Adaptive Beamforming with a Focal-Fed Offset Parabolic Reflect Or Antenna

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

Jason Duggan

A thesis submitted to the Department of Elect rid and Computer Engineering in conformity with the requirements for the degree of Master of Science (Engineering)

Queen's University Kingston, Ontario, Canada

April 1997

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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 may 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 may 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 may 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 may to a plane wave arriving from a given angle. While the steering vector for a uniform linear may 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 may processing can be performed by using the may 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 beamforming. Specifically, I would like to thank Chun Loo of CRC and Mark Rollins, formerly of CRC for their assistance. Thanks also goes to all 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 Ante~aArrays ...... 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 Summqof 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 Bhd Spatid Filtering Algorithms ...... 5.3.1 Cyclic Bhd 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.ll 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/Dp= 0.625) ...... 64 3.13 Effect of lateral displacement of the feed (F/Dp= 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 MBA optimum combining: case 2 ...... 81 MBA optimum combining: case 3 ...... 82 MBA optimum combining: case 4 ...... 83 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.4 Transformation of Cartesian coordinates: step 2 ...... 146 B.5 Transformation of Cartesian coordinates: step 3 ...... 147 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 liintermsofiand& ...... 152 B.10fintermsoffiand~ ...... 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 ith interferer's received signal vector

correlation matrix of the noise vector

xii 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

Pcul Eularian angIe

direction cosines

propagation constant

Eularian angle

direction 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 systern) 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 dinate 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 direction) 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 heor 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

Ef complex electric field of source i

Et E field in transmitting situation

E coordinate transformation matrix

EzVtl Fourier coefficients

ET edge taper e(x7Y) describes the E-field transverse variations in a transmission Line or waveguide

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 waveguide 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)

radial variable in a spherical coordinate system (corresponding to the primed cartesian coordinate system)

radial variable in a spherical coordinate system (corresponding to the

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 transformation 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 direction cosine of expected main beam position weight vector statistically optimum weights direction cosine direction cosine of expected main beam position 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 ith 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 ~roducethe 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 may 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 dow 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 mature 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 due. 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 may 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, Mdett 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 [Il] 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 (my)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 may 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 antennas, 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 sirnulation. The reflector ante~aanalysis 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 ante~aanalysis 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 development 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 array, 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 statistically 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 (DMI) algorithm, is performed demonstrating its performance on such an antenna.

Chapter 5 looks at blind spatial filtering algorithms based on exploitation of cyclostationarity 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 may geometry and the antenna patterns of the individual antenna elements. Most of the beamforming literature deals with a direct radiating amy(DRA) which is simply a number of antenna elements asranged in some arbitrary configuration. The most common DMconsidered 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 Mher apart than that, grating nulls can form [49]. If the ante~aelements 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~umptionfrequently made in the antenna array literature is that the antenna elements are isotropic meaning that they radiate I...

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 DM.

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"ftwhich 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 may 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 direction (Od, 4d)and frequency f, vd,(Bdr4d, 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 signals 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 ith (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 will 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,

jid ( 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, ud is the steering vector of the desired user (i.e. the response of the antenna in the direction 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 fc 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 received 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 addenotes a N' x NE matrix cded the correlation matrix of the desired 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 cZ1Pi + Pn

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

f Data-lndepemdent Statistically 0 timum Bearnforming BeamformP ng determination of the weights usin the statistics of the data to based on predetermined find & wei- which give the locations of maximums and nulls sbtistkal~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: 0verview 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 response 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 independent 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 statistics. 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 may 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 8dB. 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 (24, fiom section 2.2, with knowledge of the direction of anid 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 uw, 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 bedonning 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 geostationary satellite communications. Reflectors provide the large antenna size necessary to acheive sdlicient gain. The gain of an ante~ais 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 ante~atheory 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:

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

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

x, K &r

Source Y. Y'

Figure 3.3: The coordinate systems used in the reflector antenna analysis develop ment . 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 as 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 equivalent. 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 cded 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 electromagnetics 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 reproduced 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 eduated at a large distance fiom the reflector (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 approximation.

