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Wijnholds, S.J., Ivashina, M., Maaskant, R., Warnick, K.F. (2012) Polarimetry with phased
array antennas: sensitivity and polarimteric performance using unpolarized sources for
calibration. IEEE Transactions on Antennas and Propagation, vol. 60, no. 10, pp. 4688-4698.
http://dx.doi.org/10.1109/TAP.2012.2207340
VERSIONJUNE1,2012 1
Polarimetry with Phased Array Antennas:
Sensitivity and Polarimetric Performance Using
Unpolarized Sources for Calibration
Stefan J. Wijnholds Senior Member, IEEE, Marianna V. Ivashina Member, IEEE, Rob Maaskant Member, IEEE,
Karl F. Warnick Senior Member, IEEE
Abstract—Polarimetric phased arrays require a calibration upgrade of the Westerbork Synthesis Radio Telescope in The
method that allows the system to measure the polarization state Netherlands[5],theAustralianSKAPathfinder(ASKAP,PAF)
of the received signals. In this paper, we assess the polarimetric
intheWesternAustralianDesert[6]andtheLongWavelength
performance of two commonly used calibration methods that
Array (LWA, AA) in the US [7].
exploitunpolarizedcalibrationsources.Thefirstmethodobtains
apolarimetricallycalibratedbeamformingsolutionfromthetwo The goals of these instruments for future radio astronomy
dominanteigenvectorsofthemeasuredsignalcovariancematrix. research require high system sensitivity for detecting weak
We demonstrate that this method is sensitivity equivalent to the radio signals and accurate reconstruction of the polarization
theoretical optimal method, but suffers from an ambiguity that
state of these signals. A polarimetric antenna system, capable
hastoberesolvedbyadditionalmeasurementson(partially)po-
of sampling the incident field by two orthogonally polarized
larizedsourcesorbyexploitingtheintrinsicpolarimetricquality
oftheantennasystem.Theeasy-to-implementbi-scalarapproach receptors, can fully reconstruct the polarization state of the
assumesthat thefeedsystem consistsof twosetsof orthogonally field. This reconstruction can be done by inverting a 2 2
×
orientedantennaelements,eachassociatedwithonepolarization. transfer matrix that relates the two output signals of the
We assess its sensitivity and polarimetric performance over a
receiversystemtothepolarizationstateofthefieldreceivedby
wide field-of-view (FoV) using simulations of a phased array
the system[8].Inpolarimetricsystems, calibrationshouldnot
feed system for the Westerbork Synthesis Radio Telescope. Our
results indicate that the sensitivity loss can be limited to 4.5% only compensate the gain differences between the receiving
and that the polarimetric performance over the FoV is close elements to provide maximum sensitivity at the beamformer
to the best achievable performance. The latter implies that the outputs, but also ensure that the covariance between the two
intrinsic polarimetric quality of the antennas remains a crucial
beamformeroutputsignalsallowsproperreconstructionofthe
factor despite the development of novel polarimetric calibration
polarizationstateoftheincidentfield.Inarecentpaper[9],the
methods.
authorshavedevelopedasystemmodelandusedittorelatethe
IndexTerms—phasedarrayantennas,polarimetry,calibration,
astronomicalperformancecriteriatostandardIEEEdefinitions
beamforming, far-field radiation pattern
forpolarimetricantennasandtofindabeamformingalgorithm
that simultaneously optimizes for minimum system noise and
I. INTRODUCTION polarimetric accuracy. In this paper, we use this framework
Polarimetric phased array antenna systems are commonly to assess the polarimetric performance and sensitivity of two
used in imaging radar applications [1], [2], but are relatively commonly used calibration methods that exploit unpolarized
new in the field of radio astronomy. The radio astronomical sources. This is particularly relevant, because the majority
community is developing the Square Kilometre array (SKA) of extragalactic radio sources with a continuum spectrum,
[3], a future radio telescope that is envisaged to be an order which are typically used for calibration of radio telescopes,
of magnitude more sensitive than present-day instruments. are weakly polarized or unpolarized and hence calibration
Phasedarrayantennaswillplayacrucialroleeitherasaperture methods based on polarized reference sources are usually not
array (AA) or as phased array feed (PAF) for reflector anten- applicable.
nas.Severalprecursorandpathfindersystemsusingphasedar- A recently proposed method exploits the two dominant
rayantennasarecurrentlyinuseorbeingdeveloped.Examples eigenvectors of the signal covariance matrix measured on an
of such instruments are the Low Frequency Array (LOFAR, unpolarizedsourcewithadual-polarizedarray[10],whichwe
AA) inEurope[4],theaperturetile-in-focus(APERTIF,PAF) will refer to as eigenvector method. This method may cause
a large unknown change of the orientation of the polarization
S.J. Wijnholds is with the Netherlands Institute for Radio Astronomy
axesoftheinstrument,aswedemonstrateinthispaper,andis
(ASTRON),P.O.Box2,NL-7990AA,Dwingeloo, TheNetherlands, Email:
[email protected] therefore unsuitable for application in actual systems without
M.V. Ivashina is with Chalmers University of Technology, SE-412 96 furtherpolarimetriccorrections.Thesepolarimetriccorrections
Gothenburg,SwedenandwiththeNetherlandsInstituteforRadioAstronomy
eitherrequireadditionalmeasurementsonpolarizedsourcesor
(ASTRON),Email:[email protected]
R. Maaskant is with Chalmers University of Technology, Email: relying on the intrinsic polarimetric quality of the system.
