Table Of ContentAstronomy&Astrophysicsmanuscriptno.holler6879 (cid:13)c ESO2008
February5,2008
A 6–12 GHz Analogue Lag-Correlator for Radio Interferometry
ChristianM.Holler1,2,TakKaneko1,MichaelE.Jones3,KeithGrainge1,andPaulScott1
7
0
0 1 CavendishLaboratory,CambridgeUniversity,CambridgeCB30HE,UnitedKingdom.
2 e-mail:[email protected]
n 2 Ludwigshohenweg11,83253Rimsting,Germany.
a e-mail:[email protected]
J 3 Sub-DeptofAstrophysics,UniversityofOxford,DenysWilkinsonBuilding,KebleRoad,Oxford,OX17RH,UnitedKingdom.
3
2 ReceivedDecember6,2006;acceptedDecember22,2006
1
ABSTRACT
v
4
Aims.Wedescribea6–12GHzanaloguecorrelatorthathasbeendevelopedforuseinradiointerferometers.
4
6 Methods.We use a lag-correlator technique to synthesis eight complex spectral channels. Two schemes were considered for sampling the
1 cross-correlationfunction,usingeitherrealorcomplexcorrelations,andwedevelopedprototypesforbothofthem.Weoptedforthe“addand
0 square” detection schemeusing Schottky diodesover themorecommonly usedactive multipliersbecause the stabilityof thedevice isless
7 critical.
0 Results.We encountered an unexpected problem, in that there were errors in the lag spacings of up to ten percent of the unit spacing. To
/ overcomethis,wedevelopedacalibrationmethodusingastronomicalsourceswhichcorrectstheeffectsofthenon-uniformsamplingaswell
h
p asgainerroranddispersioninthecorrelator.
-
o
r
t Keywords.instrumentation:interferometers–techniques:interferometric
s
a
:
v
1. Introduction torshavebeenusedinseveralrecentinterferometersdedicated
i
X to CMB observations (O’Sullivanetal. 1995; Loetal. 2001;
High brightnesssensitivity is essential for many observations
r Padinetal.2002;Leitchetal.2002;Watsonetal.2003).
a in radio astronomy, for example, observations of the cosmic
microwave background (CMB). Once the angular resolution To achieve the low spectral resolution typically required
requiredhasbeenfixedbythesizeoftheinstrumentinwave- (of order ten spectral channels, rather than the hundreds
lengths,thebrightnesssensitivityisdeterminedbytwofactors: or thousands typical of digital correlators) there are two
the system temperatureand the observingbandwidth.System basic solutions. One is to split the broadband signal di-
temperatures have been reduced dramatically by the use of rectly into smaller bands with filter banks or multiple down-
cryogenicHEMTamplifiers,tothepointwheretheatmosphere converters (eg. Padinetal. 2001). The other is to use a
ratherthantheamplifiermaybethedominantnoisesourcein single broad intermediate frequency (IF) band and imple-
thesystem, andthuslittle furtherimprovementis possiblefor ment a Fourier transform spectrometer, or lag correlator.
ground-basedinstruments. Increased continuum bandwidth is This forms the cross-correlation between antennas at multi-
thereforeanimportantroutetoincreasedsensitivity. ple delays, and the outputs are Fourier transformed to es-
Radio interferometers with any significant bandwidth re- timate the cross-frequency spectrum. In this paper, we de-
quire the band to be broken up in to sub-bands,both because scribe the development of an analogue 6 GHz bandwidth
spectralresolutionmaybeintrinsicallyrequiredfortheobser- lag correlator with eight complex bands. The correlator is
vation, and to overcome the effects of chromatic aberration. currently operational in the Arcminute Microkelvin Imager
Digital correlatorsare commonlyused since they providethe (AMI; see, for example Kneissletal. 2001; Jones 2002;
mostconvenientwayofgeneratinghighspectralresolutionand Kaneko&theAMICollaboration2006),a new radiointerfer-
a large range of time delays. However, for very high bright- ometer operatingover the 12–18GHzband.AMI was specif-
ness sensitivity continuum observations which demand max- ically designed to carry out a survey for clusters of galaxies
imum possible bandwidth with only modest spectral resolu- throughtheSunyaev-Zel’dovicheffect(Sunyaev&Zel’dovich
tion, analogue correlators are competitive. Analogue correla- 1972). Broadband analogue correlators with similar concepts
have been described by Harris&Zmuidzinas(2001); Lietal.
