics of Off-Focus Feeds

One of the properties of offset reflector antennas that we are most interested in is the

scanning properties of the reflector. In other words, we wish to study the effect on

the secondary pattern of a lateral displacement of the feed from the focal point. This

is of central importance. In order for us to place a beam in a direction other than

along the axis of the reflector we have to either physically move the reflector or move

the feed off of the focal point. Of course, when we have an array feed only a single

feed can be on the focal point.

When a feed is laterally moved off of the focal point some interesting effects are

observed. First of d, and most importantly, the beam generated by the reflector

is scanned away from the axis of the reflector. Therefore, by properly choosing the

location of the feed in the vicinity of the focal point, we can control where the re-

sultant beam points. To figure out where a beam is going to point for a given feed

displacement, the familiar rule &om physical optics that the angle of incidence equals

the angle of reflection cao be applied. In practice, the boresight of the beam points

away from the point calculated in this way, by a factor appropriately called the beam

deviation factor (BDF). Figure (3.10) shows a feed displaced from the focal point

of an offset parabolic reflector. According to the angle of incidence equals angle of

reflection rule, OF = OB where OF is the feed tilt angle and OB is the beam scan angle

as defined in figure (3.10). The BDF is defined as

@B BDF = -. OF

Reference [55] gives charts for the BDF for an offset reflector. Knowing the BDF

and the bcation of the feed we can determine where the boresight of the main beam

will be due to that feed. When an array of feeds is used, each of the feeds due to

its unique location on the feed plane, generates its own beam in a unique direction.

The closer we can place the feeds to each other, the closer the beams can be placed

Figure 3.10: The angles OF and Bs for a feed displaced fiom th; focal point of an offset reflector with focal length F, aperture diameter D, and angle to the center of the aperture of the reflector Qc.

together. The resultant pattern of the antenna is the superposition of these beams.

Conversely, when a plane wave is incident on the antenna, the direction of the

plane wave will dictate where on the feed plane the radiation is concentrated. To

give a concrete example, if a feed is placed such that it will generate its main beam

in the direction 4 = 0' and 9 = 2.5" then if an incident plane wave arrives fiom that

direction more power will be received at that feed than at any of the others and the

feeds in the near vicinity of that feed will generally receive more power than those

that axe further away.

Now, let us discuss further what the antenna pattern looks like when a feed is

laterally displaced. As a feed is displaced and the resultant beam is scanned there

is degradation in the beam. The maximum directivity drops, the sidelobes rise, the

main beam widens and the shape of the pattern becomes more and more distorted.

At a certain scan angle the sidelobes start to join the main beam. The physical

justification for this degradation can be understood by first thinking about a feed on

the focal point. The distance &om the focal point, to a point on the reflector, to the

plane which is perpendicular to the axis of the reflector and includes the focal point

is the same for all points on the reflector. Therefore the waves will all be essentially

in phase in that plane. That's the defining characteristic of the parabola and the

motivation for its use. The waves will add up constructively. As a feed is moved

off of the focal point this distance relationship for each point on the reflector no

longer holds and there is some destructive interference. This is a somewhat simplified

physical explanation but it is valuable in terms of getting a better understanding of

the operation of the reflector.

The following graphs, generated by the antenna analysis program, show how the

beam is degraded as it is scanned away from the axis of the reflector. Two configu-

rations were studied to demonstrate how the scanning properties vary with the focal

Length of the reflector. The first configuration has F = 150h, a = 50X, and doB = 70X.

This leads to F / D = 1.5 and F / D p = 0.625. According to [55] the puameter F / D p

characterizes the scanning performance of the reflector better than FID. The feed

has ET = -10 dB. Figures (3.11) and (3.12) are results for this configuration. Figure

(3.11) shows how the beam changes as the beam is scanned 2 and then 4 beamwidths

away. Figure (3.12) shows the more drastic changes as the beam is scanned much

further away with the last beam being about 13 beamwidths away.

Larger F / D ratios lead to better scan performance [&I. Figure (3.13) shows

a second configuration where a smaller focal length is used. The parameters are

F = 96X, a = 50A, dO8 = ?Oh and ET = -10 dB. Here, F / D = 0.96 and F / D p = 0.4.

Figure (3.13) shows the severe degradation in scan performance for this reflector.

Comparison of figure (3.13) with figure (3.12) supports the claim that a larger F/D

ratio leads to better scan performance. Unfortunately, there are trade offs involved in

increasing the focal length to achieve th6 superior scan performance. First of all it is

structurally more difEcult to implement, and secondly a longer focal length requires

a more directive feed to obtain the same edge taper. The Cassegrain dual reflector

Lateral Displacement of the Feed (FID, = 0.625)

8 in degrees

Figure 3.11: Lateral displacement of feed from focal point. F = 150A, a = 50X, d f l = 70X7 F/ 4 = 0.625, x-polarized feed with ET = -10 dB. Displacement of feed in the x, direction by (a) OX, (b) -4X, (c) -8A.

Lateral Displacement of the Feed (FID, = 0.625)

0 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2

8 in degrees

Figure 3.12: Lateral displacement of feed from focal point. F = 150X, a = 50X, do* = 70X, F/ Dp = 0.625, s-polarized feed with ET = -10 dB. Displacement of feed in the I, direction by (a) OX, (b) -6A, (c) -12X, (d) - M A , (e) -24A.

Reference Polarkation Dlrectlvily in dB .&

VI 8 C: 0 I 8 R B 0

antenna, which is commonly used in Earth station antennas, is one design that has

been developed to increase the focal length of the antenna to try and acheive better

scan performance.

3.13 Extension to an Array Feed

In this thesis we use an may of feeds to illuminate the reflector. There are two ways

of obtaining the far field pattern of the reflector with an array feed, both based on

superposition. The first method is cded primary field superposition In primary

field superposition the fields of each of the individual feed elements are weighted and

added up at the reflector surface. Then a method like the Fourier-Bessel Method

with the physical optics approximation may be used to calculate the secondary far

field. The second method of computing the electric field of the array is to find

the secondary electric field due to each element separately, and then weight and

add up the secondary fields. This method is called secondary field superposition.

Both methods are equivalent. The secondary field superposition method is the one

used in this thesis. It is more efficient in our situation because we wish to vary

the weights frequently. With the secondary field superposition method the lengthy

time to compute the secondary pattern due to each feed is done up front and then

these patterns are stored. To find the final secondary pattern we weight and add

up the stored patterns. We repeat this procedure for each of the reference and cross

polarizations.

Let us express this process mathematically. It is possible to combine the feed

patterns described in equations (3.87) to (3.95) in two equations by introducing the

quantities as, bsl and 11, [lo]. The two equations, which we will specify as the feed

pattern for feed i, are

~ ; ( r . ) = (Jij) [e. w*(&) (a, sin c$= - bseid. cos c&) .-ikr.

+Au,$~) (a. cos #s + b d 9 * sin &)] -- (3.101) r s

The quantities as, b., and tit. define the polarization of the source. The polarization

parameten are normalized such that

2 a, + bt = 1. (3.102)

For a x-polarized source, as = L,6. = 0, $s = 0. For a y-polarized source, a, = 0, b, =

1, , = 0 A LHCP source can be derived by setting the polarization parameters to

a, = &, 6, = -& $. = 90' and a RHCP source has the polarization parameters set

1 1 to a, = T , b s = 7+ = -90'.

When feed i is excited by a unit-amplitude source, the secondary field due to feed

i may be expressed

where Fi(k) is the amplitude vector

the feed polarization, we describe the

of the secondary field of feed i. As we did for

polarization of the secondary field by three real

polarization parameters (a:, b:, $:). Ludwig's 3rd definition [lo, 521 of the reference

and cross-polarization vectors is given by

R = 8(a: cos 4 + b;@# sin 4) + &(-a, sin @ + bS&" cos 4) (3.104)

c = 9 (a: sin 4 - 6:e-j*' cos #) + &a, cos q5 + bse-j4~ sin 4). (3.105)

These two vectors are orthogonal and unitary meaning that R . iL' = c - e' = 1 and

k- C* = c - iL' = 0. Lam states [lo] that the polarization parameters of the secondary

pat tern (a:, b:, +:) are related to the polarization parameters of the primary pattern

(a,, bs, $,) for a single reflector offset in the x-direction by

In other words the phase of E, is changed by 180° upon reflection. Therefore a LHCP

field becomes a RHCP field and vice versa. The component of the secondary field

due to feed i in the reference polarization is given by

where the component of the amplitude vector of the secondary field for feed i in the

reference polarization is given by

The secondary field of the reflector with an array feed is a linear superposition of

the secondary field of each of the feeds. Introducing the complex weight vector,

the secondary field of the reflector in the reference polarization is

This may be written as a vector multiplication by introducing the vector F(k, R) as

where the ith element of the vector represents the component of the amplitude vector

of the ith element in the direction k and in the reference polarization. Equation

(3.111) may now be written as

A normalized plot of [wtF(k, R)[* (which is the reference polarization radiation in-

tensity of the reflector antenna in Wattslsteradian) gives the antenna pattern of the

reflect or (in the reference polarization).

3.14 Calculation of the Directivity

As mentioned, the program calculates the amplitude vector F(k) which describes the

angular variation of the electromagnetic radiation. We have defined the radiation

intensity and stated that the antenna pattern of the reflector is a normalized plot of

the radiation intensity. Often, the directivity is plotted. The directivity is a measure

of how the antenna preferentially radiates power more in some directions than others.

A plot of the directivity differs from a plot of the normalized radiation intensity only

in its maximum d u e . The shape of the patterns is the same. A normalized plot of

radiation intensity has its maximum value at 0 dB whereas a plot of the directivity

has its maximum value at the maximum directivity. As defined in section 3.1, the

directivity is defmed as the radiation intensity of the antenna in a given direction to

the radiation intensity of an isotropic antenna in that same direction. This section

outlines briefly how the directivity is calculated.

The directivity in the reference polarization and in the direction k is given by (see

Appendix A)

where Pmd is the total time-averaged power radiated by the may. So in order to

calculate the directivity we need to know the vector F ( ~ , R ) , the weight vector w,

and the total time-averaged power radiated by the array. The theory to this point

has allowed us to calculate the vector F(k, R) by evaluating &(k, R) for each feed

i. It hasn't been demonstrated yet how to evaluate the total time-averaged power

radiated. One way to do this would be to integrate the radiation intensity over a

sphere around the antenna. An alternative has been presented by Lam [lo] (whose

derivation was based on the work of Rahmat-Samii and Galinddsrael [ll]) which is

va.Iid provided the feeds are aIl on a common plane called the feed plane and all of

the feeds have the same polarization. Figure (3.14) shows the feed plane.