[email protected] A second commonly used method exploits the fact that
K.F.WarnickiswiththeDepartmentofElectricalandComputerEngineer-
feed systems usually consist of two sets of orthogonally
ing,BrighamYoungUniversity,459ClydeBuilding,Provo,UT84602,USA,
Email:[email protected] oriented antenna elements. For such systems, it is common
VERSIONJUNE1,2012 2
E(r,t)=Eu(r,t)uˆ+Ev(r,t)vˆ The antenna output signals are amplified to form the N-
element output voltage vector v and are subsequently com-
bined into the beamformer output voltages v and v using
1 2
the N 1 beamformer weight vectors w and w .
1 2
×
At a fixed position r, E (r,t) and E (r,t) are complex
x-orientedantennaelements y-orientedantennaelements u v
random processes in the phasor or complex baseband repre-
sentation. The polarization state of the field is determined by
the correlation matrix of the two field components, which is
1 2 N
Σ= σuu σuv = |Eu|2 hEuEv∗i , (2)
··· ··· σvu σvv  DE∗E E E 2 
(cid:20) (cid:21) h u vi | v|
v  D E 
where denotes the expected value. For an unpolarized
h·i
source, we have Σ=I.
We will model the phased array antenna signal output
in terms of the voltage response vectors of the array, v
w w u
1 2
and v , containing voltages at the array receiver outputs
v
before beamforming induced by unit intensity, linearly po-
larized waves having their polarization aligned with u and
v1=wH1v v2=w2Hv v respectively. Theoretically, these voltages can be measured
using a reference source producingtwo signals with perfectly
b
Fig. 1. A radio polarimeter comprised of a dual-polarized actively beam-
orthogonal polarizations.
formedreceiving antenna array. b
For an arbitrary polarized wave, the array signal voltage
response vector can be formulated in terms of v and v as
u v
practiceto calibratethe two sets independentlyandto applya
v =E v +E v . (3)
singlepolarizationcorrectionattheend.Thisbi-scalarmethod s u u v v
imposesadditionalconstraintsonthesystemdesign,including The array output signal covariance matrix is
manufacturing tolerances. We assess the stringency of these
constraints by comparison with the theoretically optimal per- Rs = vsvsH =VΣVH, (4)
formance using theoretical analysis and simulations.
where we have introduc(cid:10)ed V (cid:11)= [v ,v ]. Assuming that the
u v
This paper is organizedas follows. We start by introducing
phased array system noise can be characterized by a noise
the system model and performancemetrics used to assess the
covariance matrix R , the covariance matrix of the array
n
performance of the antenna system. In Sec. III we introduce
output voltage v can be described as
the calibration methods and discuss their theoretical perfor-
mance. We then introduce a simple analytic dipole model in R= vvH =Rs+Rn =VΣVH +Rn. (5)
Sec.IVtodemonstratetheimpactoftheunitaryambiguityin-
The noise cova(cid:10)riance(cid:11)matrix can be determined using an off-
troducedwhen calibratingon anunpolarizedsourceandshow
source measurement on an empty part of the sky, i.e., a part
how the polarimetric quality of the methods exploiting the
of the sky without significant source structure. The noise
intrinsic polarimetric quality of the antenna system depends
covariancematrix includesfactorssuch as noise couplingand
on the orthogonality between the feed systems. In Sec. V,
spillover noise.
we present simulations for an actual PAF system to assess
The beamformeroutputcovariancematrix is obtained from
the impact of using a bi-scalar instead of a full-polarimetric
the beamformer output voltages, v = wHv and v =wHv,
approachonthe sensitivityandpolarimetricperformanceover 1 1 2 2
by
the field-of-view (FoV). We show that the sensitivity loss can
btheelFimoViteidstcoom4.p5a%rabwlehitloetthheebpeosltaaricmhieetvriacblpeerpfeorrfmoramncaencoev.er Ro =*(cid:20) vv12 (cid:21)(cid:20) vv12 (cid:21)H+= Dh|vv1∗1v|22iE h|vv12v|2∗2i . (6)
II. THEORY  D E 
Intermsofthesignalandnoisecovariancematricesintroduced
A. System model
earlier, this can be written as
Figure 1 shows an N-element polarimetric phased array.
wHRw wHRw
Theantennasystemisassumedtobeilluminatedbyapartially Ro = w1HRw1 w1HRw2
polarizedsource.Theelectricfieldintensityvectorradiatedby (cid:20) 2 1 2 2 (cid:21)
H
such a source is = w1 w2 R w1 w2
= WH VΣVH +R W
E(r,t)=E (r,t)u+E (r,t)v, (1) (cid:2) (cid:3) (cid:2)n (cid:3)
u v
= R +R , (7)
where u and v are orthogonal unit vectors according to o,s (cid:0) o,n (cid:1)
a certain specific Ludwig’s defibnition [11] brelative to the where R and R are the beamformer output covariance
o,s o,n
coordinate system of the array. matrices due to the signal and the noise, respectively.
b b
VERSIONJUNE1,2012 3
provides a system level view of the phased array antenna
J J−1 system. The polarimetric properties of the source are defined
by the source covariance matrix Σ. This is transformed to
instrument pol. correct.
