Polarimetry With Phased Array Antennas: Theoretical Framework and Definitions
ABSTRACT For phased array receivers, the accuracy with which the polarization state of a received signal can be measured depends on the antenna configuration, array calibration process, and beamforming algorithms. A signal and noise model for a dualpolarized array is developed and related to standard polarimetric antenna figures of merit, and the ideal polarimetrically calibrated, maximumsensitivity beamforming solution for a dualpolarized phased array feed is derived. A practical polarimetric beamformer solution that does not require exact knowledge of the array polarimetric response is shown to be equivalent to the optimal solution in the sense that when the practical beamformers are calibrated, the optimal solution is obtained. To provide a rough initial polarimetric calibration for the practical beamformer solution, an approximate singlesource polarimetric calibration method is developed. The modeled instrumental polarization error for a dipole phased array feed with the practical beamformer solution and singlesource polarimetric calibration was 10 dB or lower over the array field of view for elements with alignments perturbed by random rotations with 5 degree standard deviation.

Conference Paper: Advances in phased array systems for radio astronomy
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ABSTRACT: Phased array antenna systems are a defining technology in many current and planned radio astronomical instruments. Phased array technology enables new observing modes due to their large fieldofview, multibeaming capability and rapid response. The radio astronomical application provides interesting challenges such as stringent requirements on system temperature and system calibration. In this paper, we discuss the motivation for using phased array systems in radio astronomy and provide an overview of recent advances and results in this area.IEEE International Symposium on Antennas and Propagation, Orlando (Fl.); 07/2013  SourceAvailable from: M. V. Ivashina
 SourceAvailable from: Benedetta Fiorelli[Show abstract] [Hide abstract]
ABSTRACT: Polarization purity is one of the most stringent requirements for future radio telescopes. To evaluate polarization ratio levels of large phased arrays involving thousands of elements, a full wave simulation approach is very time consuming, when feasible. Commonly, large arrays are approximated by infinite array approach or studied by single (isolated) element analysis. Being simplified approaches, part of the effect of the array configuration is missed resulting in analysis error. A study of this effect for regular arrays based on the intrinsic crosspolarization ratio (IXR) is presented.Antennas and Propagation (EuCAP), 2013 7th European Conference on, Gothenburg, Sweden; 04/2013
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or image pixel can be adjusted on the fly by changing beam
former coefficients. If array output correlations are computed
Proof
phased array receiver. This is accomplished in Section II.
The treatment leads to the general problem of beamforming
for ranktwo signals and the concept of a polarimetric array
beam pair. In Section III, the standard IEEE definitions for
polarimetric figures of merit are related to the beam pair
Jones matrix. In Section IV, the optimal beam pair solution
is derived for a perfectly known instrument, and a practical
beamforming method based on signal correlation matrix eigen
IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 60, NO. 1, JANUARY 20121
Polarimetry With Phased Array Antennas: Theoretical
Framework and Definitions
Karl F. Warnick, Senior Member, IEEE, Marianna V. Ivashina, Stefan J. Wijnholds, Member, IEEE, and
Rob Maaskant
Abstract—For phased array receivers, the accuracy with which
the polarization state of a received signal can be measured de
pends on the antenna configuration, array calibration process, and
beamforming algorithms. A signal and noise model for a dualpo
larized array is developed and related to standard polarimetric
antenna figures of merit, and the ideal polarimetrically calibrated,
maximumsensitivity beamforming solution for a dualpolarized
phased array feed derived. A practical polarimetric beamformer
solution that does not require exact knowledge of the array
polarimetric response is shown to be equivalent to the optimal
solution in the sense that when the practical beamformers are
calibrated, the optimal solution is obtained. To provide a rough
initial polarimetric calibration for the practical beamformer
solution, an approximate singlesource polarimetric calibration
method is developed.The modeled instrumental polarization error
for a dipole phased array feed with the practical beamformer
solution and singlesource polarimetric calibration was
or lower over the array field of view for elements with alignments
perturbed by random rotations with 5 degree standard deviation.
dB
Index Terms—Array signal processing, phased array antennas,
polarimetry.
I. INTRODUCTION
T
(PAFs) for large reflectors [1]–[4]. Accurate polarization state
measurements for observed sources is critical to the science
goals for current and planned phased array instruments. With
a traditional waveguide feed, the polarization properties of the
receiver are fixed at the time of manufacture and unwanted
instrumental polarization can be calibrated by observing
sources with known polarization parameters. For a phased
array receiver, the polarimetric properties of each formed beam
HE radio astronomy community is currently developing
polarimetric aperture arrays and phased array feeds
Manuscript received January 12, 2011; revised June 20, 2011; accepted June
21, 2011. Date of publication September 15, 2011; date of current version Jan
uary 05, 2012.
K. F. Warnick is with the Department of Electrical and Computer En
gineering, Brigham Young University, Provo, UT 84602 USA (email:
warnick@ee.byu.edu).
M.V.IvashinaiswiththeDepartmentofEarthandSpaceSciences,Chalmers
University of Technology, S41296 Gothenburg, Sweden (email: marianna.
ivashina@chalmers.se).
S. J. Wijnholds is with the Netherlands Institute for Radio Astronomy (AS
TRON), NL7991 PD, Dwingeloo, The Netherlands (email: wijnholds@as
tron.nl).
R. Maaskant is with the Department of Signals and Systems, Chalmers
University of Technology, S41296 Gothenburg, Sweden (email: rob.
maaskant@chalmers.se).
Digital Object Identifier 10.1109/TAP.2011.2167926
and stored, one set of observation data can be processed with
multiple sets of beamformer coefficients tuned to optimize
sensitivity, sidelobe level, or polarimetric accuracy. Exploiting
this flexibility and achieving best possible system performance
requires the development of a theory for polarimetric phased
arrays, including figures of merit, optimal beamformer solu
tions, and practical calibration strategies.
Key questions that must be answered by this theory include
the following:
• How do astronomical performance criteria relate to the
standard IEEE definitions for polarimetric antennas?
• What beamforming algorithm will simultaneously opti
mize for high SNR and polarimetric accuracy?
• Which requirementsshould beset on the antenna arrayand
beamformer design to achieve optimal performance?
• How can a polarimetric array be accurately and efficiently
calibrated?
This paper will consider the first two questions in detail and ad
dresses the third empirically through a numerical study. An ap
proximate singlesource calibration scheme is presented to ad
dress the fourth issue. A full treatment of polarimetric calibra
tion is beyond the scope of this paper and will be addressed in
future work.