Lam [lo] has shown that the total power radiated by the may of feeds can be

z 4

Towards the center of the reflector

' Y

02

Feed Plane

Figure 3.14: The feed plane. The distance and angle of feed n with respect to feed n are shown as p,, and &,,.

expressed by:

where A is an NE x lVE Hermitian matrix with elements given by

and where the values C, , BLT, BL:), DL?, DL!) are respectively given by the equa-

tions

Co = 1 + 2a.b. cos & sin 24- + (a3 - a:) cos 24,,, (3.118)

~ ( ~ 1 rnn = 2"~-1r(uH) Jy, ( b r n n ) ( b m n ) " ~ '

and

In equation (3.118), as, b,, and ?,bS are the polarization parameters of the feeds (as-

sumed to be the same for all feeds). The parameters p,,, and &, represent respec-

tively the distance and angle of feed m with respect to feed n (see figure (3.14)). If

feed rn is located by the Cartesian coordinates (xm, y,) on the feed plane, and feed

n by (s,, y,) then the parameters p,,,, and +,, are given by

Pmn = \/(xm - xn)* + ( ~ m - fi)2 (3.123)

In equation (3.119), (3.120), (3.121), and (3.122) the parameters us and v~ have

been introduced. They are functions of the q values of feed m and n and are defined

and

Chapter 4

Beamforming With An Offset Parabolic Reflector Antenna

4.1 Introduction and Overview

This chapter brings together many of the ideas and results from previous chapters.

The objective of this thesis is to demonstrate the performance of adaptive algorithms

on an offset reflector antenna with an array feed, to this point referred to a s a multiple

beam antenna (MBA). I . chapter 2, a signal model was developed which was general

enough to include both the direct radiating array (DRA) and the MBA. The difference

between a DRA and a MBA is that each produces different steering vectors for a given

direction o i arrival. The elements of the steering vector represent the response of

each of the antenna elements to a plane wave signal arriving from the angle ( 9 , 4 ) and

frequency f . This signal is received with a gain and a phase shift. It is this gain and

phase shift which determine the steering vector. In section 2.2, it was demonstrated

that the steering vector of a d o r m h e a r array (ULA) with isotropic elements is

determined by a unit gain and a phase shift. Knowing the direction of arrival of a

plane wave, the separation of the antenna elements, and the frequency of the plane

wave, we can determine analytically the interelement phase shift. For the case of a

MBA it isn't as simple. We can not determine the gain and phase shift presented to

a plane wave from direction (8 ,d ) aoalyticdy because it is not simply a phase shiR

between the elements. In the next section of this chapter it will be shown how the

steering vector can be determined numerically. The theory of offset reflector antenna

analysis and the implementation of that theory in a computer program described in

chapter 3 is used, along with the principle of reciprocity, described in Appendix A,

to determine numerically the steering vector. Once we know the steering vector the

rest of the beamforming theory described in chapter 2 applies. After describing how

to determine the steering vector for a MBA, optimum combining with a MBA will

be demonstrated. This will be followed by a description of some adaptive algorithms

and then the simulation results for one of those algorithms when used with a MBA.

Finally, some of the qualitative differences between beamforming with a D M and an

MBA will be discussed.

4.2 Bedorming With a Multiple Beam Antenna

In chapter 2, the narrowband signal model was derived. In that model the steering

vector represented the response of the antenna to a unit-amplitude plane wave arriving

from an angle (8 ,4) . In order to study adaptive bedonning with an offset parabolic

reflector we need to determine the steering vector. In other words we need to h o w

the response of each of the antenna elements to a unit-amplitude plane wave arriving

from direction (8 ,#) .

In chapter 3, the theory of offset reflector antenna analysis was developed and the

result was that we could find the electric field of an antenna at the far-field point

described by the vector r = (r, 0,4). The electric field (secondary field) due to feed

i, when excited by a unit-amplitude source, may be described by

The computer program based on the reflector antenna analysis theory in chapter

3 evaluated the amplitude vector Fi(k). The amplitude vector, Fi(k), is a vector

which is independent of distance r and depends only on the angles (0,#). We de-

fine a reference and cross polarization described by the orthogonal vectors R and c

respectively. The component of the amplitude vector in the reference polarization,

F(k, R), is given by

F(k, R) = F(k) iL'. (4-2)

We may similarly define the component in the cross polarization as

Reciprocity is now used to find the ith element in the steering vector which r e p

resents the response of the ith antenna element to a unit-amplitude plane wave.

Reciprocity is a theorem that relates the transmitting properties of an antenna to the

receiving properties. We will consider the two situations illustrated in figures (4.1)

and (4.2).

A

Figure 4.1: An antenna transmitting when excited by a unit amplitude source

The fist situation is an antenna excited by a unit-amplitude source. It transmits

into space the electromagnetic fields,

Figure 4.2: An antenna receiving a plane wave in the reference polarization and with propagation vector k

Now consider that same antenna receiving a plane wave with propagation vector -k

and in the reference polarization (with polarization &). The incident plane wave

may be expressed by

where C is the amplitude of the plane wave in ~ " ~ I r n . Reciprocity states that the

received amplitude, b,, under the matched load condition (a, = 0) is given by [55],

Thus, assuming that the antenna is under the matched load condition and that the

antenna is lossless (all power incident on the antenna is radiated) the value of the

amplitude vector in the reference polarization, numerically evaluated by the methods

in chapter 3, may be used to find the response of the antenna to an incident plane

wave. Under these assumptions, the response of the ith element to a unit-amplitude

plaae wave, and therefore the ith element of the steering vector is

where, as defined in section 3.13, Fi(k, R) is the component of the amplitude vector

for element i in direction k and in the reference polarization. This is how the steering

vector is evaluated for a MBA.

4.3 Optimum Combining With an Offset Reflector Antenna

In section 2.7, optimum combining was demonstrated with a uniform hear array.

Optimum combining assumes that we've reached the optimum solution in some way.

Adaptive algorithms, through their adaptation, approach the optimum solution and

the rate at which the algorithm approaches this optimum solution is called the con-

vergence rate of the adaptive algorithm. In this section optimum combining will be

demonstrated for a MBA. This demonstration is not meant to be a system study or

a comprehensive study of optimum combining on a MBA. Instead the intention is to

give an example which demonstrates the capability of the simulation tool developed.

The configuration used was the same as that used in [lo] where the use of a cluster

feed was investigated to maximize the directivity in a given direction. The reflector

used has F = 94.87X, a = 54.08X, and do* = 70.94X. An array of 7 feeds will be

used in the hexagonal configuration shown in figure (4.3) with each feed being linearly

x-polarized and having q,g = 3.6 and q~ = 2.8. The center feed is displaced from the

focal point and placed at x = -5.53X. Each of the feeds is placed an equal distance

apart with that spacing being set to d = LA. The frequency is selected to be 5 GHz.

The main beam peak will be in the vicinity of (4 = 0°, B = 3'). The location of the

main beam peak has been determined by figuring out the angle OF in figure (3.10)

from the geometry and then setting OB = OF. In other words, we are neglecting the

beam deviation factor.

Figure 4.3: The hexagonal configuration of the array feed on the feed plane.

This simulation was run for four different cases. In the first case we had only

a single desired user and in each subsequent case an interfering user was added. A

unique angle of arrival was selected for each of the users. Since we only want to look

at Zdimensional plots of the antenna pattern we fix the # arriving angle to 0'. The

value of 8 was then varied. The desired user wived fiom 8 = 3.0'. Interferer one

wived from 8 = 2.4', the second interferer &om 0 = 4.2O, and the third interferer

from 0 = 2.0". Each of the users was selected to have a signal to noise ratio of 8 dB

which was chosen as a typical value.

The theory on optimum combining was derived in chapter 2. Looking at equation

(2.58) we can find the optimum weights to within a scalar constant by determining

the inverse of the correlation matrix and the steering vector of the desired user.

The theory behind finding the steering vector when the antenna is a MBA was

presented in section 4.2. The steering vector is found by using the computer program

developed kom the theory in chapter 3 to find the amplitude vector of feed element

i (i = 1, . . . , Ns) , &(k, R) , and then these values were used with equation (4.7) to

find each of the NE elements of the steering vector. It should be noted that in the

optimum combining simulation we know the angle of arrival of each user since we

assume that we have reached the optimum solution. We are investigating what that

optimum solution gives us in terms of SINR and antenna pattern.

The correlation matrix, 0, is determined from equation (2.53) to within a scalar

constant with knowledge of the steering vector and SNR for each of the users. With

knowledge of the correlation matrix to within a scalar constant, we also know the

inverse of the correlation matrix to within a scalar constant. In summary, we can

determine the inverse of the correlation matrix to within a scalar constant, and the

steering vector of the desired user and with these two components the optimum weight

vector to within a constant.

Knowing the optimum weight vector to within a constant allows us to determine

the optimum output SINR (since the optimum SINR is unchanged if the weight

vector is scaled from the optimum weight vector - see section 2.4) and the antenna

pattern. Equation (2.34) was used to find the SINR. The antenna pattern (plot of

reference polarization directivity) was found using equation (3.115) with the total

time-averaged power radiated being determined using equations (3.116) - (3-126) and

the vector F(k, R) being determined with the use of the antenna analysis computer

program.

The first case considered was simply the case of a single desired user. The user

has pd = 8 dB which, as described in chapter 2, is the notation for the SNR of

the desired user. The desired user arrives from direction (0 = 3", 4 = 0"). The

statistically optimum solution for the weights was found and figure (4.4) shows the

resulting reference polarization directivity of the MBA. The antenna pattern shows

a maximum in the direction of the desired user. The SII\TT(. is 21.06 dB. We can

use this value as an upper bound for cases with interferers. The closer the SINR is

Optimum Combining With A MBA

Figure 4.4: Optimum combining with a MBA: case # 1. Desired user at 3' and with pd = 8 dB. No interferers.

to this d u e the better the antenna array is canceling out the effects of interferers.

Note that the SINR won't necessarily reach this bound but it gives an idea as to how

well the array is nulling out the effect of interfering users. The maximum reference

polarization directivity was calculated to be 48.59 dB.

Now let us add an interferer. The interferer has p, = 8 dS (SNR of interferer 1)

and arrives &om angle 2.4'. Figure (4.5) shows the resulting antenna pattern after

the optimum solution is found. The antenna pattern continues to show a maximum

in the direction of the desired user and now a null is placed in the pattern at the angle

that the interferer arrives, 2.4'. The SLNR is 20.48 dB, a drop of 0.58 dB &om the

21.06 dB acheived with no interferers. The null that is created at 2.4* is very deep.

The maximum reference polarization directivity was found as 48.40 dB, a slight drop

from case 1.

Case 3 will include a second interferer on the other side of the main beam. In-

terferer 2 arrives from 4.2' and has p2 = 8 dB. Figure (4.6) shows the results. Once

again the MBA is abIe to null out the interfering users and place a maximum in the

direction of the desired user. The SINR acheived is 20.05 dB, another drop of about

0.43 dB. Both nulls at the positions of the interferers are very deep. The maximum

directivity dropped once again to 48.07 dB.