Σ Ro Σ′ the beamformer output covariance matrix Ro (equal to Ro,s
in the noise free case), that is used for further processing
Fig.2. Asystemlevel view ofaradiopolarimeter. ′
to produce a reconstructed source covariance matrix Σ that
should ideally be proportional to Σ. An ideal system does
B. Useful concepts not require polarimetric correction, since it has J=I, where
I denotes the identity matrix, i.e., it leaves the covariance
1) Sensitivity equivalence: The combined sensitivity, ex-
matrix of the input signal unchanged and does not introduce
pressed as the ratio of effective aperture area A and system
e so-called instrumental polarization. Since J = WHV, the
temperature T , of the two beamformer outputs, the beam
sys instrumentalpolarizationintroducedbytheantennasandother
pair sensitivity, is determined by a linear combination of the
analogelectronicscanbecompensatedforin thebeamformer.
two output signals. This can be obtained from [8]
3) Max-SNR beamforming: The highest signal-to-noise ra-
Ae kbB kbB aHWHRsWa tio (SNR) for the u- and v-polarizedsignalis achievedby the
= SNR= , (8)
T S S aHWHR Wa max-SNR beamformer [12]
sys sig sig n
where a=[a ,a ]T is an arbitrary vector other than the null W =R−1V. (12)
1 2 SNR n
vector, S is the power flux density of the received signal
sig
Although this beamformer does not calibrate the polarimetric
expresedin W/m2, k is the Boltzmann constantand B is the
b
responseofthearray,itdoesprovidethemaximumsensitivity.
bandwidth.
4) Optimalbeamforming: Ifweknowthevoltageresponse
As shown in [9], the beam pair sensitivity is bounded by
vectors to two perfectly orthogonally polarized signals, v
u
k B A k B
b λmin e b λmax, (9) and vv, and the noise covariance matrix, Rn, we can derive
Ssig ≤ Tsys ≤ Ssig optimalweightsforthebeamformerthatensuresminimization
where λ and λ are the smallest and largest eigenvalue of the noise in the measurementand perfectreconstructionof
min max
of thepolarizationpropertiesofthesource.Thisoptimizationcan
C=VHW WHR W −1WHV. (10) be formulated as the constrained minimization problem
n
By replacing W with W(cid:0)′ = WA w(cid:1)here A is an arbitrary Wopt =argmin tr WHRnW subject to WHV=I.
W
invertible 2 2 matrix, it is easily seen that C is independent (13)
× (cid:0) (cid:1)
of a linear transformationof the beam subspace and therefore It can be shownthat the solution to this optimizationproblem
ofthepolarimetriccalibration.Wewillrefertobeampairsthat is given by [9]
are related by such a transformation as sensitivity equivalent
beam pairs. This transformation signifies that all sensitivity Wopt =R−n1V VHR−n1V −1. (14)
equivalent beam pairs lie within the same two dimensional
This pair of beamformer we(cid:0)ight vectors(cid:1)minimizes the re-
subspace of the N-dimensional space of complex valued N-
sponsetothesystemnoisewhileconstrainedtobepolarimetri-
elementvectors.Therearemanypossibleranktwosubspaces,
callycalibrated.Wewillrefertothiscalibrationmethodasthe
but only one that includes the maximum sensitivity beam- optimalmethod.Thefactor VHR−1V −1 canbeinterpreted
former. Within each two dimensional subspace, there is one n
as a Jones matrix that applies a polarimetric correction to
unique beam pair that is polarimetrically calibrated. (cid:0) (cid:1)
the max-SNR beamformer weights. If the Jones matrix is
Transformationsappliedafterbeamformingnecessarilystay
invertible, an assumption that should hold for polarimetric
within one two dimensional subspace. Sensitivity equivalence
arrays,the optimalbeamformeris sensitivity equivalentto the
implies that the beamformer weights can be described as
max-SNR beamformer. Hence, the optimal and the max-SNR
WJ ,whereJ isa2 2matrixdescribinga polarimetric
cor cor
× beamformer weigths span the same subspace and within this
correction.Ifabeamformercanbeformulatedinthisform,the
subspace, the optimal beamformer picks the weight vectors
polarimetriccorrectioncanbedoneafterbeamformingwithout
that provide the optimal polarimetric response. The same
affecting the sensitivity bounds of the beamformer given by
responsecanbeachievedwiththemax-SNRbeamformeronly
(9).
with an additional polarimetric correction after beamforming.
2) Jones matrices: In the noise free case or with noise
5) XPDandXPI: FollowingstandardIEEEdefinitions,the
estimated and subtracted, (7) reduces to
cross-polarizationdiscrimination(XPD)andcross-polarization
R =WHVΣVHW=JΣJH, (11) isolation (XPI) can be expressed in terms of Jones matrix
o
elements as [13]
wherewehaveintroducedthe2 2JonesmatrixJ=WHV.
This Jones matrix represents th×e transfer function of the in- J 2 J 2
11 22
XPD = | | XPD = | | (15a)
strumentincludingantennas,receiverchainsandbeamforming u J 2 v J 2
21 12
scheme, that transforms the two input voltages adhering to | | | |
2 2
J J
a suitable polarimetric definition into two output voltages, XPI = | 11| XPI = | 22| . (15b)
possibly adhering to another polarimetric definition. Figure 2 u J12 2 v J21 2
| | | |
VERSIONJUNE1,2012 4
6) Intrinsic cross-polarization ratio: Since the goal of To compare this beamformer with the optimal solution
polarimetric phased arrays is to reconstruct the polarimetric presented in the previous section, we note that the voltage
properties of the source signal, preservation of these polari- response vectors v and v must span the same subspace as
u v
metricpropertiesbytheJonesmatrixJoftheinstrumentneed the eigenvectors v and v . This implies that
1 2
not be a design goal by itself as long as the source coherency
can be reconstructed by a polarimetric correction J−1. This V=VeigJeig, (19)
idealedtothedefinitionoftheintrinsiccross-polarizationratio wherethe2 2JonesmatrixJ describesthetransformation
eig
×
(IXR). The IXR provides a measure for the reconstructability from v and v to v and v . The weights obtained by the
1 2 u v
of Σ (invertibility of J) and is defined as [13] eigenvector method can thus be described as
κ(J)+1 2 W =R−1VJ−1. (20)
IXR= , (16) eig n eig
κ(J) 1
(cid:18) − (cid:19) Substitution in (14) gives
where κ(J) denotes the condition number of J.