The first question arises because antenna engineers assess the
polarimetric performance of antenna systems in terms of the
axial ratio, crosspolarization discrimination (XPD), and cross
polarization isolation (XPI), while astronomers judge instru
ment performance and express system requirements in terms of
Stokes parameters [5], [6]. Another challenge is that the stan
dard IEEE definitions of the ARP, XPD and XPI have been
established for single port systems, and these figures of merit
must be extended to phased array systems that are capable of
forming multiple dualpolarized beams simultaneously. Astro
nomical antenna applications also have unique constraints be
causeradiationinterrestrialcommunicationsystemsistypically
highlypolarized,whereasastronomicalsourceshaveasmallbut
important polarized component of a few percent or less relative
to the total signal flux density.
The starting point for answering these questions is the
development of a signal and noise model for a polarimetric
0018926X/$26.00 © 2011 IEEE
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metric beamforming array comprised of two groups of antenna
elements with nominally orthogonal polarization. The antenna
system is assumed to be illuminated by a point source radiating
partiallypolarized fields. The electric field intensity vector at
the point
radiated by such a source can be approximated in
the neighborhood of the receiver by the incident plane wave
Proof
are orthogonal unit vectors according to one
of Ludwig’s polarization definitions relative to the coordinate
system of the array [7], and
is the wave vector corresponding
to the angle of arrival. Since
andare defined with respect
to the coordinate system of the array, we are neglecting in this
treatment a rotation of the polarization state from the astronom
ical coordinate system on the sky to the coordinate system of
the array.
2IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 60, NO. 1, JANUARY 2012
Fig.1. Aradiopolarimetercomprisedofadualpolarizedactivelybeamformed
receiving antenna array.
vectors is considered for partially characterized polarimeters.
An important result is that if sample estimation error (the error
incurred by computing the array output correlation matrix from
a finite number of voltage samples) is neglected, the practical
eigenvector maxSNR method is equivalent to the optimal
solution when polarimetrically calibrated. The performance of
practical beamformers are compared to the ideal solution using
a numerical model in Section V.
Voltage and field quantities are phasors with the
tion. An overbar
is used to denote threedimensional field
vectors, whereas vectors of voltages are typeset in boldface
The superscript
designates the complex conjugate and
conjugate transpose.
denotes expectation over time.
conven
.
the
II. POLARIMETRIC PHASED ARRAY MODEL
The purpose of a radio polarimeter is to measure the polariza
tionpropertiesofanincidentelectromagneticwaveasafunction
of the angle of arrival. Fig. 1 illustrates an
element polari
(1)
where
and
In the following theoretical development, we will consider
the polarimetric calibration problem for one beamsteering di
rection.Thecalibrationprocessmustbeappliedtoeveryformed
phased array beam for each desired beam direction, as the nu
merical results make clear. Array calibration is accomplished
with point sources, rather than extended sources, so all signal
responsequantitiesareassumedtoarisefromapointsource.For
animagingarray,eachpixelrepresentsadifferentbeamsteering
direction and set of array beamforming coefficients. The figures
of merit and beamforming procedures developed in this paper
apply independently to each image pixel.
The source of interest is assumed to be a point source, and
all results for figures of merit are calculated at the beam center.
Since extended astronomical sources are common, beam polar
ization patterns are important, but this aspect of phased array
polarimetry will be considered in future work.
The antenna output signals are amplified to form the
ment output voltage vector , which is subsequently combined
into the output voltages
and
vectors
andrespectively, each of which is a column
vector of size
. Together,
metric beam pair for a given sky pointing direction1.
The electric field components
plexrandomprocessesinthephasororcomplexbasebandrepre
sentation.Thepolarizationstateoftheplanewaveisdetermined
by the covariance matrix of the two field components, which is
the
Hermitian matrix
ele
using the beamformer weight
andconstitute a polari
andare com
(2)
The covariance matrix has two real and one complex degrees
of freedom, or four real degrees of freedom. The timeaverage
power flux density of the incident wave is
(3)
where
the matrix trace operation.
Some authors rearrange the covariance matrix to form the
coherency vector [6]
is the characteristic impedance of free space and tr is
(4)
where
of
determine the degree of polarization and the polarization state
of the polarized part of the wave.
is the vector obtained by stacking the columns
. The relative magnitudes and phases of these quantities
A. Array Signal Response
With reference to Fig. 1, we will model the antenna array
output signals
in terms of the voltage responses of
the array to the unit intensity, timeharmonic, linearly polarized
waves
1In the less general biscalar method,
and are only associated to the  and oriented antenna elements,
respectively.
andare weight vectors of size
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IEEE
output voltage covariance matrix can be described as
Proof
polarization ratio [11] can be applied to polarimetric phased ar
rays but will not be considered further in this paper.
WARNICK et al.: POLARIMETRY WITH PHASED ARRAY ANTENNAS: THEORETICAL FRAMEWORK AND DEFINITIONS3
(5a)
(5b)
where
by the two waves at the array receiver outputs before beam
forming. Even though
thogonal due to possible nonideal polarimetric response of the
receiver.
The voltage responses
and
in practice, but are useful in developing a system model
for a beamforming phased array antenna. We will show in
Section IV.A that if
andwere known, the beam pair
can be exactly calibrated polarimetrically. Experimental
procedures for calibrating the array in the case that
are not known are discussed in Section IV.B.
Since
and are orthogonal, it follows by linearity that for
an arbitrarily polarized wave, the array signal voltage response
vector can be written using
and
andare vectors of the respective voltages induced
andmay not be or
are not exactly known
and
as
(6)
The
array output signal voltage covariance matrix is
(7)
which is of rank one for a fully polarized wave and of rank
twoforapartiallypolarizedorunpolarizedincidentwave.Upon
introducing the
matrix
(8)
we can write (7) in the more compact form
(9)
Assuming that the phased array system noise can be charac
terized by the noise covariance matrix
, the complete array
(10)
This expression provides a signal and noise model for a polari
metric array. The precise form of the noise response
array receiver is not important here, but is considered in detail
in [8]–[10].
for an
B. Beam Pair Signal and Noise Response
Afterbeamforming,thetwooutputvoltagesobtainedwiththe
beamformer pair are (cf. Fig. 1)
(11a)
(11b)
The covariance matrix
of the two beam outputs is
(12)
In terms of the array signal and noise covariance matrices, the
beam pair output covariance matrix is
(13)
Introducing the
vector
(14)
leads to the more compact expression
(15)
where
(16a)
(16b)
This result provides a signal and noise model for the polari
metric beam pair
.