Case 4 includes a third interferer at 2.0' and with f i = 8 dB. Figure (4.7) shows

the resulting directivity as a function of 8. Here the MBA hasn't been able to null all

the interferers. The level at the position of the 3rd interferer at 2.0' is substantially

lower but at the cost of the deep null at 4.2'' the position of the 2nd interferer.

This degradation in performance is matched by the much lower SINR at 14.32 dB.

The maximum directivity was 48.18 dB, slightly up from case 3. It is diEcult to

explain why the performance degraded so severely when the 3rd interferer was added.

It is possible that the 3rd interferer was placed too close to the 2nd interferer to

simultaneously null them or it is possible that the axray has run out of degrees of

freedom in the 4 = 0 plane.


Figure 4.5: Optimum combining with a MBA: case # 2. Desired user at 3' and with p d = 8 dB. 1 interferer at 2.4' and with pl = 8 dB.


Figure 4.6: Optimum combining with a MBA: case # 3. Desired user at 3 O and with pd = 8 dB. Interferer # 1 at 2.4' with pl = 8 dB and interferer # 2 at 4.2' with pz = 8 dB.


Figure 4.7: Optimum combining with a MBA: case # 4. Desired user at 3' and with p d = 8 dB. Interferer # 1 at 2.4' with pl = 8 dB, interferer # 2 at 4.2" with pz = 8 dB, and interferer # 3 at 2.0' with p3 = 8 dB.

4.4 Adaptive Algorithms

In chapter 2, the statistically optimum solution for the weight vector was given by

equation (Ul),

wort = +-LS~om, (4.8)

where + is the correlation matrix of the input data vector and S,, is the correlation

vector which represents the correlation of the input data vector with a reference signal.

This solution assumed that the input data was wide-sense stationary and also that

the second-order statistics of the input data are known. In practice, the data is not

wide-sense stationary and we do not know the second order statistics. To get around

this problem we generally estimate the secondorder statistics with the assumption

that the underlying time-series is ergodic. We use adaptive algorithms to track the

optimum solution as the environment changes.

There are many different adaptive algorithms that have been proposed in the Liter-

ature each having their strengths and weaknesses. There are three general approaches

taken in adaptive algorithms: (1) direction finding followed by classical beamforming,

(2) reference-signal based algorithms, and (3) blind spatid filtering algorithms.

The first method uses a direction finding algorithm to find the directions of arrival

of d of the users of the system and then uses a data-independent type of b e d o r m -

ing algorithm. This method therefore places n d s and maximums in the antenna

pattern with the locations of those nulls aod maximums being found by one of the

many location finding algorithms, such as MUSIC (Multiple Signal Classification)

[62], available in the literature. It should be noted that this isn't a statistically o p

timum approach. However, this does provide the advantage of being able to control

characteristics of the antenna pattern such as the side lobe levels. The disadvantage

of using this method is the huge computational cost of both finding the direction

of the users of the system and forming the beams with this knowledge. A further

disadvantage is sensitivity to errors in the estimates of the directions of aaival.

The second approach taken in adaptive algorithms is the use of a reference signal.

The advantage of the referencesignal method is it's simplicity. Knowing a reference

signal that is corre1ated with the desired user and uncorrelated with interferers we

find estimates of the correlation matrix and correlation vector and use equation (4.8)

to find the optimum weights. The disadvantage is the need to provide a reference

signal. This may be done by giving each user a code which is orthogonal to the code

of the other users [54]. The code is placed in a preamble which is used to form the

beam. The problem is that the need to transmit a preamble increases the bandwidth

and power required of the system. This is where the third approach to adaptive

algorithms enters the picture, blind spatial filtering.

Blind spatial filtering dgorithms attempt to form beams without use of a training

signal or prior knowledge of the directions of arrival of the users [29, 31, 32, 34, 35,

36, 381.

In this section we consider a simple adaptive algorithm based on the reference

signal approach. In chapter 5 we consider a special class of blind spatial filtering

algorithms which exploit the cyclostationary properties of the signal.

4.5 Direct Matrix Inversion

Direct matrix inversion (DMI) [3] is a reference signal based technique and therefore

requires a reference signal r( t ) . It is sometimes referred to in the literature as sample

matrix inversion (SMI). Direct matrix inversion forxns an estimate of the correlation

matrix &om equation (4.8) using N, samples of the input vector, x(l),

and it estimates the correlation vector by

where r( l ) is the lth sample of the reference signal. The optimum weight vector is

then calculated, as in equation (4.8) to be

DM1 is a block adaptation approach in that a block of Np symbols is read in before

the optimum solution is found.

Reed, Mdett and Breman [3] originally studied the DM1 approach and found

that the algorithm converged with roughly 2NE samples where NE is the number

of a n t e ~ a elements. This has a much better rate of convergence than the well-

known least mean squares (LMS) algorithm and, moreover, its convergence rate is

independent of the spread of the eigendues of the correlation matrix (which is a

problem with LMS). The difEculty with DM1 is that it is computationally intensive

as matrix inversion is an order (NE)3 process. This compares to the LMS algorithm

which is an order NE process. An adaptive dgorithm with the quick convergence of

DM1 and the low computational cost of LMS is still an elusive goal in the field of

adaptive algorithms.

4.6 Simulation of the Direct Matrix Inversion Al- gorit hm

In this section a baseband simulation of the direct matrix inversion algorithm is de-

scribed and the results presented. Simulations of this sort with uniform linear mays

have been performed in the literature [54]. This simulation demonstrates the perfor-

mance of DM1 on a multiple beam antenna. This section is also a precursor to the

next chapter where an adaptive algorithm which takes advantage of the cyclostation-

ary properties of the signal is described and a similar simulation is performed. What

this simulation demonstrates is that adaptive algorithms can be performed on MBAs

without any changes to the algorithm - that is, as long as the algorithm doesn't

impose any geometrical constrictions on the antenna array.

The simulation is a baseband simulation with 100 trials being performed. The

signals used were mutually independent BPSK waveforms with a square pulse shape.

Time is normalized to the sampling period. We let each symbol have a duration

of 5 sample periods and 1000 samples are taken (giving a total of 200 symbols).

There will be a single desired user with an S N R of 10 dB (10 dB chosen as a typical

d u e ) arriving from 3 O . A single interfering user will also have an SNR of 10 dB

and will arrive from 2.4". The geometry of the a n t e ~ a and the configuration of the

antenna array will be the same as that of section (4.3). That is, the antenna will

have F = 94.87X, a = 54.08X, and do* = 70.94X. An m a y of 7 feeds will be used in

the configuration shown in figure (4.3) with each feed being linearly x-polarized and

having q~ = 3.6 and q~ = 2.8. The center feed is displaced from the focal point and

placed at x = -5.53X. Each of the feeds is placed an equal distance apart with that

spacing being set to 1X. The frequency is selected to be 5 GHz.

For each user a random bit sequence is generated. In addition, a random initial

phase is selected for each user, where the phase is uniformly distributed in the range 0

to 2n. We assume that a reference signal is a d a b l e which has the same bit sequence

as the desired user's signal and is in phase with that signal. This ensures that the

reference signal will be correlated with the desired user.

At each sample the received signal at the antenna array was determined. This

was done by adding up the contribution of the desired signal, the interferer, and the

noise. The desired user's and the interferer's received signal across the array were

found by multiplying the appropriate BPSK waveform by the steering vector of the

user. Equation (4.7) was used along with the reflector antenna analysis program to

derive the steering vectors. To generate the received noise vector complex random

noise samples were generated using a noise Mtiance of 1. Adding the desired user's

signal with that of the interferer and adding the noise samples gives the resulting

received signal at the array. With the desired user's bit sequence and intid phase the

reference signal available to the DM1 algorithm was determined at each sample.

As the resulting received signal at the array and reference signal were determined

at each sample, the sample correlation matrix and sample correlation vector were

updated using equations (4.9) and (4.10). Every 25 samples (5 symbols) the opti-

mum weight vector was determined by inverting the sample correlation matrix and

then multiplying it by the conjugate of the sample correlation vector as in equation

(4.11). Using these weights the output SINR was calculated using equation (2.33). In

equation (2.33) the correlation matrices used were not sample correlation matrices,

but the actual correlation matrices calculated knowing the steering vectors, signal to

noise ratios, and the noise variance.

The graph in figure (4.8) shows the output SINR as it varies with the number

of symbol periods considered. This graph shows the very quick convergence of the

DM1 algorithm. Reed, Mdet t , and B r e ~ a n [3] examined the convergence of DM1

and found that the weights converged such that the SINR was within 3 dB of the

optimum within 2NE - 1 samples. This claim is supported by the simulation results

presented here. Quick convergence is important when the beamforming situation is

changing frequently and rapidly and the new beams must be created just as quickly.

4.7 Discussion

To this point it has been shown how beamforming can be applied to a MBA as well as

on the DRA. In this section some of the differences between these two situations will

be discussed. The key difference between a MBA and a DRA is of course the presence

of the reflector. What the reflector does is to act as an initial spatial filter (which is a

continuous filter). The reflector attenuates greatly signals arriving from wide angles.

For this reason a reflector antenna with m a y feed is suitable for a geostationary

application where the angular range of vision is somewhere in the region of 5-10

degrees about the boresight. An antenna beam from a reflector scanned far out is

greatly distorted and attenuated. A reflector antenna could not be used in a low or

medium earth orbit. It just doesn't have the scanning ability. A DRA on the other

Convergence of the DMI Algorithm on a MBA

0 25 50 75 100 125 150 175 200

Symbol Number

Figure 4.8: Convergence of the DM1 algorithm when performed on a MBA (same an- t e ~ a and feed configuration as in optimum combining demonstration) with a desired user (pd = 10 dB, Od = 3.0') and a single interfering user (pl = 10 dB, = 2.4')

hand can scan out to wide angles quite easily.

One of the effects of the reflector is to focus a signal arriving kom an arbitrary

angle. In other words a signal arriving from some direction (8,# will be received with

the most amplitude in an associated local region of the focal region of the reflector.

This effect was described when we studied the effect of moving a feed off of the focal

point in chapter 3. The result of this is that we only need to combine the signals

from some of the antenna elements to form a beam. In a DRA all of the antenna

elements must be combined to form a beam. Therefore the DRA requires much more

hardware, many more computations and therefore more power. On a satellite where

weight and power usage axe critical factors the reduction in hardware required for the

MBA is a great advantage.

In the next chapter cyciostationary adaptive algorithms ase described and demon-

strated to work on a MBA.

Chapter 5

Cyclic Beamforming Algorithms on a Multiple Beam Antenna

5.1 Introduction and Overview

This chapter considers cyclic beamforming algorithms. Cyclic beamforming a l p

rithrns are a class of blind spatial filtering algorithms which exploit property restoral

techniques to restore known properties of the desired signal in the output signal of the

array. The key advantage of these blind spatial filtering algorithms is that they don't

require a training signal which takes up valuable bandwidth and power resources.