Sendoffprintrequeststo:ChristianM.Holler (2004);Robertsetal.(submitted).Afeaturethatdistinguishes
2 ChristianM.Holleretal.:A6–12GHzAnalogueLag-CorrelatorforRadioInterferometry
ourdesignfromthesecorrelatorsisthedetectors.Ratherthan (see eg. Thompsonetal. 2001). To satisfy this requirement
active multipliers, we chose the “add and square” detection for an instrument with centre frequency ν = 15GHz and a
RF
scheme using zero-biased Schottky diodes. They are simpler maximum ratio of baseline to dish diameter D/d ≃ 10 we
tooperatebecausenobiascurrentisrequiredandtheirstabil- chose to split the full 6GHz bandwidth into eight sub-bands
ity is notas criticalas itis foractive devices.Itistechnically of0.75GHzeach.
challengingtodesignIFcomponentsthatspanovermorethan ThebasiclayoutoftheLagorFouriertransformcorrelator
oneoctaveinfrequencysowechoseanIFbandof6–12GHz. isillustratedinFig.1a.ThesignalsA(t)andB(t)fromthetwo
Toreducecostsandcomplexity,weimplementedthecorrelator antennasofabaselinearecorrelatedatdiscretetimedelaysτ
i
using microstrip technologyand each baseline was fabricated withstepsizeδτ.Thesetofcorrelatedsignalsformthecross-
onasinglesubstrate. correlationfunction:
Section2ofthispaperdiscussesthebasicrelationshipsfor
abroadbandlagcorrelatoranddiscussesrealvscomplexcorre- 1 T
R(τ)= A(t)B(t+τ)dt, (4)
latorsand phase-switchedmultipliers.Section 3 describesthe i T Z i
0
detectorcircuitinourdesign.Section4thendescribestheover-
allhardwareimplementation,andSec.5thecalibrationscheme where T is the integration time. This type of correlator mea-
implemented to recover accurate spectra from the lag data in sures the temporal coherence function of a signal, where the
the presence of dispersion and lag spacing errors. Section 6 coherence function is an indirect measure of the signal’s fre-
describesissuesrelatedtothemanufacturingandpracticalim- quencyspectrum.Thecross-powerspectrum(hereafterabbre-
plementationofthecorrelator,withconclusionsinSec.7. viated to the spectrum)can be recoveredby applyingthe dis-
creteFouriertransform(DFT)tothecross-correlationfunction;
2. CorrelatorOverview
S(νk)= R(τi)e−2πjνkτi. (5)
2.1.FourierTransformCorrelator Xτ
i
The response of an interferometer baseline with a single real
Figure2 illustratestypicalcross-correlationfunctionsofa
correlatorhavinga rectangularpassbandofwidth ∆ν is given
flat-spectrumpointsourceatthetelescope’sphasecentrefora
by
narrowbandandbroadbandsystemwithrectangularpassband.
sin[π∆ν(τ −τ)] The functionisgivenbyEq.2 with τ = 0 andτ asthe vari-
g i g i
R=R cos 2π(ν τ −ν (τ −τ) +φ (1)
0 π∆ν(τ −τ) " LO g IF g i LO# able.ThecosinetermhasaperioddeterminedbythecentreIF
g i (cid:17)
frequencyν andthesincenvelopeisdefinedbythebandwidth
IF
where R0 is a normalisingfactor, τg and τi are geometricand ∆ν.Asthesourcemovesacrossthesky,thecosinepatterndrifts
instrument delays (i.e. the delays introduced at the radio fre- in instrument delay space. The envelope encodes the spectral
quency (RF) and IF respectively), νLO is the local oscillator information and its Fourier transform gives the spectrum of
(LO) frequency, νIF is the IF frequency and φLO is the phase the observed signal. The desired information is contained in
differencebetweentheLOateachaerial.Thecosinetermde- theenvelopeanditmustbesampledatNyquistsamplingstep,
scribes the fringesthatchange with changingdelay.The path δτ =1/(2∆ν).