III. POLARIMETRIC DEFINITIONS
In this section, we consider the relationship between IEEE
definitions for polarimetric figures of merit and the Jones
and Mueller matrix formulations that are common in radio
astronomy and remote sensing. We also consider the problem
of quantifying the joint sensitivity of a polarimetric beam pair.
A. IEEE Definitions for Single Port Antennas
The IEEE definitions for polarization terms for single port
antennas are as follows [5]:
Crosspolarization discrimination (XPD) is the ratio of the
power level at the output of a receiving antenna, nominally
copolarized with the transmitting antenna, to the output of a
receiving antenna of the same gain but nominally orthogonally
polarized to the transmitting antenna.
Crosspolarization isolation (XPI) is the ratio of the wanted
powertotheunwantedpowerinthesamereceiverchannelwhen
thetransmittingantennaisradiatingnominallyorthogonallypo
larized signals at the same frequency and power level.
Other figures of merit such as axial ratio and intrinsic cross
B. Definitions for Phased Array Antennas
With reference to the polarimetric phased array model devel
oped in Sec. II, the signal response of an array antenna to co
and crosspolarized incident fields
the two beamformer output voltages
andis described by
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IEEE
Crosspolarizationisolationisdefinedastheratioofpowers
received at the same beamformer output, 1 or 2, due to orthog
onallypolarized incident fields
and:
Proof
the form
4 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 60, NO. 1, JANUARY 2012
(17)
(18)
where
elements in response to pure
vectors
and
defined such that they optimally receive or at least approximate
the optimal reception of
and
some specified criterion to be discussed in Section IV. This im
plies that the polarizations of the two formed beams are nom
inally aligned with the
and
defined as real vectors in (1), the definitions in this section hold
for any pair of waves
and
whether linear, circular, or elliptical.
Ideally, one expects the two output voltage vectors to be di
rectly proportional to
and
polarization leakage caused by imperfections in the element ge
ometry or mutual coupling between the array elements, in par
ticular when the antennas are placed in a finite array exhibiting
strongtruncationeffects.Inthelattercase,thepolarizationchar
acteristics of the embedded element patterns differ significantly
from each other. The degree of polarization leakage is depen
dent not only on the array geometry and mechanical construc
tion, but also on the values of the beamformer coefficients.
To quantify the beamdependent polarization leakage, we ex
tend the standard definitions in Section III.A as follows:
Crosspolarization discrimination is defined as the ratio of
powers received at the beamformer outputs 1 and 2 due to the
same incident field, i.e.,
or
andare the output voltage vectors of the receiving
and
of the two beamformers are assumed to be
signals. The weight
, respectively, according to
directions. Whileand are
with orthogonal polarizations,
, but this may not hold due to
:
(19)
(20)
where
ance matrices in response to pure
intensity.
andare the signal covari
andsignals of unit
(21)
(22)
C. Jones Matrix Formulation
Unlike radio communications, for which antennas are fabri
cated to meet a fixed polarization purity requirement, applica
tions such as radio astronomy and remote sensing that rely on
accurate measurement of wave polarization states also require
operational polarimetric calibration. When modeling system ef
fectsthatcontributetothemeasuredwavepolarizationstate,itis
convenient to use the Jones matrix formulation [12]. The Jones
formulationallowsmatrixtermsforvarioussystemcomponents
to be chained together into an overall Jones matrix that must be
modeled or measured in order to infer the incident wave polar
ization state.
For a phased array, each polarimetric beam pair has an asso
ciated Jones matrix. The general relationship between the beam
outputs and the incident electric field intensity vector is
(23)
where the two by two matrix on the right side of this equation
is the beam pair Jones matrix, which we will denote in the fol
lowing as
. Using the notation developed above, for an array
this relationship becomes
(24)
The Jones matrix can be identified as
(25)
where
(26)
Usingthisresult,thebeam pairoutputsignal voltagecovariance
matrix (16a) can be written as
(27)
Thegoalofpolarimetriccalibrationistotransformagivenbeam
pair
to a new beam pair
are steered to the same angle of arrival as the original beam pair
but having Jones matrix
as close as possible to the identity
matrix. If
, the crosspolarization figures of merit are
large in value. If the initial Jones matrix
it can be readily seen that the beam pair
for which the radiation patterns
is known,
(28)
is ideally polarimetrically calibrated and the realized Jones ma
trix is
.
If the beam pair output signal coveriance matrix
sured for one signal with a known polarization state
relationship (27) does not uniquely fix the beam pair Jones ma
trix. Given known input and output polarization parameters, by
substitution it can be shown that the general solution to (27) has
is mea
, the
(29)
where
only a single calibrator source, there remains a 2
degree of freedom in the Jones matrix. The physical interpreta
tion of the unitary degree of freedom is discussed in [6], [13].
The ambiguity can be removed using knowledge of the nom
inal element polarizations (see Section IV.C), measurements of
is an arbitrary unitary matrix. This shows that with
2 unitary
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IEEE
For an unpolarized wave,
.
A Stokes polarimeter can be represented by the relationship
Proof
larimetric calibration means measuring the beam pair Mueller
matrix
and using that information to transform the weight
pair
,oringeneraladaptthebeampaircovariancematrix
such that the effective Mueller matrix becomes proportional to
the identity matrix.
Polarimetric accuracy can be quantified by the error in the
measured Stokes parameters for a given source or by the devia
tion of the Mueller matrix from the identity. The relative RMS
Stokes error is
WARNICK et al.: POLARIMETRY WITH PHASED ARRAY ANTENNAS: THEORETICAL FRAMEWORK AND DEFINITIONS5
additional calibrator sources, or correlation with a second cali
brated polarimetric antenna.
In terms of the elements of the Jones matrix, the crosspolar
ization figures of merit (19)–(22) are [11]
(30a)
(30b)
These expressions show that the standard antenna polarization
figures of merit are measures of the magnitudes of the offdiag
onal elements
andrelative to the diagonal elements
and.
D. Mueller Matrix Formulation
While Jones matrices operate in the voltage phasor domain,
4 Mueller matrices represent transformations on wave po
larization states in the correlation or Stokes parameter domain.
Since the Mueller matrix formulation is commonly used in the
astronomical literature, the treatment will be rehearsed here and
placed into the mathematical framework developed in earlier
sections.