There are two property restoral approaches that have been suggested in the Litera-

ture. The fist is the constant modulus algorithm which takes advantage of the low

modulus variation of most communication signals. The second property exploited in

property restoral algorithms has been cyclostationarity. There are many communi-

cat ion signals which exhi bit cyclost ationarity and this cyclost at ionarity implies that

the signal is spectrally self-coherent. In other words, many communication signals

are highly correlated with frequency shifted (and possibly conjugated) versions of

themselves. Therefore by properly weighting and summing up frequency-shifted ver-

sions of the received signal, a desired signal may be extracted from an environment

of spectrally incoherent interference.

Cyclic beamforming algorithms are not without their disadvantages and limita-

tions. First of all, the cyclic beamforming algorithms suggested to date either suffer

from a slow convergence rate or a large number of computations. The other key

disadvantage of cyclic beamforming algorithms is that they impose limitations on

the modulation techniques employed. Certain modulation techniques exhibit more

cyclostationarity than others. Despite these disad~i~ntages and limitations, cyclic

beamforming algorithms are very interesting. These algorithms are a fairly recent

addition to the field of adaptive beamforming and there is still a great deal of room

for improvement and innontion.

This chapter focuses on the use of cyclic b e a m f o ~ g algorithms for a multiple

beam antenna. Before exhibiting the performance of a cyclic beamforming algorithm

on a MBA this chapter will discuss the theory behind cyclostationary signal analysis,

and then introduce a number of cyclic blind spatial filtering algorithms which have

been proposed in the literature.

5.2 Cyclostationary Signal Analysis

The theory of cyclostationary signal analysis has largely been developed by William

A. Gardner and his graduate students. Gardner's 1987 text "Statistical Spectral

Analysis: A Non-Probabilistic Theory" [30] was the first full development of the non-

statistical theory of cyclostationary timeseries In addition, Gardner has written an

excellent tutorial paper on cyclostationary signals titled "Exploitation of Spectral

Redundancy in Cyclostationary Signals" [38] which was published in the April 1991

edition of IEEE Signal Processing Magazine. More recently, Gatdner has edited

the book UCyclostationarity in Communications and Signal Processingn [31] in 1993.

This book covers some of the most recent research in the field of cyclostationary signal

processing. In this thesis the non-statistical version of the theory of cyclostationarity,

as developed by Gardner [30, 31, 33, 34, 35, 36, 381, will be used. Only the key

definitions and ideas of cyclostationary signal processing will be presented. The reader

is referred to these other treatments for greater detail.

The key quantity in this chapter is the cyclic autocomlation function (CAF) of

x ( t ) defined by

rm(r) = ( z ( t + r/2) z*(t - ~ / 2 ) e - j ~ % ~ ' )a (5-1)

where T is a time lag, a is a value called the cycle fiqueney and the infiniteduration

timeaveraging operation has been used

We may also define the cyclic conjugate-correlation /unction of x ( t ) defined by

The CAF is a quadratic nonlinear transformation. If the CAF of a time-series x( t )

is nonzero for some value cr and time lag r then the signal x ( t ) is said to be second-

order cyclostationary. Note that for a = 0, the CAF reduces to the conventional

autocorrelation hc t ion which is

R&(r) may be thought of as a generalization of the autocorrelation function where

a cyclic weighting factor e-jzrat is included. Note that the CAF may be rewritten as

By defining the two functions u( t ) and v ( t ) by

equation (5.5) may be written as a conventional cross-correlation function

When a signal is multiplied by e+jTat it is translated in frequency by 4 2 . There-

fore u( t ) and u ( t ) represent frequency shifted versions of x( t ) by -a/2 and 4 2

93

respectively. Since the CAF may be written as a cross-correlation fimction of u(t)

and v ( t ) it follows that the CAF of x ( t ) is nonzero only if u(t) and v( t ) are correlated.

Therefore x( t ) is second-order cyclostationary if and only if x( t ) exhibits spectral self-

coherence for frequency separation a. Note that if the cyclic conjugatecorrektion

fuoction is nonzero for some value of cr and r then x ( t ) is said to be spectrally conju-

gate self-coheeent for frequency separation a.

The introduction of u(t ) and v( t ) also dows us to introduce an appropriate nor-

malization of the CAF. If u(t ) and v ( t ) are zero mean then the cross-correlation

function defined above is equivalent to the cross-covariance function

The appropriate normalization factor is the geometric mean of the two temporal

variances

Therefore the normalized quantity called the cyclic temporal correlation function (also

called the spectral self-coherence function in the literature) is defined by

The magnitude of the cyclic temporal correlation function, ly&(r) 1, mies between 0

and 1 and represents the strength of the correlation. It is referred to as the feature

strength and it is aa important quantity in determining the convergence of cyclic

adaptive beamforming algorithms.

Before proceeding to a discussion of cyclic blind spatial filtering algorithms one

modification of the cyclic autocorrelation function definition has to be made for the

situation where we have a vector of data as we do in array signal processing. If there

are NE elements in the m y then the CAF is defined as an NE x NE matrix

and the cyclic conjugate-correlation function of x(t ) is defined as

5.3 Cyclic Blind Spatial Filtering Algorithms

There are many cyclic blind spatial filtering algorithms that have been introduced

in the Literature in the past few years. In the next section a brief survey of these

algorithms will be presented and then this will be followed by a more detailed discus-

sion of one of the simplest of these algorithms, called LS-SCORE. LS-SCORE is an

algorithm in the spirit of the reference signal based algorithms such as direct matrix

inversion. The only difference is that LS-SCORE gets its reference signal blindly. In

other words, LS-SCORE extracts a reference signal that is correlated with the desired

user and uncorrelated with the interferers &om the incoming data. Other than that

LS-SCORE is exactly Like any other reference signal based algorithm.

5.3.1 Cyclic Blind Spatial Filtering Algorithms - A Brief Overiew

The initial work on cyclic blind spatial filtering algorithms was performed by Gad-

ner, Agee and Schell in the late 1980's [32]. They developed a set of algorithms

collectively referred to as the SCORE family of algorithms where SCORE refers to

Spectral Coherence REstomL Their basic idea is as follows. A signal which exhibits

cyclostationarity is spectrally self-coherent. This spectral self-coherence is degraded

by the addition of interfetence that is not spectrally self-coherent at the same value

of frequency shift. So, their approach is to restore the spectral self-coherence of the

signal of interest and thus the name Spectral Coherence Restoral. There are three

main SCORE algorithms: Least-Squares SCORE (LS-SCORE), Cross-SCORE, and

Auto-SCORE. Each has a different cost function based on some measure of spectral

self-coherence at the output of the spatial filter.

Least-squares SCORE [32] uses the familiar Ieast-squares cost function

where, y (t) = wt -x(t) is the output of the spatial filter, < - >T denotes timeaveraging

over the interval [0, TI, and r( t) is a reference signal derived from the data and given

by

r (t ) = ct - xc*) (t - T ) dzXQt. (5.17)

where c is a control vector (kept fixed) and the optional conjugation ('1 is applied

only if conjugate self-coherence is to be restored.

C ross-S C 0 RE [32] maximizes the strength of the temporal cross-correlation coe f- 2

Jicient, l s ( r ) 1 , between the output signal y(t) and the reference signal r ( t ) (from

equation (5.17)). This is done by adapting both the weight vector w and the control

vector c. The cost function becomes

Cross-SCORE has a better convergence rate than LS-SCORE because the control

vector c is also adapted. This imploved convergence rate is achieved at the cost of

increased computational complexity.

Unlike LS-SCORE which resembles conventional adaptive algorithms, and Cross-

SCORE which is really just an extension of LS-SCORE, Auto-SCORE [32] is a pure

property restoral algorithm. Auto-SCORE maximizes the spectral or conjugate self-

coherence strength at the output of the beamformer. In other words, the cost function

is given by

One of the disadvantages of the SCORE family of algorithms is their computa-

tional complexity. There have been several attempts at achieving an algorithm with

a reduced computational cost but similar performance to the SCORE algorithms.

Wu asd Wong [39, 401 have presented a family of algorithms c d e d CAB, short for

cyclic adaptive beamforming. CAB is a variant on Cross-SCORE. Instead of maxi-

mizing ly>(r) f , CAB attempts to maximize the cyclic sample comIation given by

I (Y ( t )r' (t )), I*. Several different variants on both the CAB and SCORE algorithms

have been suggested in the literature with varying computational requirements and

rates of convergence.

This section has very briefly gone over a few of the cyclic beamforming algorithms

proposed in the literature. There are several more, many of which are variants on the

ones discussed above. The next section will go into the LSSCORE algorithm in more

detail. The essential god of this chapter is to demonstrate that cyclic beaxdorming

will work on a multiple beam antenna. LS-SCORE was the chosen algorithm because

it is very similar to the algorithms already discussed and yet it demonstrates the

exploit at ion of the cyclost ationarity inherent in the signal. In other words, LS-SCORE

is the perfect dgorithm to build our understanding upon.

5.3.2 LS-SCORE

In this chapter LS-SCORE is the cyclic adaptive beamforming algorithm which we

concentrate our attention upon. As expressed in equation (5.16), LS-SCORE involves

a least-squares cost function

with r(t ) as the reference signal given by equation (5.17),

The value of the control vector c is kept fixed as we vary the weights. Recall that the

optional conjugation is only used if we are interested in restoring conjugate spectral

coherence. Reference [32] shows that the reference signal contains a component that is

correlated with the desired signal and a corruption term that is uncorrelated with both

the desired signal and the interference and noise. In fact, [32] goes on to show that

the square of the feature strength, 17z(r) 12, is a measure of the relative strength of

x (t ) contained within s(*) (t - T ) &"% Let us consider using a direct matrix inversion

approach to LS-SCORE with (5.21) as the reference signal. From the incoming data

we form the sample correlation matrix, as in equation (4.9),

and the sample correlation vector, as in equation (4.10),

and then form the optimum weights (equation (4. L I))

As the number of samples approaches infinity

Provided the noise and interference are not spectrally coherent at cycle frequency

cu then

R;h(r) = u~&,~(T) (5.27)

where ud is the steering vector of the desired user and R&(T) is the cyclic autocorre-

lation function of the desired user's signal. Therefore,

where e is a constant. Equation (5.29) indicates that we come to within a scalar

constant of the optimum weights. A closer look at the scalar, e, applies a condition

that the control vector may not be orthogonal to the steering vector of the desired

user. Therefore, since scaling of the weights doesn't change the SINR, we've reached

the optimum SINR solution for the weights.