N
compensator inserts an instrument delay τ which (approxi-
i Apointsourceattheedgeofthefieldofviewwillintroduce
mately)balancesoutthegeometricaldelay.Thesinctermpro-
an additionalgeometrical time delay ∆τ into one arm of the
g
ducesa fringe envelope,which maximises the response if the
correlator.Theadditionaldelaywilloffsetthecross-correlation
residualdelay,(τ −τ),issmall.Forsimplicity,thephasedif-
g i function away from the centre lag. We define a narrowband
ferenceintheLOatthetwomixers,φ ,canbedroppedasit
LO correlator to be one where this delay ∆τg is shorter than the
is easily removed by calibration. Substituting for the LO fre-
Nyquistsamplingstepδτ .Inordertoavoidchromaticaberra-
N
quencyν usingtherelationshipν −ν =ν forlowerside-
LO LO IF RF tion,noadditionallagsarenecessary.Howeverinabroadband
bandreception,
system,where∆τ isgreaterthanδτ ,theoffsetcannotbene-
g N
sin[π∆ν(τg−τi)] glected. For our design parameters, ∆τg can be up to ±6δτN
R=R0 π∆ν(τ −τ) cos2π νRFτg+νIFτi . (2) sothecross-correlationfunctionhastobesampledoverawide
g i (cid:16) (cid:17) interval.Wesample16lagsat−8δτ ,−7δτ ,...+7δτ .After
N N N
Chromaticaberrationisaproblembecauseoftheextentof applyingthe DFT, this splits the passbandinto eightcomplex
theobservingfieldofview.Whileitispossibletointroducethe sub-bandsof0.75GHzeach.Mostofthesignalpoweriscon-
correct instrumental delay to compensate for the geometrical centrated in the centre of the cross-correlation function’s en-
delayatthecentreofthefield,aradiosourceasmallangle∆θ velope. Its position in instrumentdelay space dependson the
away from the field centre will have an additional geometric position of the sourcein the field of view andmust be within
delay D∆θ/c, where D is the baseline length projected on to therangeofthesampledlags.Figure3showssimulationsofthe
theplaneperpendiculartothesourcedirection.Ifitisrequired amplituderesponseforeachoutputchannelofa16-lagcorrela-
to image with minimal loss of sensitivity to the edge of the torasafunctionofthecentrepositionofthecross-correlation
primarybeamofaninterferometerwithanantennadiameterd function. By using 16 lags for the correlator, its sensitivity is
∆ν/ν notsignificantlyreducedforsourceswithinthetelescope’sfield
π RF ≪1 (3)
ofview.
d/D
ChristianM.Holleretal.:A6–12GHzAnalogueLag-CorrelatorforRadioInterferometry 3
1
F
C 0.5
C
d
an 0
b
d
oa −0.5 ∆τ
Br g
−1
1
F
C
C 0.5
d
(a) Cross-correlationfunction n
a 0
b
w
1 o
Broadband CCF arr −0.5
N
−1
0
−500 −250 0 250 500
Instrument Delay /mm
Instrument Delay /mm
−1
Fig.2. The cross-correlation functionsfor a broadband(ν =
−250 −125 0 125 250 IF
1.5 1.5 9GHz, ∆ν = 6GHz) and a narrowband system (∆ν =
1 1 0.75GHz).Thehorizontalscalesareinstrumentdelayinelec-
Real 0.5 mag 0.5 trical length. The gray envelopes are the bandwidth patterns
0 I 0 andtheopencirclesarethelagpositions.Asourceattheedge
−0.5 −0.5
of the field of view will introduce an additional geometrical
−15 0 15 −15 0 15
Frequency /GHz time delay ∆τg and offset the cross-correlation function over
(b) RealCorrelator therangeofthearrow.
Fig.1. (a) The lag correlatorcross-correlatesthe signalsfrom
1.4
a pair of antennas at discrete delay steps (denoted by δτ).
