Fish-Eye Observing with Phased Array Radio Telescopes

220 Pages·2010·10.66 MB·English

Fish-Eye Observing with
Phased Array Radio Telescopes
PROEFSCHRIFT
terverkrijgingvandegraadvandoctor
aandeTechnischeUniversiteitDelft,
opgezagvandeRectorMagnificusprof.ir.K.C.A.M.Luyben,
voorzittervanhetCollegevoorPromoties,
inhetopenbaarteverdedigenopdinsdag2maart2010om12:30uur
door
StefanJeroenWIJNHOLDS
natuurkundigingenieur
doctorandusindesterrenkunde
geborenteGroningen.
Ditproefschriftisgoedgekeurddoordepromotor:
Prof.dr.ir.A.J.vanderVeen
Samenstellingpromotiecommissie:
RectorMagnificus voorzitter
Prof.dr.ir.A.J.vanderVeen TechnischeUniversiteitDelft,promotor
Prof.dr.ir.J.Biemond TechnischeUniversiteitDelft
Prof.dr.ir.M.Verhaegen TechnischeUniversiteitDelft
Prof.dr.A.G.deBruyn RijksuniversiteitGroningen
Prof.dr.W.N.Brouw RijksuniversiteitGroningen
Prof.dr.Ir.M.Moonen KatholiekeUniversiteitLeuven,Belgie¨
Prof.dr.M.Viberg ChalmersUniversityofTechnology,
Go¨teborg,Zweden
Copyright c 2010byS.J.Wijnholds
(cid:13)
All rights reserved. No part of the material protected by this copyrightnotice may
bereproducedorutilizedinanyformorbyanymeans,electronicormechanical,in-
cludingphotocopying,recordingorbyanyinformationstorageandretrievalsystem,
withoutthe priorpermissionofthe author. An exceptionis madeforretrievalfrom
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PrintedinTheNetherlandsbyWo¨hrmannPrintService
TypesetbytheauthorwiththeLATEXDocumentationSystem.
Authoremail:[email protected]
PublishedanddistributedbyS.J.Wijnholds
ISBN978-90-9025180-6
Fish-Eye Observing with
Phased Array Radio Telescopes
Stefan J. Wijnholds

Contents
Preface ix
1 Introduction 1
1.1 Phasedarraytechnology . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 LOFAR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Hierarchicalsystemarchitecture . . . . . . . . . . . . . . . . . . . . 8
1.4 Scopeofthisthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.5 Outlineandsummaryofmainresults . . . . . . . . . . . . . . . . . . 11
1.6 Publicationsandothercontributions . . . . . . . . . . . . . . . . . . 14
2 CalibrationandimagingwithLOFAR 17
2.1 Coordinatesystems . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.1.1 Directionofarrival . . . . . . . . . . . . . . . . . . . . . . . 18
2.1.2 Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.1.3 Antennalocations. . . . . . . . . . . . . . . . . . . . . . . . 20
2.2 Highlightsfromearlyprototypes . . . . . . . . . . . . . . . . . . . . 20
2.2.1 TheTenHeterogeneousElementTestArray . . . . . . . . . . 20
2.2.2 TheLOFARPrototypeStation . . . . . . . . . . . . . . . . . 22
2.2.3 TheInitialTestStation . . . . . . . . . . . . . . . . . . . . . 24
2.2.4 CoreStation10 . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.3 Stationcalibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.3.1 Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.3.2 Stationcalibrationpipeline . . . . . . . . . . . . . . . . . . . 33
2.3.3 Demonstrations . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.4 Fish-eyeimaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.4.1 DFTversusleastsquaresimaging . . . . . . . . . . . . . . . 39
2.4.2 Statisticalperformance . . . . . . . . . . . . . . . . . . . . . 41
2.4.3 All-skyleastsquaressynthesisimaging . . . . . . . . . . . . 43
v
vi CONTENTS
3 Datamodel 47
3.1 Physicalmodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.2 Signalprocessingmodels . . . . . . . . . . . . . . . . . . . . . . . . 53
3.2.1 Shorttermcovariancematrix . . . . . . . . . . . . . . . . . . 53
3.2.2 Polarimetry . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.2.3 Commonlyuseddatamodels . . . . . . . . . . . . . . . . . . 57
3.3 Noisesubspaceandsignalsubspace . . . . . . . . . . . . . . . . . . 58
3.4 Narrowbandassumption . . . . . . . . . . . . . . . . . . . . . . . . 61