For a signal characterized by the field coveriance matrix (2),
the Stokes vector containing the four Stokes parameters is de
fined as [14, pp. 97–98]
4
(31)
Some authors define the Stokes parameters with a factor of
, whereis the intrinsic impedance of space, so that the
Stokes parameters have units of power density (W/m .
It can be shown that [15, p. 29]
(32)
The degree of polarization is
(33)
(34)
where
wave given by (31) and
The vector of Stokes parameters referred to the beam output
voltages is
is a vector of the Stokes parameters of the incident
is the Mueller matrix of the system.
(35)
The beam pair is said to be polarimetrically calibrated if the
Stokes parameters of the source, (31), and the measured Stokes
parameters,(35),areidentical,apartfromaconstantgainfactor.
In the following, we will include the gain calibration with po
larimetric calibration, and consider the system to be polarimet
rically calibrated when
.
The Mueller and Jones matrix formulations are connected by
known formulas, which we will review here. Stokes parameters
are related to the coherency vector (4) by
(36)
where
(37)
The inverse of this transformation matrix is
(38)
Using this relationship for the incident wave and the beam
output voltages
(39)
Using the properties of the Kronecker product, (27) can be re
arranged in the form
(40)
In view of (4), this becomes
(41)
Combining these results shows that the system Mueller matrix
is related to the Jones matrix by (see also [16])
(42)
In the present context, the Jones matrix
dependent, so that, by using (25) along with the properties of
the Kronecker product
is weight
(43)
If the polarimeter is ideally calibrated, then
, and.
For a realistic system configured as a Stokes polarimeter, po
,
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IEEE
the beamformer coefficients control the beam output SNR and
antenna sensitivity. For one beam, it is straightforward to de
termine the beamformer coefficients that maximize sensitivity
given a knowledge of the array signal and noise responses. For
a beam pair, the sensitivity of both beams should ideally be as
highaspossible.Inparticular,thebeamsensitivitiesofanuncal
ibrated beam pair
and the polarimetrically calibrated beam
pair
areingeneraldifferent,andtherearemanypossiblepo
larimetrically calibrated beam pairs that have lower sensitivity
than the classical maximumSNR beamformer solution. There
fore,itisofinteresttocharacterizethesensitivityofbothoutputs
of a beam pair with a metric that is independent of the beam pair
polarimetric calibration.
For one beam
, the sensitivity is
Proof
where
andare the eigenvalues of
, where is an arbitrary invertible 2
can be seen that the matrix
mation of the beam subspace and hence of the beam pair polari
metric calibration.
Since the response
to orthogonal polarized waves is not
directly available for a phased array in practice, it is of interest
to express the sensitivity bound in terms of measurable array
6IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 60, NO. 1, JANUARY 2012
(44)
This metric can be applied to computed Stokes parameters be
fore the instrument is polarimetrically calibrated, to determine
the raw or uncalibrated Stokes error, or it can be applied after
calibration, to assess the performance of both the calibration
procedure and the native instrumental polarization properties.
As with the Jones formulation, sourceindependent measures
of polarimetric accuracy are also desirable. By the definition of
the induced operator norm, we have the bound
(45)
For weakly polarized sources,
imately
, and we have approx
(46)
The quantity
(47)
therefore is an approximate upper bound on the relative RMS
Stokes error.
might be referred to as the Stokes instru
mental polarization bound. This bound is derived for the case of
a noisefree system. Since estimation error due to noise can be
reduced to arbitrarily low levels by integration, (47) is adequate
for many purposes, including antenna design optimization, but
when analyzing astronomical observation data in practice, the
additional effect of noise on polarimetric accuracy should be
considered.
E. Beam Pair Sensitivity
The above treatment has considered traditional polarimetric
formulations for the case of a phased array antenna. Another
important aspect of beamforming for polarimetric arrays is that
(48)
where
mann’s constant, and
dent wave in W/m . As required by the definition of effective
area, the incident wave is polarization matched with the polar
izationofthebeam.Thesignalcorrelationmatrixforatimehar
monic incident wave can be expressed as
is the system noise equivalent bandwidth,
is the power flux density of the inci
is Boltz
(49)
where
is matched to the beam when
occurs for the incident field state
. The polarization of the incident wave
is maximized, which
(50)
where the scale factor follows from (3). Using these results in
(48) leads to
(51)
This same result can be obtained from (48) for an unpolarized
wave
with
placed by half the power flux density of the unpolarized wave.
To define the beam pair sensitivity, we seek a figure of merit
that is independent of the beam pair polarimetric calibration.
This means finding bounds on the beam sensitivity (48) with
an arbitrary linear combination of
sensitivity is
in the leading scale factor re
and. Using (51), the
(52)
where
is an arbitrary vector. With
, where
(53)
(52) becomes
(54)
This shows that the ratio of quadratic forms in (52) lies within
the field of values of the matrix
the eigenvalues are real and the field of values is an interval
on the real line. It follows that for any beamformer
subspace spanned by the beam pair the sensitivity is bounded
by
[17]. Sinceis Hermitian,
in the
(55)
. By substituting
2 matrix, it
is independent of linear transfor
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IEEE
inordertounderstandtheultimateperformancelimitsofagiven
beam pair, as well as the practical case for which polarization
properties of a beam pair must be calibrated empirically using
observations of sources with known polarization parameters.
To avoid sacrificing observation time for polarimetric calibra
tion,onecanobservewith nonpolarimetricallycalibratedbeam
pairshaving imperfectpolarizationdiscrimination.Therefore,it
is important to find an approach for obtaining best possible po
larimetric beam pairs using the singlesource calibration data
required for nonpolarimetric array beamforming, and to char
acterize the polarization purity of beam pairs that are obtained
without additional effort at polarimetric calibration.
Proof
of
is
WARNICK et al.: POLARIMETRY WITH PHASED ARRAY ANTENNAS: THEORETICAL FRAMEWORK AND DEFINITIONS7
outputquantities.IntheAppendixitisshownthatthesensitivity
bound can be expressed in the form
(56)
The calibrationindependence of this form of the bound follows
from the properties of the matrix trace.
We will refer to two beam pairs
equivalent if they lead to the same upper and lower sensitivity
bounds in (55) or (56). By the above derivation, if the beam
pairs are related by a linear transformation, they are sensitivity
equivalent, and a beam pair is sensitivity equivalent to its po
larimetrically calibrated counterpart
serve that a polarimetrically calibrated beam pair does not nec
essarily achieve the upper bound in (55). The degree to which
polarimetriccalibrationreducesbeamsensitivitywillbestudied
empirically in Section V.
andas sensitivity
. Finally, we ob
IV. POLARIMETRIC BEAMFORMING
For a polarimetric phased array, calibration has two aspects:
1) determiningthebeamformer pair
other beam figures of merit are optimal, and;
2) determining the beam pair Jones matrix or the Mueller ma
trix so that the incident wave polarization state can be in
ferred from the beam pair outputs.