The above development has shown that LSSCORE approaches the optimum s e

lution. The reference signal in (5.21) contains a component that is correlated with

the desired signal and a corruption term which is uncorrelated with both the desired

signal, the noise and the interference [32]. As one might suspect the performance

of LS-SCORE is poorer than when we have a reference signal supplied to us (via a

training signal or separate signalling channel) that is perfectly correlated with the

desired user. The advantage is that since the reference signal was derived from the

incoming data signal through the exploitation of the cyclostationarity inherent in the

desired signal, we don't require a training signal or a separate signalling channel which

consume precious bandwidth. In the next section the cyclostationarity inherent in a

BPSK signal is examined and this is then followed up with a simulation of LS-SCORE

with BPSK signalling.

5.4 Cyclostationarity of BPSK

In the next section the simulation of LS-SCORE performed on a focal fed reflector

a n t e ~ a (a MBA) is described. The simulation is a baseband simulation and the

signalling method selected was BPSK. This is equivalent to a PAM signal which

takes the form

where {a, = a(nT,)} is a sequence of random variables and p ( t ) is a deterministic

finiteenergy pulse. A square pulse shape has been used in the simulation. Gardner

[31] has shown that if we assume that the input sequence {a,} is stationary, uncor-

related and unit power then x ( t ) exhibits cyflostationarity at a! = &rn/To where rn

is an integer. Moreover, the feature strength is strongest for rn = 1 and for r = T,/2

(for a square pulse shape). These are termed baud rate features. Therefore signds

with different baud rates will exhibit cyclostationarity at Werent values of a and T.

This allows the cyclic adaptive algorithm to distinguish between signals with different

baud rates.

A second type of cyclostationarity may be created by offsetting each signal from

the center of the reception band. In other words, each user which shares a fiequency

channel has a unique carrier ofbet. Gardner, Schell and Murphy state that a signal

offset by Af and with baud rate f6 will exhibit conjugate spectral coherence for

a = f 2 A f f. m fa where m is an integer [29] . This is maximized for m = 0 and at

T = 0. The signal is said to exhibit carrier rate features.

The simulation in the next chapter will demonstrate LS-SCORE for both baud

and carrier rate features.

5.5 Simulation of LS-SCORE

A baseband simulation of LS-SCORE operating on a MBA was performed. The

pulse shape chosen was a squaxe pulse. The simulation follows along the same Lines

as the one performed with the DM1 algorithm in section 4.6. The same antenna

configuration is used. The antenna has F = 94.87X, a = 54.08& and dd = 70.94A.

The m a y consists of 7 feeds in the same configuration as in the DM1 simulation.

Each feed is Linearly x-polarized and has q~ = 3.6 and g~ = 2.8. As before, the

feeds are spaced l h apart with the center feed displaced horn the focal point along

the x-axis at a distance of -5.53X. Also as in the DM1 simulation we have a single

desired user and a single interferer. The desired user arrives from 3.0' and has SNR

of 10 dB- The intederer arrives from 2.4" and also has a SNR of 10 dB.

Two simulations were performed. The first simulation demonstrated baud rate

features, the second carrier rate features. Almost all of the details of the simulation

are identical to that of the DM1 simulation. 100 trials were performed. The signals

used were mutually independent BPSK waveforms with a square pulse shape and

once again, time was normalized to the sampling period. For each user a random bit

sequence was generated as was a uaiformly distributed initial phase in the range of 0

to 27r. The cyclostationary simulation differs fiom the DM1 simulation in two respects.

First of all, the reference signal is now extracted from the input data signal rather than

assuming that a perfect reference signal is supplied. The second difference between

the D MI and cyclostationary simulations is that in the cyclostationary simulation the

signals of the two users have to exhibit cyclostationarity at different cycle frequencies

in order to extract them. This means that they must have different baud rates if we

wish to exploit baud rate features, or they must have different carrier offsets in order

to take advantage of carrier rate features.

For the baud rate simulation we let the desired user have a symbol period of 4

samples while the interferer has a symbol period of 5 samples. Therefore, as discussed

in section 5.4, we will set the cycle frequency to a = l/Td = 0.25 in order to extract

the desired signal. Td represents the symbol period for the desired user's signal. Also,

as discussed in section 5.4, we will set the time lag parameter, r, to Td/2 = 2. 8000

samples (giving 2000 desired signal symbol periods) were taken for each of the 100

t rids.

For the carrier rate simulation we set the symbol period of both users to 4 samples

per symbol. This time, each user has a distinct carrier offset. The desired user's carrier

offset was selected to be h fd = 0.0208 and that of the interfer was set to A fi = 0.0417.

One key point is that in order to take advantage of carrier rate features we must look

for conjugate self-coherence. We set the cycle frequency to a = 2A fd = 0.416 and

the time lag r = 0 in order to extract the desired user's signal (see section 5.4). As

in the baud rate simulation, 8000 samples were taken for each of the LOO trials.

The reference signal for both simulations was formed using equation (5.17) with

the control vector set to c = [ l o 0 - - 0IT.

At each sample the received signal vector at the antenna array was determined.

This was done by adding up the contribution of the desired signal, the interferer,

and the noise. The desired user's and the interferer's received signal across the array

were found by multiplying the appropriate BPSK waveform at the sample by the

steering vector of the user. Equation (4.7) was used along with the reflector antenna

analysis program to derive the steering vectors. To generate the received noise vector

complex random noise samples were generated using a noise variance of 1. The desired

user's signal was added with that of the interferer and the noise samples giving the

received signal at the array. With the received signal vector the reference signal

was calculated using the appropriate d u e of cycle frequency, a, and time lag, r,

to extract the desired user's signal. As well, if conjugate self-coherence was being

exploited, as it was in the carrier rate simulation, then the optional conjugation was

used in equation (5.17).

As the resulting received signal at the array and reference signal were determined

at each sample, the sample correlation matrix and sample correlation vector were

updated using equations (4.9) and (4.10). Every 40 samples (10 symbols) the optimum

weight vector was determined by inverting the sample correlation matrix and then

multiplying it by the conjugate of the sample correlation vector as in equation (4.11).

Using these weights the output SINR was calculated using equation (2.33).

The results of both the baud rate and the carrier rate simulation are shown in

figure (5.1). The results of the simulation are quite revealing. First of all note that the

convergence time of the LS-SCORE algorithm, whether baud or carrier rate features

are being exploited, is much longer than that of the DM1 algorithm which has a

perfect reference signal. Second of all, note that the convergence with carrier rate

features is much superior to that with baud rate features. This is due to a much

larger feature strength for carrier rate features. Perhaps the most important point to

note from these simulations, as far as this thesis is concerned, is that cyclic adaptive

beamforming algorithms do work on multiple beam antennas and no changes need

to be done to the algorithms in order to get them to work. This has only been

Convergence of LS-SCORE (Baud and Carrier Features)

- 6aud Rate Features

#--I - - = m-D Carrier Features 1-1

Symbol Periods of Desired User's Signal

Figure 5.1: Convergence of the LS-SCORE algorithm when performed on a MBA for baud and carrier rate features (same antenna and feed configuration as in DM1 simulation) with a desired user (pa = 10 dB, Bd = 3.0') and a single interfering user (p , = 10 dB, 4 = 2.4")

demonstrated for LSSCORE but the principle is the same and this fact carries over

to other cyclic adaptive b e d o r m k g algorithms.

Chapter 6

Conclusions

This chapter completes the thesis with a list of conclusions along with a List of rec-

ommendations for further study of this topic.

6.1 Conclusions

The general conclusion is that this thesis has unified several methods and theoretical

concepts in the development of an efficient digital computer based simulation tool

to study st at istically optimum bedonn ing in a multi-user digital communication

system that involves a multiple beam antenna. In support of this general conclusion

the following conclusions are made.

A narrowband signal model has been developed which is general enough to include

both the direct radiating array and the multiple beam antenna. The key quantity

required to study beamforming was shown to be the steering vector which represents

the response of the array of antenna elements to a unit-amplitude plane wave.

A digital computer program was composed in C++ that provides modem reflector

antenna analysis. The program is based on the Fourier-Bessel technique and uses the

physical optics approximation. The beam pattern for an offset parabolic reflector can

be computed in approximately 30 sec per feed element on a SUN Sparc 20 workstation.

Secondary field superposition was found to be a very efficient method of finding the

antenna pattern of the reflector with an may feed. This is due to the fact that in

secondary field superposition the secondary pattern of each feed is first computed and

then stored. The stored pattern can then be quiddy weighted and superimposed in

the far field. Secondary field superposition is particularly suited to a beamforming

simulation since such a study involves frequent variations of the weight vector.

The reflector antenna analysis computer pro- was used to briefly study some

of the properties of offset parabolic reflectors. The effect of the edge taper of the

feed and the scanning properties of the offset reflector were investigated. As the

edge taper of the feed is increased from 5 dB to 20 dB the 3 dB beamwidth of the

secondary field widened by 0.2 dB, and the first sidelobe level dropped by 21 dB.

For an offset refiector with focal length to parent paraboloid diameter ratio 0.625, a

beam was scanned approximately 4.2 beamwidths for each displacement of a feed by

6 wavelengths. The scan was associated with a degradation of the beam. The peak

directivity dropped steadily with the beam scanned out to 17 beamwidths having a

lower peak directivity than the on-axis beam by 3 dB.

A formulation of the reciprocity theorem was used to relate the transmitting prop-

erties of the reflector antenna to the receiving properties. This dowed the steering

vector for a multiple beam antenna to be found numerically with the use of the am-

plitude vector of each of the feeds in the array. The amplitude vector for each of the

feeds was found using the computer program based on the reflector antenna analysis

theory developed in chapter 3. An example of statistically optimum beamforming

with a MBA was demonstrated. In addition, the simulation of two different adaptive

algorithms were performed with a MBA. The direct matrix inversion algorithm was

simulated and it's convergence properties were demonstrated to conform to theoretical

expectations. The LS-SCORE algorithm, with exploitation of both baud and carrier

rate features was simulated. Both the baud and carrier rate versions of LS-SCORE

were much slower to converge to the optimum solution than DMI. DM1 converged to

within 3 dB of the optimum SINR within 5 symbols while carrier-rate LS-SCORE

converged in approximately 75 symbols. Baud-rate LS-SCORE was much slower than

both DMI and carrier-rate SCORE with convergence within 3 dB of the optimum in

just under 2000 symbol periods.

6.2 Recommendations for Further Study

A number of assumptions have been made in the a n t e ~ a analysis program which

Limits its effectiveness as an analysis tool for a real reflector antenna. This opens

up the possibility of expanding and improving upon the analysis program to make it

more realistic. In particular, the program could include:

a More accurate feed models.

a The effect of mutual coupling between the feed elements.

a The effect of surface distortion and deviation fiom a perfect paraboloid.

a A more accurate evaluation of the antenna away fiom the main beam by im-

proving upon the physical optics approximation.