Applying a discrete Fourier transform to the set of lag data el 1.2
n ch 2−7
givesthecomplexpowerspectrum.Therealcorrelatorsamples n
ha 1
at16lags.(b)Thecross-correlationfunctionandthespectrum. c
h
The gray envelope is the bandwidth pattern and the open cir- ac 0.8 ch 1 & 8
e
clesarethepositionsofthelags. Fora sourceatthecentreof of 0.6
the field of view and no geometricdelay,the imaginarycom- e
d
ponentof the spectrum is zero. In general, the real correlator plitu 0.4
hascomplexspectraatpositiveandnegativefrequencies. m
A 0.2
0
2.2.Signalrecovery 0 2 4 6 8 10
Position of peak from lag centre /lag
Anarrowbandsignalofcentrefrequencyνcanbesampledby
a simple single-lag correlator.The signalreceivedat each an- Fig.3. The amplitude response of each frequency channel
tennaisarealquantitybutitisconvenienttoexpressitincom- of the correlator to the broadband cross-correlation functions
plexnotation.Thesignalsatthetwoantennasaretherealcom- (Fig.2)positionedawayfromthelagcentre.Thisisequivalent
ponentsof topointsourcesawayfromthepointingcentre.Thesensitivity
of the correlator is fairly constant over the field of view, cor-
A(t)=aej(2πνt+φ) and B(t)=bej2πνt,
responding to a maximum of ±6 lags. The structures and the
where φ is the signalphase which dependson the position of reduced sensitivity in the edge sub-bands 1 & 8 will be dis-
thesourceandgeometryoftheinterferometer.Aftercorrelating cussedinSec.5.2.
thesesignals,theoutputofthecorrelatoris
Re[A(t)B∗(t)]=Re[abejφ]=ab cosφ. (6) Thesecondmeasurementwillbe
We take the complex conjugate of B(t) because we are inter- Re[A′(t)B∗(t)]=Re[abej(φ+π/2)]=ab sinφ. (8)
estedintheproductoftherealparts.Thequantitieswewantto
From these in-phase(equation3) and quadrature(equation4)
measurearetheamplitudeabandthephaseφ.Tomeasurethem
components,wecansimultaneouslymeasuretheamplitudeand
both simultaneously, we would need two measurements. We
phase of the signal. As mentioned earlier, the Fourier trans-
couldmakethe secondmeasurementbyinsertinga 90◦ phase
formed output of the lag correlator is a complex spectrum.
shiftinonearmsothat
This is a way of expressing the amplitude and phase of each
A′(t)=aej(2πνt+φ+π/2). (7) sub-band.Similarly,wecouldexpresstheamplitudeandphase
4 ChristianM.Holleretal.:A6–12GHzAnalogueLag-CorrelatorforRadioInterferometry
measuredby the single-lagcorrelatoras a complexvalue.We
stressthatthecross-correlationfunctionisa realquantityand
the complexnotation is for mathematicalconvenience.To re-
flectthisnotation,wewillrefertothein-phaseandquadrature
componentsasrealandimaginary.
Inamultiple-lagcorrelator,however,wehavetwooptions.
We could either sample just the in-phase components of the
cross-correlationfunction(realcorrelator–seeFig.1a)orboth
thein-phaseandquadraturecomponents(complexcorrelator–
see Fig. 4a). The choice will affect the form of the spectrum
S(ν ).Buttheinformationwerecoverisequivalent.
k
2.2.1. Realcorrelator (a) ComplexCorrelator
Here,2nrealcomponentsofthecross-correlationfunctionare
sampledatNyquistratewithstepsizeδτ ;anelectricallength
N 1
of 25mmor an equivalentsampling frequencyof 12GHz for Real CCF
a 6-GHz bandwidth signal. The cross-correlation function is
0
sampledfrom−n·δτ to+(n−1)·δτ ,wherenisthenumberof
N N
complexsub-bands1.Thesymmetrybetweenthepositiveand
−1 1
negativehalvesofthespectrummeansthathalftheinformation
Imag CCF
inthespectrumisredundant.16independentmeasurementsof
thecross-correlationfunctiongive8independentcomplexsub- 0
bands(seeFig.1b). Inst. delay c τ /mm
i
−1
−250 −125 0 125 250
2.2.2. Complexcorrelator 1.5 1.5
1 1
Thesecondmethodsamplesboththerealandimaginarycom- Real 0.5 mag 0.5
ponents of the cross correlation function but at only half- 0 I 0
Nyquistrate(6GHzsamplingfrequency,oranequivalentelec- −0.5 −0.5
−15 0 15 −15 0 15
tricallengthof50mm).Weexpressthemeasurementsasaset
Frequency /GHz
ofncomplexnumbers.Thestepsizeis2δτ andsamplesspan
N (b) Cross-correlationfunction
from−n·δτ to+(n−2)·δτ .Theresultingspectrumcontains
N N
onlypositivefrequencies(Fig.4b).Tomeasuretheimaginary Fig.4.(a)Thecomplexcorrelatorsamplesatonlyeightlagsbut
component,onearmofinputsignalhasabroadband90◦phase withtwomeasurementsperlagandtwicethedelaysteps.Using
shifter2. 8 complex measurements will give 8 complex sub- the 90◦ phase shifters, it samples the in-phase and quadrature
bands. The single-sided spectrum explains why half-Nyquist components of the cross-correlation function at each lag. (b)
sampling is sufficient. It can also be understood in terms of Thecross-correlationfunctionandthespectrum.Ingeneral,the
information content: The information in 8 complex measure- spectrumrecoveredbythecomplexcorrelatorissingle-sided.