TheIXRprovidesanupperlimitontherelativeerrorinthe Wopt = R−n1VeigJeig VHR−n1VeigJeig −1
reconstructed Stokes vector S that is given by [13] = Weig VHW(cid:0)eig −1 (cid:1) (21)
∆Ss 4√IXR ∆M ∆So,s This shows that the two(cid:0)beamform(cid:1)ers are sensitivity equiv-
k k / 1+ + k k + k k ,
kSsk 1+IXR ···!(cid:18) kMk kSo,sk (cid:19) alent. This implies that one beamformer can be transformed
(17) into the other by an additional polarimetric correction after
where denotes the Euclidian norm, ∆M denotes the beamforming.
k · k k k
calibration error in the Mueller matrix and ∆S denotes Unfortunately, a single measurement on an unpolarized
k k
the measurement error in the Stokes vector. Note that the source does not provide sufficient information for full polari-
term involving the IXR describes an increase of the relative metric calibration. This can be demonstrated by substitution
Stokeserrorcomparedtoasystemwithperfectlyorthogonally of (19) in (4) with Σ=I. This gives
polarized feeds. For example, if the noise on the observation
R =VIVH =V J JH VH . (22)
causes a relative error (uncertainty) on the measured Stokes s eig eig eig eig
vector of 1% and the relative error in the instrument model If we replace J by J′ = J U, where U is an arbitrary
eig eig eig
(due to, e.g., calibration errors) is 1% as well, such a system unitary matrix, we get
would have about 1.4% relative error in the reconstructed
R = V J′ J′HVH
Stokes vector. An IXR of 25 dB may (note that (17) gives s eig eig eig eig
an upper bound) cause a 22% increase in this error, which = VeigJeigUUHJHeigVeHig
may therefore increase to 1.7%. = V J JH VH , (23)
eig eig eig eig
i.e.,wegetexactlythesamesolution.Thisshowsthatourmea-
III. CALIBRATION USING AN UNPOLARIZED SOURCE
surement cannot discriminate between J and J′ , which
The majority of continuum extragalactic sources, which eig eig
implies that the relation between V and V given by (19)
are typically used for calibration, are weakly polarized or eig
has a unitary ambiguity.
unpolarized.Calibration of radio telescopesthereforerequires
This polarimetric calibration problem has been studied in
methods that exploit unpolarized reference sources. In this
detail by Hamaker [14]. He found that the physical signifi-
section, we discuss three proposed methods. We will see that
cance of the aforementioned unitary ambiguity is described
wecannotdoafullpolarimetriccalibrationofthesystemonly
by polconversion and polrotation. Polconversion is the effect
usinganunpolarizedsource.Twooftheproposedmethodsrely
that part of the unpolarized power is detected as (“converted
on the intrinsic polarimetric characteristics of the instrument.
to”) polarized power. Polrotation refers to a rotation or axial
ratio change of the polarizationellips. This unitary ambiguity
A. Eigenvector method
can be resolvedby additionalmeasurementson two distinctly
For a polarimetric array, neglecting estimation error, the polarizedsourcesorbyimposingadditionalconstraintsonthe
signal covariance matrix R measured on an unpolarized instrumental Jones matrix as discussed in [14], [15]. Relying
s
sourcehasrank2.Ifthesystemconsistsoftwosetsoforthogo- on the intrinsic polarimetric purity of the receiving elements
nally polarized feeds, there is hardly any correlation between as discussed in the next sections is an example of such an
the receiving elements in the two sets when measuring an additional constraint. In Sec. IV, we demonstrate the impact
unpolarizedsource.Intuitively.the two dominanteigenvectors of ignoring this ambiguity using a simple dipole model.
will each be associated with one set of feeds. This naturally
suggeststhe use of the two principaleigenvectors,v and v ,
1 2 B. Eigenvectormethodwithbi-scalarapproximatecalibration
to form the maximum SNR eigenvector beamformer weight
Since the eigenvector method produces a pair of beam-
vectors [10]
W =R−1V , (18) former output signals that are only polarimetrically correct
eig n eig up to a unitary matrix, we would like to find a correction to
where V = [v ,v ]. We will refer to this approach as the ensure well-defined polarimetric characteristics at the beam-
eig 1 2
eigenvector method. former output. This can be achieved by assuming that the
VERSIONJUNE1,2012 5
system consists of two sets of feeds that have an orthogonal implications of ignoring this term, we note that, in general,
polarimetricresponse as proposedin [9]. For analysis of such the voltage output of the beamformer is given by
a system, it is convenient to partition the signal covariance
wH wH
matrix as WHv = u,u u,v v
R = Rs,uu Rs,uv . (24) (cid:20) wvH,u wvH,v (cid:21)
s R R wH 0 0 wH
(cid:20) s,vu s,vv (cid:21) = u,u + u,v v,
0 wH wH 0
ThesubmatricesareN/2 N/2matricesandwehaveassumed (cid:18)(cid:20) v,v (cid:21) (cid:20) v,u (cid:21)(cid:19)
× (29)
that the first N/2 elements are optimally matched to u-
polarizedsignalswhilethesecondN/2elementsareoptimally where w denotes the weight vector assigned to the u-
u,v
matched to v-polarized signals. The matrices polarizedelementsfor the beamformeroutputassociated with
v-polarizedsignals and similar definitionsfor w , w and
R 0 0 0 u,u v,u
Rs,u = s0,uu 0 and Rs,v = 0 R , (25) wv,v.