In addition, the program could be incorporated as part of a larger program which

evaluates the fields on the surface of the earth, rather than just over a range of

spherical angles. Such a program would have to convert the latitude and longitude

coordinates on earth to the spherical variables (r, 8,4). As well, any such study should

include the effects of propagation between the Earth and the satellite.

Generally, the feeds in this thesis were restricted to a feed plane. A study of

whether there is a better way of placing the feeds in the region of the focal plane

would be a valuable study.

In terns of adaptive bedorming with a multiple beam antenna, a more in-depth

study of how this might be incorporated in a realistic design of a system would be

very useful. For example, although optimum combining may yield the minimum

mean-square error solution, the sidelobes created by this set of weights is often quite

high. As far as an antenna designer is concerned, that is unacceptable. Generally, the

goals of antenna design and the goals of statistically optimum beamforming aren't

necessarily the same. More must be done to find a compromise. Statistically optimum

beamforming with constraints may be a direction which offers some promise in terms

of meeting these needs.

One of the lingering problems in adaptive signal processing is the lack of an algo-

rithm which is both rapidly convergent and has a low computational complexity. One

direction which may show some promise is the use of subspace constraints. A second

area being pursued is efficient, parallel structures for implementation of algorithms.

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Appendix A

Antenna Basics

A.1 Introduction

In chapter 2 a signal model was developed which was applicable to both a DRA and

a MBA. The effect of the antenna is demonstrated through the steering vector which

represents the response of each of the antenna elements to a unit-amplitude plane wave

from a given direction. For a ULA the steering vector was derived and an example of

optimum combining was presented. This thesis is primarily concerned with the use of

a MBA. It is not analytically possible to derive the steering vector of the MBA as we

did for the ULA in chapter 2. The steering vector must be established numerically

and the development of the theory and technique of doing this is presented in chapter

3 and 4. In this appendix the groundwork of basic antenna and electromagnetic

theory is given in preparation for that development.

A. 2 Time-Harmonic Fields and Maxwell's Equa- t ions

The starting point in any electromagnetics problem is Maxwell's equations. Maxwell's

equations are expressed in terms of the following fields:

& the electric intensity (V/m) ,

R the magnetic intensity (Alrn),

P the electric flux density (C/m2),

B the magnetic flux density (Wb/m2),

3 the electric current density (A/m2), and

Q, the electric charge density (C/rn3).

All of these fields are vectors except for the electric charge density which is a scalar

field. Maxwell's equations are:

V - B = 0

V * P = Q,.

The constitutive relations specify the characteristics of the medium in which the field

exists. The constitutive relations are given by

For lineax, homogeneous and isotropic media these equations become

where,

c is the permittivity (F/m),

C1 is the permeability (Hlm), and

115

Q is the conductivity (l/Slm).

In a vacuum c = 8.854 x 10-l2 Ffm, p, = 4s x W7 Hfm, and cro = 0 (11 am). The

Poynting vector represents the density of power flux and is defined as

Therefore, the scalar quantity

is the total power leaving the region bounded by the surface of integration.

When the field quantities are timeharmonic they can be represented as complex

quantities. We may define the complex electric intensity, E, such that

Similar definitions are made for the other field quantities: H, B, D, J, and Q,. We

will assume harmonic time dependence throughout. Maxwell's equations become

V x E = -jwB

V x H = jwD+J

V - B = 0

V - D = QY.

(A.14)

(A.15)

(A.16)

(A. 17)

The constitutive relationships for linear, homogeneous, isotropic media become

where 6, p, and a are in general complex. The time-average Poynting vector can be

shown to satisfy [63] - - S = E x 'fl = Re{E x H*) (A.21)

where the notation d is used to represent the tim*average of I. W e can define a

complex Poynting vector

S = E x H * (A.22)

whose real part is the time average of the instantaneous Poynting vector. Therefore,

We may similarly define the complex power leaving a region by

where the complex power leaving a region is related to the time-averaged power flow

as

A.3 The Magnetic Vector Potential

Since V B = 0 from equation (A.16) and the divergence of the curl of a vector is 0,

we may define a vector potential A called the magnetic vector potential with

and therefore,

Using this definition and equation (A. 14) we can state

Using this and the fact that the curl of the gradient of a scalar is 0 we may define an

electric scalar potential, 9, by

To obtain the equation for A, the magnetic vector potential, substitute (A.27) and

(A.29) into (from ( A X ) and (A.18)),

The result is

V x V x A = jwrp(-Vp - jwA) + pJ.

Rearranging and using the vector identity

V X V X A = V ( V . A ) - v 2 ~

we get

V(V A) - V2A - W % A = pJ - j ~ p V 9 . (A.33)

The definition of magnetic vector potential in equation (A.26) only specified the

curl of A. We axe free to choose the divergence of A. The usual choice is the "Lorentz

Gauge" given by [63]

V A = - jwepp (A.34)

Therefore, in (A.33) we get

V ~ A + k 2 ~ = -pJ

where we have used k = w@. Equation (A.35) is often called the complex wave

equation. The magnetic vector potential is expressed solely in terms of the source

current, J. Once we solve for A we use the previously derived equation (A.27),

to find the magnetic intensity. To find the expression for the electric intensity in

terms of the magnetic vector potential combine equation (A.34) and (A.29) to get

I E = -jwA + -V(V0 A).

f WECt

A.4 Solution of the Complex Wave Equation

Now let us discuss solving the complex wave equation (A.35). Initially, consider a

current I over an incremental length At. This represents a current element. Let

this current element be at the origin of the coordinate system and let the current be

z-directed. This situation is shown in figure (A.1). Because the current is z-directed

Figure A.1: A current 1 over an incremental length A0

A has only an A, component which satisfies

everywhere except at the origin. The source at the origin is a point source and A,

should therefore be spherically symmetric. Therefore A, is only a function of the

radial miable r and the equation (A.38) becomes

This equation has the two independent solutions be-jk and ;gkr. The first solution

corresponds to an outward-travelling wave and the second solution to an inward-

travelling wave. Therefore we take as the solution

where n is a constant. Now as k + 0, the complex wave equation (A.35) reduces to

Poisson's equation which has the solution [63]

Therefore

and the solution for A, is

This solution has spheres as constant phase su-4aces and therefore is called a spherical

wave. This result is the magnetic vector potential for a single current element. Now

let us generalize the result to an arbitrary distribution of electric currents. Since the

complex wave equation is lineax, we can use superposition to find A for an arbitrary

distribution of currents. There are a few changes to be made to the above results

to generalize them. First, let us change the position of the current element within

our coordinate system. Figure (A.2) shows a coordinate system with an arbirarily

located origin. The field coordinates (where we are determining A at) are specified

by unprimed coordinates

r=xli+&+ri. (A-44)

The current element is located by the source coordinates which are specified by primed

coordinates

r' = x Y + y'f + 1'2. (A.45)

Now the distance r between the source and the field point should be replaced by the

distance

Source Coordinates

Figure A.2: The source and field coordinates with respect to an arbitrary origin.

Finally, note that for an electric current density distribution J, the current element

contained in a volume Av is J Av. Therefore the general expression for the magnetic

vector potential A is, by superposition

The integration is over the volume of the source. This equation is called the magnetic

vector potential integral. Similar expressions result for the cases of surface currents

and filamentary currents with a corresponding reduction in the dimension of the

integrd. Chapter 3 applies equation (A.47) to the physical situation of a reflector

antenna.

A.5 Antenna Near and Far Fields

The radiation field fiom an antenna which is transmitting into space can be broken

into two parts: (1) the radiating field, and (2) the reactive field. The radiating field

is characterized by a real complex Poynting vector and (E, H) fields which decay at

a rate of r-I, where r is the distance of the observation point fiom the antenna. The

reactive field has an imaginary complex Poynting vector and the (E, H) fields fall off

faster than rvl. Due to the rate at which these two components of the field decay,

the space around the antema is generally divided into three regions where different

fields dominate. Figure (A.3) illustrates these three regions of space.

Reflector Surface

Antenna Feed EIement

Figure A.3: The 3 radiating regions: 1. the reactive nem-field region, 2. the radiating near-field region (the Fresnel region), and 3. the radiating fa-field region (the Frauahofer region).

The field immediately around the antenna is called the reactive near-field re-

gion. This reactive near-field region is dominated by the reactive field and the outer

boundary of this region is approximately a few wavelengths away from the antenna

feed element.

The next region away from the antenna is called the radiating near-field region.

It is sometimes referred to as the Fresnel region. In this region the radiating fields

begin to dominate. The outer boundary of this region is commonly considered to be

at a distance of approximately r = 2D2/X where D is the largest dimension of the

antenna and X is the wavelength.

The third region is called the far-field region of the antenna. Another common

term used for this region is the Fraunhofer region. In the far-field region the reactive

field is considered to be negligible and the radiating field dominates. In the fu-field

the (E, H) fields decay as r-= and have vector components only transverse to the

direction of propagation. In other words, there is no radial component of these fields.

When we speak of the radiation from an antenna we are generally speaking of the

far- field-

A.6 Plane Waves

The far-field of an antenna is locally a plane wave. When the plane wave propagates in

an isotropic homogeneous medium the plane wave can be represented by the equations

where,

k = kk is the wave vector of the plane wave (in the direction of propagation),

b is a vector describing the polarization of the plane wave,

7 = ,/& is the wave impedance of the medium, and

C is the amplitude of the plane wave. It is, in general, a complex number

and has the units of W1I2 7n-l. The quantity lC12 is the power density

of the plane wave with units of W/rn2.

The complex vector b must be unitary and transverse to the direction of propa-

gation leading to the conditions

b-b* = I

b - k = 0.

It should be noted that a plane wave in direction k and polarization b has the

same polarization as a plane wave in the direction -k and polarization b' [Xi].

A. 7 Polarization

In the previous section on the representation of a plane wave, the vector b in equation

(A.48) described the polarization of the plane wave. Here the term polarization refers

to the direction of the electric field vector (as may be seen in equation (A.48)). In

this section we examine polarization in greater detail. To do this it will be convenient

to fix the direction of propagation in the z direction. Therefore, equation (A.48)

becomes

and the vector b can be written in the form

where b, and 6, are shown to be complex. The polarization is defined within a phase

angle. In other words the vectors b and b&$ describe the same polarization. The

previous constraint on b in equation (A.49) therefore becomes

Consider for a moment the electric-field vector in the time domain and evaluated at

a fixed reference plane (i.e. z = 0). This vector is expressed

where,

This vector V(t) is a vector rotating in the x - y plane. The locus of the extremity of

this vector is, in general, the shape of an ellipse. This ellipse is called the polarization

ellipse. A line and a circle are two special cases of an ellipse and as such there are three

cases of polarization to consider: (1) linear polarization, (2) circular polarization, and

(3) elliptical polarization.