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-jb 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)+ ~(r')g~~.'d~. 4nr 11

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 ~roductof (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 TQare 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- icdy 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, trdorm 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 Jacobian. 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 + zri where (x', y', 2') may be expressed in terms of (p', #): 1 and zf = -(zR4F + - F

Now we can eduate the dot product r' - P as

+ 2dO8.pfcos 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 eosin

md wo = COSflo.

Introducing these variables into equation (3.42) gives

rt r = {[(. - u.) + (W - w0)k] 2F cos qS + (V - vo)sin 4'

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 cded 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 xu inside area y ) , the aperture A (3.79) gu(x"9yu) = { as yet undefined , in the shaded region of figure (U),

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.

3.11 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 reflector, 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(~.) = (COSQqE = 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,qH) 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 dueof 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 reference 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 symrnet 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 astema, expressed mat hemat icdy 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 resultant 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

BDF = -.@B 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/Dp = 0.625. According to [55] the puameter F/Dp 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/Dp= 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, dfl = 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)

0123456789101112 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) -MA, (e) -24A. Reference Polarkation Dlrectlvily in dB

.& 0 VI 8 C: 0 I 8 R B 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 + bd9* sin &)] -- (3.101) rs The quantities as,b., and tit. define the polarization of the source. The polarization parameten are normalized such that

a,2 + 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 to a, = T,bs1 = 7+1 = -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 of the secondary field of feed i. As we did for the feed polarization, we describe the 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 intensity of the reflector antenna in Wattslsteradian) gives the antenna pattern of the reflector (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 due. 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 equations

Co= 1 + 2a.b. cos & sin 24- + (a3 - a:) cos 24,,, (3.118)

~(~1rnn = 2"~-1r(uH) Jy,(brnn) (bmn)"~' and

In equation (3.118), as,b,, and ?,bS are the polarization parameters of the feeds (assumed to be the same for all feeds). The parameters p,,, and &, represent respectively 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 as 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 oi 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 dormhear 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 DMand 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 how 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 define 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 rep 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).

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 ~"~Irn.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 convergence 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 due 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 fi = 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. Optimum Combining With A MBA

Figure 4.5: Optimum combining with a MBA: case # 2. Desired user at 3' and with pd = 8 dB. 1 interferer at 2.4' and with pl = 8 dB. Optimum Combining With A MBA

Figure 4.6: Optimum combining with a MBA: case # 3. Desired user at 3O and with pd = 8 dB. Interferer # 1 at 2.4' with pl = 8 dB and interferer # 2 at 4.2' with pz = 8 dB. Optimum Combining With A MBA

Figure 4.7: Optimum combining with a MBA: case # 4. Desired user at 3' and with pd = 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 bedorming algorithm. This method therefore places nds 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 op 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 ante~aelements. 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)3process. 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 described and the results presented. Simulations of this sort with uniform linear mays have been performed in the literature [54]. This simulation demonstrates the performance 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 cyclostationary 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 SNR of 10 dB (10 dB chosen as a typical

due) arriving from 3O. A single interfering user will also have an SNR of 10 dB and will arrive from 2.4". The geometry of the ante~aand 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 may 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 adable 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 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). 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, Mdett, and Bre~an[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 may 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 ante~aand 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 demonstrated 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 alp 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 exhibit cyclost ationarity and this cyclost ationarity 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 versions 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 limitations. 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~ntagesand 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 beamfo~galgorithms 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 UCyclostationarityin 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 hction 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 42. There- fore u(t) and u(t) represent frequency shifted versions of x(t) by -a/2 and 42

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 conjugate self-coheeent for frequency separation a.