mentsisthesameis16realmeasurements.
To explore possible practical differences between the two
buttheir stability is oftenmorecritical. Therelative meritsof
methods,webuiltprototypesforbothschemes.
activemultipliersarediscussedinHarris(2003).
Weuseaverysimpleconcept,describedbyRyle(1952)as
2.3.Cross-CorrelationandPhase-switching “addandsquare”detection.Thetwosignalsfromtheantennas
are summedto givea+b. Thesummed signalis then passed
Thereareseveralwaysofcorrelatingtwoanaloguesignals.In
throughanon-lineardevice,inourcaseadiode,withanoutput
asinglebaselineinterferometer,twoantennasmeasurerandom
oftheform
noise-likesignalvoltagesA(t)andB(t).Wewillrepresentthese
by random variables a and b. Active multipliers will directly (a+b)2 =a2+2ab+b2. (9)
givetheproductab.Activecomponentshavehighersensitivity
Thetotalpowertermsofeachantennaa2 andb2 dominatethe
output,andthecross-correlationtermabneedstobeextracted
1 Generally, the spectrum will be complex in both positive and
before recording the data. We use phase-switching to remove
negative frequencies. These two components are Hermitian; S(ν) =
thetotalpowerterms,whichalsoreducesslowlyvaryingoffsets
S∗(−ν),wheretheasteriskdenotesthecomplexconjugate.Therela-
andcross-talkinthesystem.Thedetectionschemeisillustrated
tionshipcouldbeobservedfromthesymmetryinthespectrum.
2 ThephaseshiftproducesaHilberttransformoftheoriginalsig- inFig.5.ThetwosignalsaremodulatedbyWalshfunctions f
nal;forasingle-sidedspectrum,therealpartofthecross-correlation and g. These are periodic bi-valued functionswhich have the
functionistheHilberttransformofitsimaginarypart. propertythatallthefunctionsinagivenfamilyareorthogonal
ChristianM.Holleretal.:A6–12GHzAnalogueLag-CorrelatorforRadioInterferometry 5
toeachother,andtotheirproductwithanyotherWalshfunc-
tion.Phaseshiftingthesignalby180◦ appliesthemodulation.
Theoutputfromthecorrelatorwillbe
(fa+gb)2 = f2a2+2fgab+g2b2. (10)
The productsof Walsh functionswith themselves, f2 and g2,
areconstants.TheproductoftwodifferentWalshfunctions, fg,
isanotherWalshfunction.Toextractthecross-term,theoutput
isdemodulatedby fgandintegratedovertherepeatperiodT
w
oftheWalshfunction:
T T Fig.5. The “add and square” analogue correlator including
1 w 1 w
fg f2a2+2fgab+g2b2 dt ∝ ab dt.(11) phaseswitching.Thesignalfromthedetectorisamplifiedand
Tw Z0 (cid:16) (cid:17) Tw Z0 mostofthetotalpowertermsareremovedbythehighpassfil-
Phase-switchingalsoremovesanyslowlyvaryingdriftontime- ter (HPF). The HPF attenuates the direct current (DC) signal
scales longer than the repeat period. It also suppresses cross- by a factor ǫ. Any remaining total power and DC offsets are
talk because signals from other antennas will be modulated removedbyphase-switchdemodulation.h·iistheexpectation.
withanorthogonalWalshfunction.