Combining 1) and 2) enables high sensitivity polarimetry.
Beamformer coefficients applied to a phased array can be
updated dynamically to accomplish various goals, such as
sensitivity maximization as the noise environment changes,
interference mitigation, or beam pattern sidelobe control.
Consequently, the polarization properties of the array beam
outputs are not necessarily fixed. This introduces the possibility
of combining the beamformer optimization and polarimetric
calibration steps, so that the beam pair satisfies a specified set
optimality criteria and also is polarimetrically calibrated with
both the Jones and Mueller matrices approximately equal to
identity matrices.
We will consider the ideal case of a perfectly known system,
so that sensitivityand
A. Optimal Beamforming for a Perfectly Known System
Any beam pair that responds to two noncolinear incident po
larizations can be polarimetrically calibrated, but the resulting
beams may have poor sensitivity, which implies a low SNR and
inaccurate measured Stokes parameters for a given integration
time. We will show that there is a unique polarimetrically cal
ibrated beam pair that minimizes estimation error in measured
Stokes parameters for a point source at a given angle of arrival.
Measured Stokes parameters are a linear combination of the
elements of a sample estimate of the 2
given by (16a). We will denote the sample estimated ma
trix for a given integration length
radiometric detection technique,
an onsource measurement and offsource measurement of the
beam pair output coveriance matrix:
2 covariance matrix
as. With the standard
is the difference between
(57a)
(57b)
where
difference is
signifies a sampleestimated coveriance matrix. The
(58)
After polarimetrically calibrating the beam pair outputs, the
output covariance matrix
becomes
(59)
This is the same measurement equation that would be obtained
with (58) subject to the calibrated beam pair
(60)
Therefore, if Stokes parameters are computed by applying the
beam pair inverse Jones matrix
ters obtained from an uncalibrated beam pair
ment error due to system noise can be analyzed as if the beam
pair were replaced by a polarimetrically calibrated beam pair.
The goal is to find the polarimetrically calibrated beam pair
that minimizes the estimation error in (59), which we will write
with (60) as
to the polarization parame
, the measure
(61)
The matrix
bution [18], which means that
tically distributed Wishart random matrices. The mean of
the zero matrix. In the low SNR limit, the variance of the entries
is described stochastically by the Wishart distri
is the difference of two iden
is
(62)
where
to produce
minimizeestimationerrorintheStokesparameters,thediagonal
elementsof the2
2 beam pair outputnoise correlation matrix
must be minimized.
is the number of voltage samples that are averaged
and . This result shows that in order to
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systemnoisepowerwhileconstrainedtobepolarimetricallycal
ibrated with respect to a source with a given angle of arrival.
The optimal solution is related to a simple ranktwo general
ization of the classical maximumSNR beamformer [19]:
Proof
. The optimal beamformer solution
8 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 60, NO. 1, JANUARY 2012
This consideration reduces the polarimetric beamformer cal
ibration problem to the joint constrained minimization problem
(63)
(64)
Since
tions and recast the constrained minimization problem in the
form
ispositivedefinite,wecanaddthetwoobjectivefunc
(65)
The constrained optimization problem can be solved by set
ting to zero the matrix derivative
(66)
where
chosen to satisfy the constraint in (64). Evaluating the deriva
tive leads to
is a matrix of Lagrange multipliers, which will be
(67)
(68)
Usingtheconstraint
,andsolvingfor,we findthat
(69)
from which it follows that
(70)
The solution to the optimization problem is found by substi
tuting (70) in (68) to yield
(71)
This pair of beamformer weight vectors minimizes the output
(72)
This beam pair realizes two maximum sensitivity beam
former weight vectors, with the Jones matrix
(71) can be written as
(73)
which shows that the beam pair
polarimetrically calibrated but is sensitivity equivalent to the
optimal beam pair.
is not in general
B. Practical Polarimetric Array Calibration and the
Eigenvector Method
In practice, the voltage response matrix
sure for a PAF on a reflector antenna, since ideal, orthogonally
polarized astronomical sources are unavailable. A practical po
larimetric array calibration procedure using observations of un
polarized and partially polarized sources is needed.
For a dualpolarized array, neglecting estimation error, the
signal covariance matrix
is of rank two for an unpolarized
source. The latter is readily concluded from (9) by taking
, so that
is difficult to mea
(74)
which is a sum of two rankone matrices. The eigenvalues and
eigenvectors of
are defined by
(75)
Neglectingestimationerror,twooftheeigenvaluesarenonzero.
Veidt proposed to use the two principal eigenvectors
andof as conjugate field match (CFM) beamformer
weights [3]. We will refer to the beam pair
(76)
as the eigenvector CFM beamformer.
The eigenvector CFM method can be modified to form the
maximumSNR eigenvector beamformer weight vectors
(77a)
(77b)
This is the eigenvector maxSNR beamformer algorithm.
To compare this beamformer with the optimal solution pre
sented in the previous section, we note that the voltage response
vectors
andmust span the same subspace as the eigen
vectors
and. This implies that
(78a)
(78b)
which is equivalent to
(79)
Substitution in (71) gives
(80)
which has a similar form as (71). Upon introducing
and, (80) can be written as
(81)
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The practical beamforming algorithms, (83b)–(83e), are not
polarimetrically calibrated. The eigenvectors computed from
(75) are arbitrary up to a scale factor, and the resulting beam
pair has a unitary degree of freedom according to (29). In order
to give meaning to crosspolarization figures of merit, these de
grees of freedom must be fixed by some concrete, repeatable
algorithm.
Since each beam pair can be viewed as the output of a dual
polarized single pixel feed, any polarimetric calibration tech
nique that isusedfor conventional feedscould be applied to cal
ibrate the beam pair. Standard polarimetric calibration methods
require observations of multiple sources with known Stokes pa
rameters or tracking a polarized source over time as it rotates
relative to the feed. Since these observations would have to be
repeated with the telescope steered so that the source was at
the center of each formed phased array beam, a straightforward
implementation of standard polarimetric calibration could re
Proof
WARNICK et al.: POLARIMETRY WITH PHASED ARRAY ANTENNAS: THEORETICAL FRAMEWORK AND DEFINITIONS9
This result shows that the eigenvector maxSNR beam pair is
sensitivity equivalent to the optimal beam pair.