The introduction of u(t) and v(t)also dows us to introduce an appropriate normalization 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 mythen 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 discussion 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 0RE [32] maximizes the strength of the temporal cross-correlation coef- 2 Jicient, ls(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 cded 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 ation 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 autocorrelation 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 se 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 ante~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, uncorrelated 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 = f2 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 may 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 lh 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 hfd = 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 = [lo0 - - 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 due 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 bedormkg algorithms. Chapter 6 Conclusions

This chapter completes the thesis with a list of conclusions along with a List of recommendations 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 stat istically optimum bedonning 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 properties 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 amplitude 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 ante~aanalysis 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 improving 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 algorithm 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. Bibliography

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Adaptive Null Steering By Reflector An- tennas. In Proceedings of the 2nd hternationd Conference on Antennas and Propagation, York, England, pp. 168173, 1981. [44] J.T. Mayhan. Area Coverage Adaptive Nulling From Geosynchronous Satel- lites: Phased hays Vers;us Multipie-Beam Antennas. lEEE Transactions on Antemas and Propagation, Vol. AP-34, No. 3, pp. 410-419, March 1986. [45] K.M. Soo Hoo, and R.B. Dybdal. Adaptive Multiple Beam Antennas for Com- munications Satellites. In Proceedings of the 1990 hternational IEEE/AP-S Symposium, pp. 1880-1883, May 1990. [46] K. Sudhakar Rm et d. Development of a 45 GHz Multiple-Beam Antenna for Military Satellite Communications. ZmE Traasactions On Antennas and Propagation, Vol. 43, No. 10, October 1995. [47] R.A. Shore. Adaptive Nulling in Hybrid Reflector Antennas. Rome Laboratory - h-House Report, September 1992. [48] W.H. Press et al. Numerical Recipes in C: The Art of Scientific Computing, 2nd Edition. Cambridge University Press, New York, NY,1992. [49] R.T. Compton, Jr. Adaptive Antennas: Concepts and Performance. Prentice Hall, Englewood Cliffs, NJ, 1988. [50] C. Scott. Modern Methods of Reflector Antenna hdysis and Design. Artech House, Boston, 1990. [51] G. Maral, and M. Bousquet. Satellite Communications Systems, 2nd Edition. John Wiley & Sons, Toronto, 1993. [52] A. W. Love, editor. Reflector Antennas. IEEE Press, New York, NY, 1978. [53] R. A. Monzingo, and T. W. Miller. Introduction to Adaptive hays. John Wiley & Sons, Toronto, 1980. (541 A.M. Pasolin. Performance of Adaptive Antenna Arrays in a Fast Packet Network. M.Sc. thesis, Department of Elect rid and Computer Engineering, Queen's University, Kingston, Ont., 1994. [55] Y.T. Lo, and S.W. Lee, editors. Antenna Handbook: Theory, Applications, and Design. Van Nostrand Reinhold Company, New York, NY,1988. [56] A.D. Olver et al. Microwave Horns and Feeds. IEEE Press, England, 1994. [57] S. Haykin. Adaptive Filter Theory. Prentice Hall, Englewood Cliffs, NJ, 1986. [58] J.D. Kraus. Antennas, 2nd edition. McGraw-HilI Book Company,Toronto, 1988. [59] S. Silver, editor. Microwave Antenna Theory and Desip. Peter Perernus Ltd., London, UK, 1984. [60] H. Goldstein. CIassicaf Mechanics. Addison-Wesley, Reading, MA, 1953. [61] J. Karimi. Persoad Eland-Held Communications Via GBand CDMA-Based Geostationary Bedomzbg Satellites. M.Sc. thesis, Department of Electrical and Computer Engineering, Queen's University, Kingston, Ont., Feb. 1996. [62] R.O. Schmidt. A Sipaf Subspace Approad to Multiple Emitter Location and Spectra[ Estimation. P hD thesis, Stanford University, S tmdford, CA, Nov. 1981. [63] R.F. Hmington. Time-Harmonic Electromagnetic Fields. McGraw-ELill Book Company, Toronto, 1961. [64] R.E. Colin, and F.J. Zucker, editors. Antenna Theory: Part 1. McGraw-Hill Book Company, Toronto, 1969. [65] A. JefEiey. Handbook of Mathematical Formulas and integrals. Academic Press, Toronto, 1995. [66] M. Abramowitz, and I.A. Stegun, editors. Handbook of Mathematical Func- tions. Dover Publications, New York, 1965. 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 W7Hfm, 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 (A.14)

VxH = jwD+J (A.15)

V-B = 0 (A.16)

V-D = QY. (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. We can define a complex Poynting vector S=ExH* (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 (AX) and (A.18)),

The result is V x V x A = jwrp(-Vp - jwA) + pJ.