s,vv
(cid:20) (cid:21) (cid:20) (cid:21) Sincetheweightsw needtobeappliedtotheu-polarized
u,v
have rank one with principal eigenvectors vu and vv respec- elementsalthoughtheyareassociatedwithv-polarizedsignals,
tively.ThesevectorscanbeusedtofindtheapproximateJones we can exploit the ability of typical PAF systems to form a
matrix e e cross-polarization beam for the signals from the u-polarized
J=WH V, (26) elementsandsimilarlyforthev-polarizedelements[16].Once
eig
both co- and cross-polarized beams are formed, we can add
whereV=[vu,vv].ThiesJonesmaetrixcanbeusedtocalibrate the appropriate beam signals to reconstruct the output signals
the beam pair to obtain of a full polarimetric beamformer. This approach doubles the
e e e W =W J−H. (27) number of beams to be formed. Since the total bandwidth of
approx eig the beamformer (number of beams times the bandwidth per
Since v and v are only determined up to a scale factor, beam) is limited by the digital hardware, we can only apply
u v e
further normalization may be required as discussed in [9]. this scheme by sacrificing half the bandwidth per beam to
allow for twice as many beams (the co- and cross-polarized
e e
beams).This reducesthe sensitivity for continuumsourcesby
C. Bi-scalar method a factor √2 and the survey speed by a factor 2. This may not
Mostastronomicalphasedarraysystemsconsistoftwo sets seemattractive,butsomeobservations,suchasinterferometric
offeeds,eachoptimallymatchedtoasinglepolarization.This observations with other telescope systems, may not need the
can be exploited by calibrating and beamforming both feed fullbandwidthofthesysteminwhichcasethisschememaybe
sets separately,whichsimplifiesthedesignofthe systemcon- appliedwithoutitsdrawbacks.Thisisanimportantconclusion
siderably,sincethisapproachrequirestwoidenticalprocessing for instrumentswith a bi-scalar design that can form multiple
systems that each have to deal with only N/2 signals, instead beams within the FoV like APERTIF. However, calculation
of a single system with N inputs, thereby simplifying signal of these weights requires knowledge of the full signal and
routing and saving half the compute power for correlation of noise covariance matrices, which implies that these systems
the input signals. This is referred to as a bi-scalar approach. still need to provide a facility that can correlate the signals
Since this method treats both sets separately, it ignores the from any feed pair in the system for calibration of the array,
cross-terms between the two sets of elements in the signal although, for example, with a reduced bandwidth.
andnoisecovariancematrices.Forouranalysis,itis therefore
convenientto partitionthese matricesas indicatedin (24). We IV. DIPOLE MODELANALYSIS
can then formulate the bi-scalar beamformer as
In this section, we present a simple dipole model and
Wbi−s =(cid:20) R−n0,1uu R−n0,1vv (cid:21)[vu,vv]. (28) uamsebiigtutiotypdriosvciudsesesdoimnethienpsirgehvtioiunsstheectiiomnpoanctthoefptohlearuimniettarriyc
performanceof the instrument.We also use this dipole model
If the two feed sets are perfectly matechede to u- and v-
to assess the impact of non-orthogonality in the polarimetric
polarized signals respectively such that R = R = 0,
s,uv s,vu response of the receiving elements for the methods that rely
the bi-scalar approach gives the same result as the optimal
on the intrinsic orthogonality of the feed sets.
method or the eigenvector method and is therefore sensitivity
equivalent. Otherwise, the bi-scalar beamformer introduces a
beamforming error due to neglecting the cross-polarization A. Description of the dipole model
response of the elements. The impact on sensitivity depends
Forthissimplemodel,weassumethateachantennaconsists
ontheantennaresponses.Using(9)wecandeterminetheloss
of two co-located ideal dipoles in the (x,y)-plane making an
in sensitivity for a specific antenna system.
angle φ with each other. The open circuit responses of the
A full polarimetric beamformer combines the u-polarized
dipoles for broadside incidence are then proportional to
power picked up as co-polarization in the u-elements and as
cross-polarization of the v-elements. The latter contribution 1 0
V , (30)
is ignored by a bi-scalar beamformer. To get a feel for the oc ∝ γcosφ γsinφ
(cid:20) (cid:21)
VERSIONJUNE1,2012 6
1.2 1.2
V (Sim) Rn (Sim)
11 11
V (Sim) Rn (Sim)
1 12 1 12
V (Sim) Rn (Sim)
21 21
0.8 V22 (Sim) 0.8 Rn22 (Sim)
V (Model) Rn (Model)
11 11
0.6 V (Model) 0.6 Rn (Model)
12 12
V V21 (Model) Rn Rn21 (Model)
0.4 V (Model) 0.4 Rn (Model)
22 22
0.2 0.2
0 0
−0.2 −0.2
0 10 20 30 40 50 60 70 80 90 0 10 20 30 40 50 60 70 80 90
φ [Deg] φ [Deg]
Fig. 3. Comparison between the elements of Voc predicted by an EM- Fig. 4. Comparison between the elements ofRn,oc predicted bythe EM-
simulation[17]oftwocrossedshortdipolesandthosegivenby(30)assuming simulation[17]oftwocrossedshortdipolesandthosegivenby(32)assuming
γ=1.ThecurveforthemodeledV22 overlaps withthesimulatedV22. γ=1and8kTRrad=1.Theresults forthetwooff-axis elements andthe
twoon-axis elements ofRn,oc areidentical, asexpected fromsymmetry.
