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IEEE

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 rank-two 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 dual-po-

larized array is developed and related to standard polarimetric

antenna figures of merit, and the ideal polarimetrically calibrated,

maximum-sensitivity beamforming solution for a dual-polarized

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 single-source polarimetric calibration

method is developed.The modeled instrumental polarization error

for a dipole phased array feed with the practical beamformer

solution and single-source 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 (e-mail:

warnick@ee.byu.edu).

M.V.IvashinaiswiththeDepartmentofEarthandSpaceSciences,Chalmers

University of Technology, S-41296 Gothenburg, Sweden (e-mail: marianna.

ivashina@chalmers.se).

S. J. Wijnholds is with the Netherlands Institute for Radio Astronomy (AS-

TRON), NL-7991 PD, Dwingeloo, The Netherlands (e-mail: wijnholds@as-

tron.nl).

R. Maaskant is with the Department of Signals and Systems, Chalmers

University of Technology, S-41296 Gothenburg, Sweden (e-mail: 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 single-source 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, cross-polarization 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 dual-polarized 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

0018-926X/$26.00 © 2011 IEEE

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IEEE

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

partially-polarized 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. Aradiopolarimetercomprisedofadual-polarizedactivelybeamformed

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 max-SNR 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 three-dimensional 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 beam-steering 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 time-average

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, time-harmonic, linearly polarized

waves

1In the less general bi-scalar method,

and are only associated to the - and -oriented antenna elements,

respectively.

andare weight vectors of size

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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 non-ideal 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]:

Cross-polarization discrimination (XPD) is the ratio of the

power level at the output of a receiving antenna, nominally

co-polarized with the transmitting antenna, to the output of a

receiving antenna of the same gain but nominally orthogonally

polarized to the transmitting antenna.

Cross-polarization 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 cross-polarized incident fields

the two beamformer output voltages

andis described by

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Cross-polarizationisolationisdefinedastheratioofpowers

received at the same beamformer output, 1 or 2, due to orthog-

onally-polarized 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 beam-dependent polarization leakage, we ex-

tend the standard definitions in Section III.A as follows:

Cross-polarization 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 cross-polarization 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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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 cross-polar-

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 off-diag-

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 maximum-SNR 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, source-independent 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 noise-free 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.Thesignalcorrelationmatrixforatime-har-

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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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 non-polarimetricallycalibratedbeam

pairshaving imperfectpolarizationdiscrimination.Therefore,it

is important to find an approach for obtaining best possible po-

larimetric beam pairs using the single-source calibration data

required for non-polarimetric 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 calibration-independence 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 non-colinear 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 on-source measurement and off-source 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 sample-estimated 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 rank-two general-

ization of the classical maximum-SNR 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 dual-polarized 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 rank-one 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

maximum-SNR eigenvector beamformer weight vectors

(77a)

(77b)

This is the eigenvector max-SNR 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 cross-polarization 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 max-SNR 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 max-SNR beamformer:

(82)

Since

two nonzero eigenvalues. The corresponding principle eigen-

values provide a maximum-SNR 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 pre-whiten the array

output signals and using the eigenvector CFM method with the

pre-whitened 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-

tiple-sourcecalibrationmethodsforphasedarrayscanbefound,

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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IEEE

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 off-boresight, 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 single-source calibrated max-SNR 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 single-source polarimetric calibration pro-

cedure is essentially equivalent to a polarimeter that uses the

nominallyorthogonallypolarizedoutputsofastandarddual-po-

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 single-source

polarimetric calibration method.

V. NUMERICAL RESULTS

The polarimetric figures of merit and beamforming algo-

rithms will be illustrated for a 19

of thin, lossless,

- and-polarized crossed dipoles spaced

apart and backed by a ground plane. The phased array

feeds a 20-meter 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 max-SNR and CFM algorithms described

above, and polarimetrically calibrated using the approximate

single-source 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 open-cir-

and

A. Perfectly Aligned Elements

The first study is a comparison of the sensitivity of the eigen-

vector max-SNR and CFM beam pairs to the optimal solution

(Fig.2).Thesingle-sourcecalibrationprocedureofSectionIV.C

is used with both beamformers. The sensitivity of the max-SNR

beam pair calibrated with the single-source procedure is nearly

indistinguishable from that of the optimal solution. For the po-

larimetrically uncalibrated case, the max-SNR 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 cross-polarizationfigures 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 single-source

calibrated CFM beam pair sensitivity also lies within its corre-

sponding bound (55), but the overall sensitivity is lower than

that of max-SNR.

Cross-polarization 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 max-SNR and CFM beam pairs

can be exactly calibrated without making use of the biscalar

transformation (87). For off-boresight 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 cross-polarization 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 single-pixel 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 cross-polarization 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 off-boresight

beams, it is apparent that an additional polarimetric calibration

beyond the approximate single-source 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 zero-mean 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

Page 12

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 Hamaker-Bregman-Sault 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 dual-polarized beamforming

array receiver has been used to develop polarization figures of

merit and a theory of ideal polarimetric beamforming. An op-

timal polarimetrically calibrated maximum-SNR 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 eigenvector-based maximum-SNR 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 maximum-SNR

beam pair is not polarimetrically calibrated, and additional

correction through estimation of the beam pair Jones matrix is

required. An approximate single-source 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 single-source 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 single-source method used in

this paper, standard polarimetric calibration techniques could

be applied on a beam-by-beam basis to achieve lower instru-

mental polarization. To avoid time-consuming 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 cross-polarization 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 calibration-independent 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

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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 Urbana-Champaign.

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 Co-Chair 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 new-generation

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 multiple-beam 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 low-noise applications, meta-material

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

integral-equation based solver for the analysis of large antenna array systems.