(I) Linear Polarization. If b, and 6, are in phase, meaning that #= = &, then

the polarization ellipse becomes a straight b e as shown in figure (A.4). The vector

Figure A.4: Linear Polarization

V ( t ) moves back and forth dong this straight line which makes an angle c with the

x axis as shown in the figure. c is defined by

which allows b to be expressed

(2) Circular Polarization. The electric field vector is circularly polarized when the

x and y components have equal magnitude = 141) and are out of phase by 90"

(9, - & = f a/2). Which component leads the other in phase selects the direction

in which the vector V ( t ) rotates. The two directions of polarization are termed left-

hand circular polarization (LHCP) and right-hand circular polarization (RHCP) and

represent the direction V(t ) rotates when looking in the direction of propagation.

Left-hand Circular Right-hand Circular Polarization Polarization

Figure A.5: Circular Polarization

These two cases of circular polarization are illustrated in figure (A.5). The vector b

becomes equal to (within a phase factor) one of the two vectors defined below which

correspond to LHCP and RHCP respectively: 1

(3) Elliptical Polarization. As mentioned earlier, b is in general elliptically po-

larized. The polarization vector b may be decomposed into its x - y components

or, alternatively, into Left- and right-hand polarized components. This decomposition

may be expressed as

where the components bL and bR are given by

Figure (A.6) illustrates the polaxization ellipse. The elliptical polarization is char-

Y

Figure A.6: Elliptical Polarization

acterized by three parameters: (a) the uis ratio (AR), (b) the tilt angle, and (c) the

sense of rotation of V(t).

The axis ratio (AR) is defined as the ratio of the semimajor to the semiminor axis

of the polarization ellipse

The tilt angle of the ellipse, c, is the angle between the x axis and a semimajor

axis of the ellipse (as shown in the figure).

The sense of rotation is determined by comparing the magnitudes lbtl and lbRI.

K (bL 1 > [bRl then b corresponds to left-hand elliptical polarization. If (bL 1 = lbRl

then b is Linearly polarized, and finally, if 1 bRl > 1 bLl we have right-hand elliptical

polarization.

Let's briefly look at how lineax and circular polarization fit within these parme-

ters. When b corresponds to linear polarization lbLl = 1 bRl and therefore AR = oo.

Clearly the sense of rotation is meaningless in hear polarization. When we have

circular polarization, either lbcl or lbRl is 0 depending on whether we have LHCP or

RRCP. Therefore AR = 1.

A.8 Far-Field Representation of the Antenna Ra- diat ion Field

The radiation field in the far-field region of an antenna is a spherical wave and this

can be represented by

where,

k = ki is the propagation vector which is in the direction of propagation of the

wave (which is in the radial direction), and

F(k) is the amplitude vector of the spherical wave. It is complex in general, is

transverse to k, and has units of w'I?

The vector F(k) can be decomposed into two orthogonal components (see figure (A.7))

where b and d are unitary vectors which are orthogonal to each other and to the

direction of propagation. These restrictions on b and d are expressed mathematically

by

An example of b and d are the unit vectors and 4 which axe unit vectors in

the directions of the variables 0 and 4 in a spherical coordinate system. Two other

Figure h.7: The field at a far field point r propagates in the direction of k and can be decomposed into two orthogonal components b and d which are also orthogonal to the direction of propagation k.

common choices for b and d are linear polarization where b and d take the form [55]

b = Bcos9-&sin$ (A.70)

ci = Bsind++cos# ( ~ . 7 l )

and circular polarization where b and d become

The scalar quantity F(k, b) is given by

F(k, b) = F(k) - b*.

In other words, F(k, b) is the component of F(k) in polarization b.

129

A. 9 Radiation Intensity and Antenna Pat t ems

We define the radiation intensity of the antenna in the direction k by

The radiation intensity has units of W/stesadian. We may also define a radiation

intensity in the direction k and polarization b by

Note that we can now decompose Imd(k) by

The total power radiated by the antenna may be found by integrating the radiation

intensity over a sphere around the antenna giving

Pmd = lo Imd(k) sin B dq5 dB.

A plot of radiation intensity as a function of observation direction (B,d) is known

as the antema pattern. Both the total radiation intensity imd(k) and the radiation

intensity in a given polarization Imd(k, b) are often plotted. An example antenna

pattern is shown in figure (A.8) where a 2-dimensional cut is taken for a fixed value of

4. The plot is of the radiation intensity in the polarization a. The plot is normalized

and plotted in decibels. The normalized radiation intensity in decibels is defined by

where, I,&,, b) represents the maximum radiation intensity (which is take to be in

the direction k,). Therefore the maximum radiation intensity point is at 0 dB. The

lobe in this direction is called the main beam while the other lobes are called side

lobes. When we refer to side lobe levels we usually are referring to the highest side

lobe which is usudy the one closest to the main beam. The width of the main beam

at the -3 dB level is called the hal/power beamwidth (HPB W).

Main Beam \ . Side Lobe Level

8 in degrees ($ fixed)

Figure A.8: An example of an antenna pattern showing the normalized radiation intensity in the e polarization as a function of 0 for a fixed value of 4.

A.10 Antenna Gains

An isotropic radiator is a fictitous antenna that radiates equal power in d directions.

In reality, antennas radiate power more in some directions than others. This is quan-

tified by a dimensionless d u e c d e d the directivity of the antenna. The definition of

directivity in direction k and for all polarizations is

intensity of the antenna in direction k D(k) =

intensity of an isotropic radiator in direction k (A.81)

where, Pmd is the total power radiated by the antenna. C u e must be taken when

discussing the directivity since sometimes the term directivity is used to refer to the

maximum directivity. The directivity isn't measurable since in reality not all of the

power delivered to the antenna is radiated into space. There is a certain amount of

dissipative loss in the antenna. This then leads to the definition of a value called the

gain (also dimensionless) which is related to the directivity by an efliciency factor

according to the equation

G(k) = w D ( k ) . (A.83)

There is a third type of gain which is sometimes necessary. This third type of gain

includes the transmission line and source in with the antenna. Just as not aU of the

power delivered to the antenna is radiated into space, not all of the power generated

by the source is delivered to the antenna due to mismatch between the transmission

Line and the antenna. We can define a power reflection coefficient denoted by / rI2,

and a realized gain by

Often we are only interested in the directional properties of the a n t e ~ a in a

certain polarization. This leads to a definition of directivity and gain in polarization

b,

These d u e s are sometimes called the partial directiuity and partial gain respectively

(but not always). Note that the total gain is the sum of the partial gains for any two

polarizations which axe orthogonal as defined in section A.8. Io other words,

where b and d are orthogonal polarizations.

Often the directivity or gain is plotted versus angle much like the normalized

radiation intensity was in section A.9. A plot of directivity or gain is also called the

antenna pattern. It has the exact same shape as the normalized radiation intensity

pattern previously described but with the maximum not at 0 dB but at the value of

the maximum directivity.

In this section we have considered solely the transmitting properties of the antenna

and the antenna pattern described is the transmitting pattern of the antenna The

antenna also has a receiving pattern. The receiving pattern is a plot of a quantity

called the receiving cross section as a function of direction. The receiving cross section

of the antenna in direction k and polarization b is dehed by

received power under matched load condition ueff (k, b) = power density of incident plane wave (A.88)

with polatization bm and direction -t

The units of received cross section ate m2 and as such the received cross section can

be interpreted as an effective area presented to an incident wave to collect energy.

An upcoming section (section A.12) on reciprocity will show that the receiving cross

section is related to the realized gain of the antenna by

This equation shows that transmitting and receiving patterns of the a n t e ~ a differ

only in their maximum dues. Therefore the normalized transmitting and receiving

antenna patterns are identical.

A . l l The Antenna As A One-Port Device

An antenna is usually fed through either a transmission line or waveguide (see figure

(A.9) which we will for convenience allow to run along the z direction. We assume that

the transmission Line or waveguide supports a single propagating mode. The total field

dong the transmission line or waveguide is a superposition of two travelling waves

with one travelling in the positive z direction and one in the negative z direction. In

other words, we may write the field as [55]

where,

Source

Transmission Line

Source

Figure A.9: An antenna may be fed through a transmission line or a waveguide.

&, is the propagation constant,

(aa, ba) axe the wave amplitudes (complex) in w112, and

(e, h) represent the transverse field miations with units of ( ~ ' l ~ r n - ~ , n-1/2rn-L).

The reference plane z = 0 can be chosen arbitrarily and the vectors (e, h) must satisfy

the nonnalizat ion condition

//(e x h*) - i d z d y = 1

where the integration is over an infinite plane which is transverse to the transmission

line or waveguide. We may define the ratio

as the E-field (voltage) reflection coefficient at the r = 0 reference plane. The power

transmitted from the source into fiee space is

Note that IbAl2 represents the power reflected back toward the source.

The above equations describe the field in the waveguide or transmission line from

a wave point of view. A circuit veiwpoint building on the wave point of view is

possible but not necessary.

A. 12 Reciprocity

Reciprocity is a theorem that relates the transmitting properties of an antenna to the

receiving properties. We will consider the two situations illustrated in figures (A.lO)

and ( A M ) . The first situation is as antenna excited by a source with amplitude at.

4 ;

i b+

Figure A. 10: An antenna transmitting when excited by a source with amplitude at.

Figure A.11: An antenna receiving a plane wave in polarization b* and with propagation vector -k.

It transmits into space the electromagnetic field

Recall that the amplitude vector may be decomposed into two orthogonal polariza-

tions b and d as expressed by equation (A.66).

Now consider that same antenna receiving a plane wave with propagation vector

-k and with polilrization b'. The incident plane wave may be expressed by

where C is the amplitude of the plane wave in w1I2/m. Reciprocity states that the

received amplitude, b,, under the matched load condition (a, = 0) is given by [55]

The receiving cross section of the antenna in direction k and polarization b was

defined in section A.10 as the ratio of the received power of the antenna under the

matched load condition to the power density of the incident plane wave with polar-

ization b* and direction -k. Mathematically,

Using equation (A.97), (A.98) becomes

Since the realized gain of the antenna in direction k and polarization b is expressed

(A. 100)

we can relate the realized gain of the antenna to the receiving cross section by

In terms of the directivity, equation (A.84) may be used to express the receiving cross

section by A2

Gff (kt b) = --(I - Ir12)f).gD(k, b) 4n (A.102)

where represents the reflection coeBcient at the boundary between the transmission

line or waveguide and the antenna. This establishes the relationship between the

transmitting and receiving pat terns of the antenna.

A. 13 Feed Approximation by cosq(8)

A common analytical feed pattern that is used when studying the characteristics

of a reflector antenna is a cosq(8) feed pattern where the parameter q controls the

shape of the beam [55, 181. The reason that this pattern is used is that it is a good

approximation to the antenna pattern of many commonly used feeds in the main

beam region. The feed pat tern for a h e a d y polarized feed is

e ~ ~ ( 8 ) cos 4 - tji ~ ~ ( 0 ) sin 4 for k polarized E(r) = A. O C ~ ( ~ ) S ~ ~ ~ + ~ C ~ ( O ) C ~ S ~ for 9 polarized (A.103)

where A, is a complex constant and

CE(B) = (COS QqB = Eplane pattern

Cw(B) = (cos O)qH = H-plane pattern

for O 5 t? 5 s/2 (CE =CH=Ofor 7r/2 < Q <r).