where γ is the gain of the second dipole assuming that the
receive voltage of the first dipole is 1 V. The open circuit betweenthedipoles,weassumethatthedipoleshavethesame
signal covariance matrix for an unpolarized source is then gain, i.e., that γ = 1. Solving for the eigenvalues using the
proportional to characteristic polynomial, we find that
1 γcosφ λ =1 cosφ. (33)
Rs,oc =VocVoHc ∝ γcosφ γ2 . (31) 1,2 ±
(cid:20) (cid:21) We can now solve for the eigenvectors associated with each
UsingRn,oc =8kTRe ZA ,whereZAdenotestheantenna eigenvalue. It is straightforward to show that for φ = π/2
impedance matrix, we fin{d tha}t, to first order, mod π, 6
1 1 1
1 γcosφ V = √2 (34)
Rn,oc ∝8kTRrad γcosφ γ2 , (32) eig 2 (cid:20) 1 −1 (cid:21)
(cid:20) (cid:21)
and that V =I if φ=π/2 mod π. Although V shows
where R is the radiation resistance. In the loaded case, eig eig
rad
we have R = QR QH and R = QR QH, where a discontinuity at φ = π/2, we will see that the response
s s,oc n n,oc
Q=Z (Z +Z )−1 withZ denotingtheinputimpedance of the system remains continuouswhen using the eigenvector
R R A R
method. Althoughthis analysis was done for a single antenna
of the receiving network. We need expressions for the imagi-
consisting of two crossed dipoles, it can easily be shown that
nary part of the antenna impedance matrix and Z to obtain
R
the results also hold for an array of identical antennas where
expressions for R and R , which will be more tedious than
s n
the couplingbetween antennapairsis negligible.This implies
the expressions for their open circuited counterparts given
that the results obtained from the analysis of a single antenna
above. For our educational example, we therefore choose to
can also be applied to a full array.
use the open circuited case, i.e., to assume that the antenna
ports are connected to voltage amplifiers.
To verify this seemingly simple model, we did an EM- C. Impact of the unitary ambiguity
simulation [17] with two crossed short dipoles making an
In this section we will demonstratethe effectof the unitary
angle φ with length L = λ/4, width w = λ/640 and inter-
ambiguity by comparing the Jones matrix and beamformer
element distance ∆z = 2w. Figure 3 compares the open
output signal covariance matrix for the optimal method and
circuit voltages obtained from the EM-simulation with those
the eigenvector method. It is straightforward to show that for
describedbytheanalyticalmodelwhileFig.4makesasimilar
the optimal method
comparison for the elements of the noise covariance matrix.
For convenience, we have assumed 8kTRrad =1 and γ =1. Jopt =WoHptV=I (35)
Theseresultsshowthatouranalyticalmodelisratheraccurate,
such that
especially for the crucial range where φ is close to 90◦. The
R =J JH =I. (36)
differences are primarily due to the finite strip width of the o,s opt opt
numerical model. Inasimilarway,wecanshowthatfortheeigenvectormethod
1 1 1−cosφ
B. Eigenvalue decomposition J =WH V= sinφ ., (37)
Fortheanalysisoftheeigenvectormethod,wecomputethe eig eig √2" 1 −1+sicnoφsφ #
eigenvectors of the signal covariance matrix given by (31). where we assumed a proportionality constant in (32) equal
Since we are interested in the impact of non-orthogonality to unity. It is interesting to note that substituting φ = π/2
VERSIONJUNE1,2012 7
in (37) gives the same result as obtained for V = I,
eig 50
i.e., the discontinuity at φ = π/2 mod π found in the eigenvector
eigenvalue decomposition vanishes when the result is applied eigenvector with correction
in the eigenvector beamforming scheme. Using (37) we find 40 bi−scalar
that the beamformer output signal covariance matrix for the
eigenvector method is given by
2sin2(φ/2) 0 dB)30
Ro,s = 0 2cos2(φ/2) . (38) R (
(cid:20) (cid:21) X20
Equation (37) shows that if φ is close to 90◦, i.e., if the I
polarimetric responses of the dipoles are (almost) orthogonal,
the magnitudesof the entriesof the Jones matrix are (almost) 10
equal. This indicates that the two input polarizations aligned
with u and v are mixed to form two output signals that
0
are aligned with two other axes. However, if we look at the 60 65 70 75 80 85 90
beamformeroutputsignalcovariancematrixforanunpolarized φ (degrees)
b b
source, it seems that the system preserves the properties of
Fig. 5. IXR as function of the angle between the two dipoles for the
the source. This counter intuitive result can be explained by
eigenvectormethod,theeigenvectormethodwithbi-scalarcorrectionandthe
the unitary ambiguity, which works on the Jones matrix, but bi-scalar beamformer.
cancels itself in the beamformeroutputsignal covariancema-
trix,i.e.,wehavedevelopedabeamformingschemethatgives
calibration shows that the eigenvector methods are not able
perfect results for unpolarized sources, but gives erroneous
to produce a pair of beamformer output signals that is more
results in observations on polarized sources.