3.5 Spatialfilteringofsourcesignals . . . . . . . . . . . . . . . . . . . . 66
3.5.1 Scalarbeamforming . . . . . . . . . . . . . . . . . . . . . . 66
3.5.2 Arrayresponse . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.6 Fromsnapshotstosynthesisobservations . . . . . . . . . . . . . . . 70
4 Cramer-RaoBound 73
4.1 DefinitionoftheCRB . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.2 AnalyticexpressionsfortheCRB . . . . . . . . . . . . . . . . . . . 76
4.2.1 PartitioningtheFIM . . . . . . . . . . . . . . . . . . . . . . 76
4.2.2 Jacobians . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.2.3 ConstituentsoftheFIM . . . . . . . . . . . . . . . . . . . . 82
4.3 Impactofgainphaseconstraints . . . . . . . . . . . . . . . . . . . . 84
4.3.1 Constraintsonthegainphasesolution . . . . . . . . . . . . . 84
4.3.2 TheconstrainedCRB . . . . . . . . . . . . . . . . . . . . . . 85
4.3.3 Singlecalibrationsource . . . . . . . . . . . . . . . . . . . . 86
4.3.4 Multiplecalibrationsources . . . . . . . . . . . . . . . . . . 88
4.3.5 Applicationexample . . . . . . . . . . . . . . . . . . . . . . 89
4.3.6 Concludingremarks . . . . . . . . . . . . . . . . . . . . . . 91
4.4 Dealingwithmissingdata . . . . . . . . . . . . . . . . . . . . . . . 92
4.4.1 Problemstatement . . . . . . . . . . . . . . . . . . . . . . . 92
4.4.2 DerivationoftheCRB . . . . . . . . . . . . . . . . . . . . . 93
4.4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5 Calibration 99
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.2 Problemstatement . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.3 Algorithmdevelopment . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.3.1 Generalizedleastsquaresformulation . . . . . . . . . . . . . 104
5.3.2 Estimationofdirectionindependentgains . . . . . . . . . . . 105
5.3.3 Sourcepowerestimation . . . . . . . . . . . . . . . . . . . . 110
5.3.4 Estimatingreceivernoisepowers. . . . . . . . . . . . . . . . 110
5.3.5 DOAestimation . . . . . . . . . . . . . . . . . . . . . . . . 112
CONTENTS vii
5.3.6 (Weighted)alternatingleastsquares . . . . . . . . . . . . . . 113
5.3.7 Computationalcomplexity . . . . . . . . . . . . . . . . . . . 115
5.4 Algorithmvalidation . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.5 Non-diagonalnoisecovariancematrix . . . . . . . . . . . . . . . . . 124
5.5.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
5.5.2 ModificationoftheWALSmethod. . . . . . . . . . . . . . . 125
5.6 Summaryofthemainresults . . . . . . . . . . . . . . . . . . . . . . 129
6 Imaging 131
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.2 Fundamentalimaginglimits . . . . . . . . . . . . . . . . . . . . . . 133
6.2.1 Imaginganddeconvolution. . . . . . . . . . . . . . . . . . . 134
6.2.2 Effectivenoise . . . . . . . . . . . . . . . . . . . . . . . . . 143
6.2.3 Implications . . . . . . . . . . . . . . . . . . . . . . . . . . 149
6.2.4 Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . 151
6.3 Leastsquaressynthesisimaging . . . . . . . . . . . . . . . . . . . . 153
6.3.1 Genericformulation . . . . . . . . . . . . . . . . . . . . . . 153
6.3.2 Scalarimagingwithknownornegligiblenoise . . . . . . . . 157
6.3.3 Scalarimagingwithunknownnoise . . . . . . . . . . . . . . 158
6.3.4 Fullpolarizedimagingwithknownornegligiblenoise . . . . 158
6.3.5 Fullpolarizedimagingwithunknownnoise . . . . . . . . . . 160
6.3.6 Treatmentofmorecomplicatednoisemodels . . . . . . . . . 161
6.A Derivationofclosedformleastsquaresimagingsolutions . . . . . . . 161
6.A.1 Scalarimagingwithknownornegligiblenoise . . . . . . . . 161
6.A.2 Scalarimagingwithunknownnoise . . . . . . . . . . . . . . 163
6.A.3 Fullpolarizedimagingwithknownornegligiblenoise . . . . 166
6.A.4 Fullpolarizedimagingwithunknownnoise . . . . . . . . . . 168
7 Conclusionsandsuggestionsforfurtherresearch 171
7.1 Summaryofthemainresults . . . . . . . . . . . . . . . . . . . . . . 171