The parameters q~ and q~ are used to control the shape of the pattern. Figure

(A. 12) plots the cosq(6) feed pattern for several d u e s of q. Paper [La] by Y. Rahmat-

Samii estimates practical d u e s of q for several different types of feeds. In chapter 3

the effect of the q parameter on the antenna pattern of the reflector was demonstrated.

0 15 30 45 60 75 90

0 (in O)

Figure A.12: The cosq(0) feed pattern for several different d u e s of q.

Appendix B

Coordinate Transformations

The problem of reflector antenna analysis involves a number of merent coordinate

systems. It is very important to be able to make transformations between these differ-

ent systems. This appendix develops all of the necessary coordinate transformations

required in this thesis. Much of the material in this appendix was adapted horn

Goldstein's classic text [60] and a paper by Rahmat-Samii [22].

B. 1 Transformation From One Cartesian Coordi- nate System to Another

Consider two sets of Castesiaa axes, one denoted by primed coordinates and the

second by unprimed coordinates. Let these two sets of Cartesian axes have a common

origin. This situation is illustrated in figure (B.l). One method of describing the

orientation of one set of axes relative to another is by specifying the direction cosines

of the primed axes relative to the unprimed axes. The I' axis can be specified by its

three direction cosines a2, a3 with respect to the x , y, z axes by

Therefore,

Figure B.1: 2 cartesian coordinate systems with a common origin but oriented arbitrarily with respect to one another

Similarly,

These equations may be expressed in matrix notation by

where the matrix B is defined as

(B. 14)

6 may be thought of as an operator acting on the unprimed system transforming it

into the primed system. There are several features of this transformation to comment

on. The &st thing to note is that this transformation may be used to relate the

components of any vector in one system to the components of that vector in the

second system. In other words, if the vector G is expressed in the unprimed coordinate

system by

G = G , 2 + G y ~ + G z i

then that vector is expressed in the primed coordinate system by

where,

In matrix notation

(B. 18)

(B.19)

The second thing to note about the t ~ o m a t i o n described in equation (B.13)

is that we may invert the process:

In matrix notation these equations become

Note that since

6-I = B=

this is an orthogonal transformation.

Before continuing with the details of bdimensional transformations between Carte-

sian coordinate systems let us briefly step aside for a moment to consider a 2-

dimensional transformation which has a bearing on the bdimensional case. Figure

(B.2) shows such a 2-dimensional transformation. As shown in figure (B.2) a 2-

dimensional transformation from one coordinate system to another corresponds to a

rotation of the axes about the origin by the rotation angle Y. Then

In matrix form

Figure B .2: 2 coordinate systems (in 2-dimensions) with a common origin but oriented with respect to one another by the angle Y

where, cos Y sin T

020 = -sinT cos T

Now it is time to get back to the 3-dimensional case. A transformation from one

Cartesian coordinate system to another can be acheived by three successive rotations.

These 3 rotations ase specified by three angles, a,d, fled, and reur. These are called

the Eularian angles. Figure (B.3) shows the first rotation. The origind (z, y, z )

axes are rotated counterclockwise about the z axis by the angle a,a resulting in the

intermediate axes (x2, yz, 22). Expressed mathematically,

where,

E =

The second step in the transformation is shown in figure (B.4). This time the

intermediate axes (s2, y 2 , ~ ) are rotated about the xz axis through an angle in

the counterclockwise direction. We denote the resulting axes (x3, y3, z3). This step of

Figure B.3: Step #1: Rotate (x, y, z ) by a..~ counterclockwise about the z axis to give (22 , Y 2 , ~ 2 ) *

the transformation can be expressed by

where,

The final step of the transfornation is shown in figure (B.5). In this final step of the

transformation we rot ate the (x3, y3, z3) axes by the angle 7cul counterclockwise about

the axis. The result is the (x', y', d) axes. The final step of the transformation is

given by

Figure B.4: Step #2: Rotate (x2, y2, z2) by PeUl counterclockwise about the 22 axis to give ( ~ 3 , ~3733)-

where, C0sye.l shyeu1 0

(B.35) 0 0 I

Performing each of these 3 transformations in succession allows us to transform

the (x, y, z ) axes into the (x', y', 2) axes. Therefore,

where,

B = CDE.

Therefore the transformation matrix B is

Figure B -5: Step #3: Rotate (23, y3, 23) by yeul counterclockwise about the z3 axis to give (x', y', 2).

where,

In summary, the primed Cartesian coordinate system is found by three successive

rot ations of the unprimed Cartesian coordinate system through the Eularian angles

PeU1, and r,.r. The components of a vector in the primed coordinate system are

found by multiplying the components of the vector in the unprimed system by the

transformation matrix B, as given by equations (B.39) - (B.48).

B .2 Transformat ions Between Spherical, Cylindri- cal and Cartesian Coordinates

Another common requirement in the work done in this thesis has been the trans-

formation between the spherical, cylindrical and Cartesian components of a vector.

Consider a vector quantity H. H may be expressed in terms of Cartesian, cylindrical

or spheric$ coordinates

These coordinate systems are illustrated in figures (B.6), (B.?), and (B.8).

B .2.1 Tkansformations Between the Rectangular and Cylin- drical Coordinates

Consider first the tramformation from cartesias coordinates to cylindrical coordinates

and vice-versa. As shown in figure (B.7)

Figure B.6: The rectangular coordinate system.

and therefore,

The i component is common to both the Caztesian and the cylindrical coordinate

system. The k and j? unit vectors may be expressed in terms of the and t$ unit

vectors. From figures (B.9) and (B.lO)

Substituting (B.57) and (B.58) into equation (B.49)

Figure B.7: The cylindrical coordinate system.

Therefore in matrix form (Z) =L (5) where,

cos q5 sin+ 0 I (B.62) 0 1

is the transformat ion matrix from rectangular to cylindrical coordinates. Note that

T, is an orthogonal matrix and therefore the inverse transformation fiom cylindrical

components to rectangular components is given by the transpose of the matrix in

where,

Figure B.8: The spherical coordinate system.

B .2.2 Tkansformat ions Between the Cylindrical and Sp heri- cal Coordinates

Now let us consider the transformation from cylindrical coordinates to spherical co-

ordinates and vice-versa. Refering to figure (B.8) we can relate (p, d, z ) to ( r , 0, 4)

by the equations

Therefore,

Figures (B.11) and (B.12) show how the i and f i unit vectors axe related to the i and

e unit vectors. Mathematically,

Figure B.9: k in terms of f i and 6.

Substituting equations (B.70) and (8.71) into equation (B.50) gives

Therefore in matrix form (g) = T s c ( $ )

where,

is the transformation matrix from cylindrical to spherical coordinates. Once again,

note that this is an orthogonal transformation and therefore the inverse of matrix Tsc

is given by the transpose allowing

Figure B.10: in terms of b and 6.

where, [ ~;e CO;S ;I T , = T ~ = T T , = (B.77)

cos 8 - sin9 O

B .2.3 Tkansformations Between the Rectangular and Spher- ical Coordinates

Finally, consider the transformation from rectangular coordinates to spherical coor-

dinates and vice-versa. Refering to figure (B.8) we can relate (x, y, z) to ( r , 0, 4)

by

and therefore,

A p axis

Figure B.11: i in terms of ? and 9 (Figure in the plane containing fi and the z axis).

The transformation from the cartesian coordinate system to the spherical coordinate

system can be viewed as a two step transformation. The first step is the transforma-

tion from the rectangulat system to the cylindricd system and the second step is the

transformation from the cylindrical system to the spherical system. Therefore,

where,

T, = TJ, sin8 0 cosd cos 4 sin4 0 cos8 0 -sin0

0 1 0 0 1

sin 8 cos # sin 0 sin $ cos 8 c o s 8 c o s ~ cos0sin+ -sin8 .

I - sin 4 cos 4 0 I

I-=.*' A ,p axis

Figure B.12: j, in terms of 3 and 8 (Figure in the plane containing fi and the a axis).

This transformation is also orthogonal. Therefore,

T T T,, = T;;' = (T,T,)-' = TsLT;;' = T,TsC = (T,T,)* = T:. (B.86)

Therefore the inverse transformation from the spherical system to the rectangular

system is given by

where,

T,, = sin 0 cos q5 cos 0 cos q5 - sin 9 sin0 sin4 cos 6 sin4 cos # . I (B.88) [ ccs0 - sin 8 0

Appendix C

Proofs in the Development of the Fourier-Bessel Met hod

C.l Proof #l

First we will prove that:

Proof:

In the Last integral make the variable change 8 = 4' - s. Now

= cos(O - a,,) cos r - sin(8 - an,) sin T

Therefore,

0

j sin{As cos(4' - a,,)) + j sin{-As cos(4' - am)} d4'. (c-7)

Since sin(-x) = -sin(x) and cos(-I) = cos(s) the imaginary terms cancel each

other out and we have

since a, is just a constant phase factor and the integration is performed over half a

period. Now we use the identity [66]

0

where Jo(z) is the Bessel function of the first kind of order 0 (see section C.3). The

final result, after applying identity (C.10), is

C.2 Proof #2

Now we will prove that

Proof:

First transform the integral via the change of variables x = As. Therefore,

Now we use the identity [65]

where J, (x) is the Bessel function of the &st kind of order n (see section C.3). Use

of equation ((2.14) and integration by pazts gives

A second integration by parts gives

Each repetition of the integration by parts contributes a -2(p- i)/A2 term out in

front of the integral, where i is one less than the number of times the integration by

parts has been performed. After p repetitions of the integration by parts, the integral

is of the form

Putting this a i l together .

That proves equation (C.12).

C.3 Bessel Function Definition

(C.18)

(C. 19)

In equation (C.10) we introduced the Bessel function of the first kind of order 0, and

in the equation (C.14) the Bessel function of the first kind of order n was introduced

(n integer). Bessel functions are defined based on Bessel's equation written as [65]

where the real parameter v determines the nature of the two linearly independent

solutions of the equation. By convention, v is a real number which is not an integer

with the u being replaced by n when the integral parameter is used. The two linearly

independent solutions of (C.20) are the called the Bessel functions of the first kind of

order v, and are symbolically represented by J,(x) and J-,(I). The general solution

to (C.20) is [65]

y = A 3 4 4 + BJ&) (C.21)

where u is not an integer. When the order is an integer (v = n) then the solutions

axe no longer independent with their dependence being described by [65]

An alternative definition of the Bessel function of the first kind of order n is by

the series expansion [65]

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