A modification of the above approach can be obtained using
the generalized eigenvalue form of the maxSNR beamformer:
(82)
Since
two nonzero eigenvalues. The corresponding principle eigen
values provide a maximumSNR beam pair. It can be easily
shownthatthisbeampairisalsosensitivityequivalenttotheop
timal beam pair (71). The same beam pair can also be obtained
by using the noise correlation matrix to prewhiten the array
output signals and using the eigenvector CFM method with the
prewhitened signal correlation matrix.
To summarize, we have so far five polarimetric beamforming
algorithms:
is rank two, the generalized eigenvalue problem has
(83a)
(83b)
(83c)
(83d)
(83e)
The first and second beam pairs require knowledge of the array
response
to orthogonally polarized incident waves, which is
not directly available in practice. The third, fourth, and fifth are
practical beamformers in the sense that they can be computed
from measurable signal output correlation matrices for unpolar
ized or partially polarized sources. Thefirst beam pair is exactly
polarimetrically calibrated, but in general the others are not.
In the absence of estimation error, all of these beam pairs
except for CFM are sensitivity equivalent. Sample estimation
error in the array output correlation matrices may affect each
beamformer algorithm differently, but the effect of estimation
error on the beamformer weights can be driven to low levels by
integrating the array output during the calibration phase for a
long period of time, so this effect is not considered here.
C. Approximate Calibration With a Single Unpolarized Source
quire many hours or even days. It is possible that efficient mul
tiplesourcecalibrationmethodsforphasedarrayscanbefound,
but a full treatment of efficient polarimetric calibration proce
dures for phased arrays is beyond the scope of this paper.
Since determining beamformer weights for a phased array
requires observations of a bright (and typically unpolarized)
source over a grid of telescope pointings at the center of each
beam [20], the goal here is to find the best possible polarimetric
calibration procedure given this already available observation
data. We will show that a single unpolarized source observation
per beam togetherwith theknownnominalelementpolarization
can be used to calibrate each beam pair approximately.
Given an uncalibrated beam pair
ibration procedure is to use the nominal polarization of array
elements to find the approximate Jones matrix. The signal cor
relation matrix has the block form
, the first step in the cal
(84)
where
polarized elements, respectively. Neglecting estimation error,
the matrices
andrepresent outputs from polarized elements and
(85)
are rank one and have principal eigenvectors
tively. These vectors are orthogonally rotated as needed to ob
tain
and, where the hat indicates that these are only ap
proximate responses to orthogonally polarized waves, whereas
andin (5) are exact. These vectors can be used directly as
beamformer weights (the biscalar method), but we will employ
them here to obtain the approximate Jones matrix
and, respec
(86)
where
to calibrate the beam pair to obtain
.ThisapproximateJonesmatrixcan beused
(87)
Since the eigenvectors
a scale factor, it remains to specify the complex scaling of the
beamformer weight vectors
Assuming that the calibrator source is unpolarized, the mag
nitudes of the weight vectors can be fixed by equalizing the
responses to the calibrator source. The overall phases of the
weight vectors are set by dividing by the phase of the largest
weight vector element, so that the largest weights are real. This
leads to the approximately calibrated beam pair
and are only determined up to
andin the beam pair.
(88a)
(88b)
In practice, the phase lengths of the receiver signal paths will
be different, but this phase can be measured using injected cali
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small random rotations away from the nominal  and polar
izations. Since only a small amount of power is received by el
ements more than one or two wavelengths away from the focal
spot associated with a given formed beam, simulations of po
larimetric accuracy for the 19 element case are expected to be
representative of results for larger arrays.
Proof
steering angle. For the beam steered in the boresight direction,
the polarimetric calibration is essentially perfect, but for beams
steered offboresight, the figures of merit decrease. Results are
shown for the
polarization. Curves for the
are nearly identical.
Error in measured Stokes parameters relative to the total
source intensity for the singlesource calibrated maxSNR and
CFM beam pairs is shown in Fig. 4. The source Stokes param
eters are tabulated values from the NRAO C Band VLA/VLBA
10IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 60, NO. 1, JANUARY 2012
bration tones or other means and incorporated in the phase con
straint. The magnitude scaling in (88) ensures that the diagonal
elements of
are unity, which through (42) implies that the
element
of the Mueller matrix is equal to unity.
This approximate singlesource polarimetric calibration pro
cedure is essentially equivalent to a polarimeter that uses the
nominallyorthogonallypolarizedoutputsofastandarddualpo
larized antenna (including a rotation from the astronomical co
ordinate system to the coordinate system of the antenna). As
with a standard antenna, coupling between phased array ele
ments and mechanical imperfections mean that the beam pair
willnotbeexactlycalibrated,andadditionalobservationsofpo
larized sourcesmay berequiredtoremoveresidual instrumental
polarization effects. Numerical results will be given in the fol
lowing section to assess the performance of this singlesource
polarimetric calibration method.
V. NUMERICAL RESULTS
The polarimetric figures of merit and beamforming algo
rithms will be illustrated for a 19
of thin, lossless,
 andpolarized crossed dipoles spaced
apart and backed by a ground plane. The phased array
feeds a 20meter reflector with
cuited element response is approximated using the analytical
expression for radiation by a dipole. Physical optics is used
to compute secondary fields scattering from the reflector. The
array mutual impedance matrix is approximated by conserva
tion of energy from the element pattern overlap integrals. The
PAF noise model includes sky noise, spillover, and receiver
noise due to low noise amplifiers with parameters
, and K. Ludwig’s first convention [7]
is used to define the
and directions, which means that
are aligned with the array  and coordinate frame.
The array is calibrated using the approach of [20] using ob
servations of an unpolarized calibrator source for each desired
beam steering direction. Beamformer weights are computed
using the eigenvector maxSNR and CFM algorithms described
above, and polarimetrically calibrated using the approximate
singlesource method of Section IV.C.