7.2 Suggestionsforfurtherresearch . . . . . . . . . . . . . . . . . . . . 174
7.2.1 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
7.2.2 Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
7.3 TowardsSKA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
A Abbreviationsandacronyms 181
B Notation 183
B.1 Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
B.2 Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
viii CONTENTS
C Matrixproductrelations 185
References 187
Samenvatting 203
Abouttheauthor 207
Preface
In some sense, this thesisis living proofthatanyplan is valid untilit is confronted
withreality. Afterfollowingacrashcourseinradiotelescopeintegrationandtesting
byJaapBregmanduringmyinternshipattheNetherlandsInstituteforRadioAstron-
omy(ASTRON),IdecidedtoapplyforajobasresearchassistantwithASTRONin-
steadofpursuingaPh.D.IthusstartedworkingonsystemtestingoftheearlyLOFAR
(LowFrequencyArray)prototypesTHETA(TenHeterogeneousElementTestArray)
andITS(InitialTestStation). Duringthisperiod,IspentevenmorehoursinJaap’s
officeto learnthe tricksof the trade. The main goalof these prototypeswasa per-
formanceevaluationofanantennaarrayinthefield. Thisnaturallyledtoanindepth
study of in situ calibration of antenna arrays, on which I gave a status update dur-
ingacalibrationworkshoponMay31,2005. Alle-JanvanderVeenwasoneofthe
attendeestothatmeetingandinvitedmeforfurtherdiscussioninDelft. Thisledto
anewseriesoffruitfuldiscussionsintroducingmetotheworldofsignalprocessing
andultimatelytothisthesis. Iwouldthereforeliketoexpressmydeepestgratitudeto
bothJaapandAlle-Janforalltheirguidance,supportandadvice.
WorkingonabigprojectlikeLOFARimpliesthatthisresearchwasnotdonein
splendidisolation. IwouldthereforeliketothankmycolleaguesatASTRONforall
inspiringconversations,theirsupportonthehardwareandsoftwareneededtodothe
measurements and for creating a pleasant environmentto work in. Taking the risk
of regretfulomissions, I would like to thank a few persons more specifically. This
research was, to a considerable extent, driven by the “features” found in the data
obtainedfromthe prototypestations. These measurementswould nothave beenso
successfulwithoutthehardwaresupportbyYdeKoopman,GijsSchoonderbeek,An-
dreGunstandMennoNordenandthesoftwaresupportbyKlaas-JanWierengaand
ChrisBroekema. ITSrequiredafocusedone-weekdebugsessiontoalignhardware
and on-lineand off-linesoftware togetherwith Michiel Brentjens, who was always
availableto reflectonnew ideasandchallengesaswell. Itwas morethanjustcon-
venienttohaveAlbert-JanBoonstra,whograduatedundersupervisionofAlle-Janin
2005, sitting in the same building to discuss signalprocessing conceptsand issues.
ix
x Preface
ThisresearchalsobenefittedfromtheinsightsofGerdeBruyn,JohanHamaker,Wim
Brouw and Jan Noordam, who shared their lifelongexperiencewith the calibration
and imaging challengesof the Westerbork Synthesis Radio Telescope (WSRT) and
fromdiscussionswithRonaldNijboer. Thelatter,togetherwithAlbert-Jan,Janand
Jaap,alsoprovidedthepersonalsupporttostartthisthesisprojectinthefirstplace.
Althoughtherehasbeenconsiderablesynergybetweentheresearchpresentedin
thisthesisandthedemandsfromtheLOFARproject,writingpapersandespecially
writingathesisformsamajordistractionfromprojectwork. Iwouldthereforelike
to thank Arnold van Ardenne, former director of the R&D department, Marco de
Vos,formerLOFARprojectmanagerandsuccessorofArnold,andDionKant,group
leaderofthesystemdesignandintegrationgroup,forsupportingmythesisworkby
allowingme tospendtimeonthis. Especiallyin thelast year,thishasbeena great
helpingettingthingsfinished.
Finally,Iwouldliketothankthecommitteemembers,JanBiemond,MichelVer-
haegen, Ger de Bruyn, Wim Brouw, Marc Moonen and Mats Viberg, for the time
investedinthoroughlyreviewingmythesis. Thishasbeenagreathelpinimproving
thequalityandreadabilityofthisthesis.
StefanWijnholds
Dwingeloo,TheNetherlands,January31,2010