suitableforreconstructionofthepolarimetricpropertiesofthe
Itwillalsoyieldwrongresultsifthetwobeamformeroutput
input signal than the bi-scalar beamformer when the dipoles
signals are correlated to the two beamformer output signals
areclosetoorthogonal.Thisindicates,thatthesemethodsrely
of another antenna system with a well-defined polarization.
onthepolarimetricqualityoftheantennasystem.Anintuitive
This situation may occur in practice when multiple telescope
explanation for this result, is that all these methods use an
systems are linked together. Hence, it is important that the
unpolarized source for system calibration and thus have to
voltageresponseofthesystemdescribedbytheJonesmatrixis
relyontheintrinsicpolarizationqualityoftheantennasystem
welldefined.Thisargumentshowsthattheeigenvectormethod
to resolve the unitary ambiguity. Since most astronomical
is not suitable for use in an actual system unless appropriate
calibration sources are not or only weakly polarized, this
corrections are made to the beamformer output signals.
impliesthatawell-designedantennasystemisinvaluable,even
If we apply the bi-scalar approximate calibration discussed
given the material presented in this paper. An IXR of 25 dB
in Sec. III-B to the dipole model, we find
limits thepotentialincrease in the relativeStokeserrorduring
1 0
J =WH V= . (39) reconstruction to 22% as indicated by (17), which requires
approx approx (cid:20) csoinsφφ 1 (cid:21) φ≥83.7◦.
This shows that if the dipoles are close to orthogonal, i.e., Since the eigenvector method with bi-scalar correction is
if φ 90◦, then J I, which is close to the sensitivity equivalent to the optimal beamformer and has the
approx
desired≈response. This shows t≈hat the unitary ambiguity can same IXR as the bi-scalar beamformer, which has lower
be resolved by the intrinsic properties of the system, but that sensitivity than the optimal method, it seems to be the best
the accuracy of that correction depends on the orthogonality method for calibration of a practical radio telescope system
in the polarimetric response of the two feeds. ononeoftheavailablecelestialcalibrationsources,whichare
usually unpolarized.
It follows from (39) and the definitions given by (15a) and
D. Impact of non-orthogonality
(15b),thatfortheeigenvectormethodwithbi-scalarcorrection
In our next simulation, we look at the impact of non-
orthogonality between the dipoles by comparing the IXR as XPD =tan2φ XPD =
u v
∞
defined in (16) of the beamformers for different inter-dipole XPI = XPI =tan2φ.
u v
angles.SinceJ Ifortheoptimalmethod,theIXRisinfinite ∞
for this method∝regardless of the orientation of the dipoles. Similarly, we find for the bi-scalar beamformer that
The IXR for the optimal method is therefore not shown in 1
XPD = XPD = φ=0
Fig.5.Asderivedintheappendix,theIXRfortheeigenvector u cos2φsin2φ v ∞∀ 6
method is conveniently described by XPI = XPI =tan2φ.
u v
∞
2
1+tan(φ/2)
IXR = . (40) Theseresultsareasymmetricinthetwopolarizations,sincewe
eig
1 tan(φ/2)
defined our dipole modelsuch that one dipole is aligned with
(cid:18) − (cid:19)
ComparisonoftheIXRforthebi-scalarmethodandforthe oneof the polarizationaxes. Since rotationof the polarization
eigenvector method with and without bi-scalar approximate axes is described by a unitary matrix, such a rotation does
VERSIONJUNE1,2012 8
1.5 14
optimal
34 35 36 37
13.5 bi−scalar
1
29 30 31 32 33 13
z)
H12.5
0.5 23 24 25 26 27 28 K/
s) 2m/ 12
egree 0 16 17 18 19 20 21 22 vity (11.5
y (d−0.5 10 11 12 13 14 15 sensiti 11
n
a10.5
e
5 6 7 8 9 m
−1 10
1 2 3 4 9.5
−1.5
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2 9
x (degrees) 5 10 15 20 25 30 35
beam index
Fig. 6. Arrangement of the 37 beams produced by the APERTIF system
Fig. 7. Average sensitivity of the two beamformer output signals for the
overtheFoVoftheWSRT.
bi-scalarmethodandtheoptimalbeamformeratthecentersofthe37beams
formed by the APERTIFsystem. The beam indices correspond to the beam
numbersgiveninFig.6.
not affect the condition number of the Jones matrix making
the IXR insensitive to such a transformation. These results
indicate thatφ 83.7◦, requiredto achievean IXR of25 dB, 60
≥
corresponds to an XPD of at least 19 dB.
55
V. BI-SCALAR BEAMFORMER PERFORMANCE
The bi-scalar approach is commonly applied in actual 50
phased array systems. However,in this paper, we have shown
that the bi-scalar beamformer is not sensitivity equivalent to B)
d
the optimal beamformer and that it relies on the intrinsic R (45
X
polarimetricorthogonalityofthefeedsystem.Wewouldthere- I
foreliketoassessthesensitivityandpolarimetricperformance 40
ofthebi-scalarbeamformerinanactualsystemtoseewhether
it provides acceptable performance. For this assessment, we
35
use results from EM-simulations from the Aperture Tile-in-
Focus (APERTIF) project. The goal of this project is to
developand build a PAF system for the Westerbork Synthesis 30
5 10 15 20 25 30 35
Radio Telescope (WSRT) located in The Netherlands to in-
beam index
crease its field-of-view (FoV) [5]. The APERTIF system will
be able to produce37 beams on the sky that will probablybe Fig.8. IXRofthebi-scalar beamformeratthecenters ofthe37APERTIF
arrangedinahexagonalpatternseparatedbyabouthalfpower beams.ThebeamnumberscorrespondtothebeamnumberingshowninFig.6.
beam width as indicated in Fig. 6.