Results are given for two array configurations: (1) perfectly
aligned elements and (2) perturbed element orientations with
2 element hexagonal array
. The opencir
and
A. Perfectly Aligned Elements
The first study is a comparison of the sensitivity of the eigen
vector maxSNR and CFM beam pairs to the optimal solution
(Fig.2).ThesinglesourcecalibrationprocedureofSectionIV.C
is used with both beamformers. The sensitivity of the maxSNR
beam pair calibrated with the singlesource procedure is nearly
indistinguishable from that of the optimal solution. For the po
larimetrically uncalibrated case, the maxSNR beam pair sensi
tivityisdifferent,butstilllieswithintheupperandlowerbounds
overarbitrarylineartransformationsofthebeampair.Thebeam
Fig. 2. Beam pair sensitivity for dipole array with perfectly aligned elements
for beam steering angles from the feed boresight to 2 half power beamwidths
(HPBW) from boresight. The cut is at an azimuth angle of 30 from the
rection in the feed coordinate system. The sensitivity bound in (55) is indicated
with gray shading.
di
Fig. 3. Beam center crosspolarizationfigures of merit as a function of steering
angle for dipole array with perfectly aligned elements.
pair sensitivity bounds are given by (55) and indicated in the
figure with gray shading. For some beam steering angles, the
sensitivity of one of the uncalibrated beams is larger than that
of the optimal beam pair. This illustrates that the calibration
constraint in (65) leads to a beam pair with sensitivity slightly
smaller than is achieved by other uncalibrated beam pairs in the
subspace spanned by the optimal beam pair. The singlesource
calibrated CFM beam pair sensitivity also lies within its corre
sponding bound (55), but the overall sensitivity is lower than
that of maxSNR.
Crosspolarization figures of merit are shown in Fig. 3. Re
sults are given for each beam center as a function of beam
polarized beams
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IEEE
It may be surprising that the perfectly aligned array, with no
receiver gain imbalances, no mechanical defects, and exact an
alytical formulas for the element response, is only exactly cal
ibrated for the boresight beam. For the boresight beam, the re
sponse vectors
andin (7) are orthogonal, and the eigen
vectors obtained from (75) are proportional to
and
at boresight the eigenvector maxSNR and CFM beam pairs
can be exactly calibrated without making use of the biscalar
transformation (87). For offboresight steered beams, due to the
depolarizing effect of the reflector,
andare not orthog
onal, and the eigenvector pair
and for the perfectly
aligned array require polarimetric calibration.
The Square Kilometer Array (SKA) design target for relative
instrumental polarization is 25 dB for general polarimetric
imaging and 40 dB for specialized applications (see [21] and
other available SKA documents). For the perfectly aligned
array, the instrumental polarization is better than 40 dB only
Proof
of the sensitivity bound (55) are no longer equal for the bore
sight beam. The crosspolarization figures of merit (Fig. 6) and
Stokes instrumental polarization (Fig. 7) are significantly de
graded. The instrumental polarization is poorer than 25 dB for
many beam steering angles. For observations that require better
polarimetric accuracy, the nominal calibration approach pre
sented in this paper is not adequate, and a further calibration
step for each beam pair similar to the methods used for tradi
tional singlepixel feeds would be required.
WARNICK et al.: POLARIMETRY WITH PHASED ARRAY ANTENNAS: THEORETICAL FRAMEWORK AND DEFINITIONS 11
Fig. 4. Relative Stokes parameter error (44) and instrumental polarization
bound (47) as a function of steering angle for dipole array with perfectly
aligned elements.
Fig. 5. Beam center sensitivity for dipole array with perturbed element orien
tations. The cut is at an azimuth angle of 30 from the
coordinate system. The sensitivity bound (55) is indicated with gray shading.
direction in the feed
Polarization Calibration Database for
Jy,
the instrumental polarization bound (47) for both beam pairs.
. The error closely follows
. Thus,
Fig. 6. Beam center crosspolarization figures of merit for dipole array with
perturbed element orientations.
Fig. 7. Relative Stokes parameter error and instrumental polarization bound
for dipole array with perturbed element orientations.
for beam steering angles near boresight. For offboresight
beams, it is apparent that an additional polarimetric calibration
beyond the approximate singlesource method of Section IV.C
is required.
B. Imperfectly Aligned Elements
Wenowstudytheeffectofmechanicalimperfectionsonarray
feed polarimetric performance. A rotation in the  plane was
applied to the orientation of each element in the array, with ro
tation angle chosen from a zeromean normal distribution with
5 standard deviation. Fig. 5 shows that the beam sensitivity is
not reduced by the perturbation, but the upper and lower limits
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IEEE
APPENDIX
Here we derive (56). By diagonalizing in (53), it can be
shown that
is bounded by
Proof
[15] H. C. v. d. Hulst, Light Scattering by Small Particles.
Dover Publications, 1981.
[16] R. J. Sault and T. J. Cornwell, “The HamakerBregmanSault mea
surement equation,” in Proc. Synthesis Imaging in Radio Astronomy
II—ASP Conf. Series, 1999, vol. 180, pp. 657–669.
[17] R.HornandC.Johnson,TopicsinMatrixAnalysis.
Cambridge Univ. Press, 1994.
[18] J. Wishart, “The generalised product moment distribution in samples
from a normalmultivariate population,”Biometrika,vol.20A, no. 1–2,
pp. 32–52, 1928.
[19] H. L. Van Trees, Optimum Array Processing.
2002.
12IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 60, NO. 1, JANUARY 2012
VI. CONCLUSION
A signal and noise model for a dualpolarized beamforming
array receiver has been used to develop polarization figures of
merit and a theory of ideal polarimetric beamforming. An op
timal polarimetrically calibrated maximumSNR beamforming
algorithm was given. For each angle of arrival, the algorithm
provides a beam pair with maximum sensitivity subject to a po
larimetric calibration constraint.
The optimal polarimetric beamforming algorithm requires
exact knowledge of the responses of the array to orthogonally
polarized waves from a given angle of arrival. In practice, the
array responses to orthogonally polarized waves are not readily
available. A practical eigenvectorbased maximumSNR beam
former solution has been shown to optimize sensitivity in the
sense that when the beams are polarimetrically calibrated, in the
absence of estimation error the beams become equivalent to the
optimal solution. In general, the eigenvector maximumSNR
beam pair is not polarimetrically calibrated, and additional
correction through estimation of the beam pair Jones matrix is
required. An approximate singlesource calibration procedure
based on the approximate orthogonality of the array element
polarizations was developed. The instrumental polarization
of the approximately calibrated beams for a perturbed dipole
phased array feed was better than 10 dB over the field of view.
The approximate singlesource method therefore provides a
rough calibration method for routine observations that do not
require high polarimetric accuracy.