Rearranging and using the vector identity

VXVXA=V(V.A)-v2~ we get

V(V A) - V2A - W%A = pJ - j~pV9. (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+ k2~ = -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' = xY + 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 region. 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 W1I27n-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 propagation 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 beas 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, - & = fa/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:

(3) Elliptical Polarization. As mentioned earlier, b is in general elliptically polarized. 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 (bL1 > [bRl then b corresponds to left-hand elliptical polarization. If (bL1 = 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# (~.7l) 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 duecded 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) = (A.81) intensity of an isotropic radiator in direction k

where, Pmd is the total power radiated by the antenna. Cue 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) = wD(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 ante~ain a certain polarization. This leads to a definition of directivity and gain in polarization b,

These dues 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 (A.88) ueff (k,b) = power density of incident plane wave with polatization bmand 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 ante~adiffer only in their maximum dues. Therefore the normalized transmitting and receiving antenna patterns are identical.

A.ll 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~rn-~,n-1/2rn-L).

The reference plane z = 0 can be chosen arbitrarily and the vectors (e, h) must satisfy the nonnalization condition

//(e x h*) -idzdy= 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 (AM). 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 polarizations 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 polarization 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 (A.102) Gff(kt b) = 4n--(I - Ir12)f).gD(k,b) 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 heady polarized feed is

~~(8)cos 4 - tji ~~(0)sin 4 for k polarized E(r) = A. e OC~(~)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

(A.12) plots the cosq(6) feed pattern for several dues of q. Paper [La] by Y. Rahmat- Samii estimates practical dues 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 dues 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 different 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+Gy~+Gzi then that vector is expressed in the primed coordinate system by

where,

(B.18)

(B.19)

In matrix notation The second thing to note about the t~omationdescribed 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,

The second step in the transformation is shown in figure (B.4). This time the intermediate axes (s2,y2,~) 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, Y2,~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 7culcounterclockwise 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 PeUlcounterclockwise 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 yeulcounterclockwise 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 Transformations Between Spherical, Cylindri- cal and Cartesian Coordinates

Another common requirement in the work done in this thesis has been the transformation 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 (B.62) 0 1 I 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 Tkansformations Between the Cylindrical and Spheri- cal Coordinates

Now let us consider the transformation from cylindrical coordinates to spherical coordinates 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 fi unit vectors axe related to the i and e unit vectors. Mathematically, Figure B.9: k in terms of fi and 6.

Substituting equations (B.70) and (8.71) into equation (B.50) gives

Therefore in matrix form (g)=Tsc($) 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~=TT,= (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 coordinates and vice-versa. Refering to figure (B.8) we can relate (x, y, z) to (r, 0, 4) by

and therefore, Ap 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 transformation 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, T, = TJ, sin8 0 cosd cos 4 sin4 0 cos8 0 -sin0 01 0 0 1 I sin 8 cos # sin 0 sin $ cos 8 cos8cos~cos0sin+ -sin8 . - sin 4 cos 4 0 I I-=.*' ,p A 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,)-' = TsLT;;' = T,TsCT T = (T,T,)* = T:. (B.86)

Therefore the inverse transformation from the spherical system to the rectangular system is given by

where, sin 0 cos q5 cos 0 cos q5 - sin 9 T,, = sin0 sin4 cos 6 sin4 cos # . (B.88) [ ccs0 - sin 8 0 I 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 ail together .

(C.18)

(C. 19)

That proves equation (C.12).

C.3 Bessel Function Definition

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 = A344 + 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]