A. Performance in the compound beam centers of the microstrip transmission line feeding the slot. Also,
The APERTIF system was simulated using a full-wave EM the antenna elements are positioned diagonally over a square
package [18]. Figure 7 shows the average sensitivity of the groundplanewhichresultsindifferentelementconfigurations
two beamformer outputs for the optimal beamformer and the per polarization at the corners of the array.
bi-scalar beamformer, while Fig. 8 shows the IXR of the bi- These results are similar to those found for an earlier
scalar beamformer. All results were obtained for the beam APERTIF prototype system [16], [18]. Measurements done
centers at 1.4 GHz. From these results, we conclude that use with the APERTIF prototype system mounted on one of the
of a bi-scalar beamformerleadsto about4.5%sensitivity loss WSRTdishesconfirmedthesensitivityloss,butalsoindicated
comparedtothe optimalbeamformerandthatthe typicalIXR that the practicalities of an actual system reduce the ratio
willbeabout38dBwithapeakvalueof59dBforthecentral of cross- to co-polarized power observed on an unpolarized
compound beam. Note that the IXR in Fig. 8 is asymmetric source to about 28 dB [16]. This was considered acceptable,
with respecttothe centralbeamwith index19.Thisis caused because the cross-polarization level can be improved by ap-
by the asymmetrical circular cavity terminating the tapered plyingappropriatepolarimetriccorrectionsto the beamformer
slot which has been bended sideways to reduce the length output signals while the sensitivity loss can be recovered by
VERSIONJUNE1,2012 9
−20
65
−1.5 −1.5
−25 60
−1 −1
55
−30
s)−0.5 s)−0.5 50
e e
e e
gr 0 −35 gr 0 45
e e
d d
y ( 0.5 y ( 0.5 40
−40
35
1 1
−45
30
1.5 1.5
25
−50
−2 −1 0 1 2 −2 −1 0 1 2
x (degrees) x (degrees)
−20 Fig.10. IXR(indB)forall37beamsproducedbytheAPERTIFsystem.
−1.5
−25
orthogonality between the two sets of orthogonally oriented
−1
antenna elements, which works very well in bore sight (the
−30
s)−0.5 central beam), but deteriorates towards the edges of the FoV.
ee The results shown in Fig. 9 indicate that an appropriate
gr 0 −35 correction for the polarimetric response of each PAF voltage
e
d
y ( 0.5 beam is required in the image processing to reconstruct the
−40 polarimetric properties of the incident waves regardless of
1 the beamformingapproachused. The reconstructabilityof the
−45 polarization state of the received signals is measured by the
1.5 IXR, which is shown in Fig. 10 for the bi-scalar beamformer.
−50 The simulations indicate that the inner 7 beams have an IXR
−2 −1 0 1 2
betterthan40dBoveralmosttheirentirebeamarea,whilethe
x (degrees)
compound beams at the edges of the FoV still have an IXR
of at least 25 dB. This gives an upper limit on the increase
Fig. 9. |J12| for all 37 beams produced by the APERTIF system for the
optimalbeamformer(top)andbi-scalarbeamformer(bottom).Thecolorscale in the relative measurement error on the Stokes vector of 4%
has units ofdBwith respect to thevalue of|J11|atthe peakofthe central and 22% for the central beams and for the edges of the FoV
compoundbeam.
respectively as predicted by (17). This should be sufficient to
allowaccuratereconstructionofpolarizedsignalswithlimited
sensitivity reduction due to image processing.
forming cross- and co-polarized beams as discussed in Sec.
III-C.
VI. CONCLUSIONS
In this paper we discussed the polarimetric and sensitivity
B. Behavior over FoV
performance of calibration schemes that exploit an unpolar-
In the previous subsection, we assessed the performance ized reference source. We demonstrated that the eigenvector
of the bi-scalar beamformer at the beam centers. Another method is sensitivity equivalent to the optimal beamformer.
concern is the behavior of the PAF voltage beams over their This implies that the eigenvector methods can exactly repro-
respectiveFoVs,sincetheimageprocessingshouldcorrectfor ducetheresultobtainedfromtheoptimalbeamformerwithan
this response. Figure 9 therefore shows J for the optimal additional polarimetric correction after beamforming.
12
| |
method and the bi-scalar method over the FoV at 1.42 GHz. We have demonstrated that the eigenvector method is not
This shows that both beamformers suffer from the direction suitable for use in an actual system without additional cor-
dependent polarimetric response of the two feed systems, but rection, because this method ignores the unitary ambiguity
that the optimal beamformer does a better job at the beam intrinsic to calibration on an unpolarized source. This am-
centers of the compound beams towards the edges of the biguity needs to be resolved either by imposing additional
FoV. For the optimal beamformer, there is always a small constraints on the system response, such as relying on the
region around the beam center in which the instrumental intrinsic polarimetric orthogonality between the feeds, or by
crosspolarizationislessthan-45dBwhilethebi-scalarbeam- additional calibration observation on two distinctly polarized
former produces some beams with -30 dB crosspolarization sources.
in their field centers. This can be explained by the fact that Thebi-scalarbeamformerisnotsensitivityequivalenttothe
the bi-scalar beamformer relies on the intrinsic polarimetric optimal beamformer and relies on the intrinsic polarimetric