In future work, more accurate polarimetric calibration
methods for phased arrays should be studied. To achieve lower
instrumental polarization than the singlesource method used in
this paper, standard polarimetric calibration techniques could
be applied on a beambybeam basis to achieve lower instru
mental polarization. To avoid timeconsuming observations
of multiple calibrator sources or long observations of a single
polarized source for each formed beam, methods for reducing
the number of required telescope pointings required to calibrate
all formed beams are needed. Beam crosspolarization response
patterns and the temporal stability of formed beam polarization
responses are also of interest.
(89)
In view of (3) and (55), the sensitivity of any beam in the beam
pair subspace is therefore bounded by the largest and smallest
values of
(90)
Using the invariance of the trace with respect to the ordering of
a product of two matrices, this can be expressed as
(91)
Using (16),
becomes
(92)
This leads directly to the bound (56). Because of the properties
of the trace, replacing
with the calibrated beam pair
does not change the value of the trace:
(93)
which shows that the trace of
to polarimetric calibration of the beam pair, and the bound in
(55) is a polarimetric calibrationindependent measure of the
intrinsic beam pair sensitivity.
is invariant with respect
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led to the definition of APERTIF—a PAF system that is being developed
Proof
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[20] B.D.Jeffs,K.F.Warnick,J.Landon,J.Waldron,D.Jones,J.R.Fisher,
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www.skatelescope.org 2010
Karl F. Warnick (SM’04) received the B.S. degree
(magna cum laude) with University Honors and
the Ph.D. degree from Brigham Young University
(BYU), Provo, UT, in 1994 and 1997, respectively.
From 1998 to 2000, he was a Postdoctoral Re
search Associate and Visiting Assistant Professor
in the Center for Computational Electromagnetics
at the University of Illinois at UrbanaChampaign.
Since 2000, he has been a faculty member in the
Department of Electrical and Computer Engineering
at BYU, where he is currently a Professor. In 2005
and 2007, he was a Visiting Professor at the Technische Universität München,
Germany. He has published many scientific articles and conference papers
on electromagnetic theory, numerical methods, remote sensing, antenna
applications, phased arrays, biomedical devices, and inverse scattering, and
is the coauthor of the book Problem Solving in Electromagnetics, Microwave
Circuits, and Antenna Design for Communications Engineering (Artech House,
2006), and the author of Numerical Analysis for Electromagnetic Integral
Equations (Artech House, 2008) and Numerical Methods for Engineering: An
Introduction Using MATLAB and Computational Electromagnetics Examples
(Scitech, 2010).
Dr. Warnick was a recipient of the National Science Foundation Graduate
Research Fellowship, Outstanding Faculty Member award for Electrical and
Computer Engineering (2005), and the BYU Young Scholar Award (2007). He
has served the Antennas and Propagation Society as a member of the Education
Committee and as a session chair and special session organizer for the Interna
tional Symposium on Antennas and Propagation and other meetings affiliated
with the Society. He is a frequent reviewer for the IEEE TRANSACTIONS ON
ANTENNAS AND PROPAGATION and Antennas and Wireless Propagation Letters.
HehasbeenamemberoftheTechnicalProgramCommitteefortheInternational
Symposium on Antennas and Propagation for several years and served as Tech
nical Program CoChair for the Symposium in 2007.
Marianna V. Ivashina received the Ph.D. degree in
electrical engineering from the Sevastopol National
Technical University (SNTU), Ukraine, in 2000.
From 2001 to 2004, she was a Postdoctoral
Researcher and, from 2004 to 2010, an Antenna
System Scientist at The Netherlands Institute for
Radio Astronomy (ASTRON). During this period,
she carried out research on an innovative phased
array feed (PAF) technology for a newgeneration
radio telescope, known as the square kilometer array
(SKA). The results of these early PAF projects have
at ASTRON to replace the current horn feeds in the Westerbork Synthesis
Radio Telescope (WSRT). She was involved in the development of APERTIF
during 2008–2010 and acted as an external reviewer at the Preliminary Design
Review of the Australian SKA Pathfinder (ASKAP) in 2009. In 2002, she
also stayed as a Visiting Scientist with the European Space Agency (ESA),
ESTEC, in the Netherlands, where she studied multiplebeam array feeds for
the satellite telecommunication system Large Deployable Antenna (LDA). She
is currently a Senior Scientist at the Department of Earth and Space Sciences
(Chalmers University of Technology). Her interests are wideband receiving
arrays, antenna system modeling techniques, receiver noise characterization,
signal processing for phased arrays, and radio astronomy.
Dr. Ivashina received the URSI Young Scientists Award for GA URSI,
Toronto, Canada (1999), APS/IEEE Travel Grant, Davos, Switzerland (2000),
the 2nd Best Paper Award (’Best team contribution) at the ESA Antenna
Workshop (2008) and the International Qualification Fellowship of the VIN
NOVA—Marie Curie Actions Program (2009) and The VR project grant of the
Swedish Research Center (2010).
Stefan J. Wijnholds (S’06–M’10) was born in The
Netherlands in 1978. He received the M.Sc. degree
in astronomy and the M.Eng. degree applied physics
(both cum laude) from the University of Groningen,
in 2003, and the Ph.D. degree (cum laude) from the
Delft University of Technology, Delft, The Nether
lands, in 2010.
After graduation, he joined the R&D Department,
ASTRON, The Netherlands Institute for Radio
Astronomy, Dwingeloo, where he works with the
system design and integration group on the develop
ment of the next generation of radio telescopes. His research interests lie in the
area of array signal processing, specifically calibration and imaging.
Rob Maaskant was born in the Netherlands on
April, 14th, 1978. He received the M.Sc. (cum
laude) and Ph.D. (cum laude, best dissertation of the
Electrical Engineering Department) degrees in elec
trical engineering from the Eindhoven University of
Technology, in 2003 and 2010, respectively.
From 2003–2010 he was employed as an Antenna
Research Scientist at the Netherlands Institute of
Radio Astronomy (ASTRON). He is currently a
Postdoctoral Researcher at the Chalmers University
of Technology, Sweden. His current research interest
is in the field of receiving antennas for lownoise applications, metamaterial
based waveguides, and computational electromagnetics to solve these types of
problems.
Dr. Maaskant received a Rubicon Postdoctoral Fellowship from the Nether
lands Organization for Scientific Research (NWO), in 2010. He won the 2nd
Best Paper Prize (Best Team Contribution) at the 2008 ESA/ESTEC workshop,
Noordwijk, and is the primary author of the CAESAR software; an advanced
integralequation based solver for the analysis of large antenna array systems.