To achieve the full signal-to-noise ratio (SNR) in such a
correlator,itisalsonecessarytoformthe“subtractandsquare” For very small input signals a bias current could be used
product (a − b)2. This can be seen by considering that each toincreasetheoutput.However,theoutputwouldbeverysen-
voltagebeingcorrelatedconsistsofasignaltermwhichiscor- sitive to changes in this current with, for example, tempera-
relatedbetweenthetwoantennasandnoisetermthatisuncor- ture,sowechosetousezero-biaseddiodes.Thedisadvantage
related.Sincethenoisetermsareuncorrelated,twoorthogonal isthattheyrequirearelativelyhighinputpowerin theregion
signalcombinationscanbemadefromthemwithuncorrelated of −13dBm or 0.05mW.But they are very stable, as long as
noises.Thenoiseonthesumsignal,a+b,isthusuncorrelated theyarekeptataconstanttemperature.
withthenoiseonthedifferencesignal,a−b.Eachcanthenbe Tradeoffshadtobemadebetweenthemostimportantper-
squaredtoformestimatesoftheproductabwithindependent formance measures of the detector. These include sensitivity
noises,andthesecanbeaddedtoformthefullsignal-to-noise overthefullband,stability,thevideobandwidth(thelowpass-
product.A full plus-minussystem is thus equivalentto a sin- filteredbandwidthafterthedetector),avoidingitbehavinglike
glerealmultiplication,andtwosuchmultiplierswouldmakea apeakdetectorandfrequencymatching.
singlecomplexcorrelator. Thesensitivitylargelydependsonthevalueoftheloadre-
Asecondcorrelatoristhusequippedwitha180◦phaseshift sistor R . A high value gives a higher output voltage for the
L
inonearm.Theoutputfromthissecondcorrelatorwillbe sameinputpower,butalsoreducesthevideobandwidthforthe
signal behind the diode. Phase-switch modulation introduces
(fa−gb)2 = f2a−2fgab+g2b. (12)
sharptransitionsinthesignal.Slewsoverthetransitionwillde-
In principle the plus and minus correlatoroutputs can be dif- gradetheorthogonalitybetweenthesignals.Forphase-switch
ferencedinhardwareandthesignalthendemodulatedandread signals with a maximum frequency of 1kHz, we required a
out.Thiscutsdownthe necessaryreadoutelectronicsbyhalf. minimumvideobandwidthofabout1MHz.Thebypasscapac-
However,ourfinalsystemdemodulatesandreadsouttheout- itor also has a large influence on the video bandwidth,butits
puts from each correlator separately. This enables individual valueshouldnotbetoolowsinceitactsasagroundfortheRF.
errorsinthecorrelatorlagpositionstobecorrectedbeforethe Undercertaincircumstances,thedetectorcouldoperateas
signalsarecombined.(seeSec.4.2). apeakdetector,wherealargevoltagepeakchargesuptheby-
Thefullschematicdrawingsoftherealandcomplexcorre- pass capacitor Cb and no more current can flow through the
latorsareshowninFigs.6aand6b.Thecomponentsforthese diode. This can be avoided if the detector is operated in the
correlatorsincludebroadband(6–12GHz)signalsplitters,slot- smallsignalregimewheretheoutputvoltageiscomparableto
line0◦and180◦phaseshifters,90◦microstripphaseshiftersfor thethermalvoltageVT.Inthisregime,thedetectorcircuitwill
the complex correlatorand the detector circuits, includingthe actasanintegrator.Additionally,whenoperatedinthatregime,
diodes.Ourdesignsofthe splitter andthe phaseshiftershave thedetector’soutputvoltageisproportionaltotheinputpower.
beendescribedinHoller&Jones(submitted). After testing several circuits the diode MSS20-146 from
MetelicswaschosentogetherwithR =10kΩandC =10pF.
L b
The results can be seen in Fig. 8. The passband is relatively
3. Detector
flat and rather upwardssloping, which is advantageous,since
The detector circuit is the core of the correlator. We use a mostothermicrowavecomponentswilladda slopeintheop-
diode power detector circuit (Fig. 7) comprising a Schottky posite direction. The features in the passband are very simi-
diodeoperatedinthesquarelawregion.ForV ≪ V ,where lar foralltested typesofdiodes,whichmeanstheycanbeat-
in T
V = k T /e ≃ 26mV is the thermal voltage at room tem- tributed to the matching circuit rather than the diode. A plot
T B 0
perature T , the outputvoltage is proportionalto inputpower of the diode response versus inputpower shows a square law
0
(Hewlett-PackardAN986). regionuptoabout30mV≃V andisconsistentwiththesim-
T
6 ChristianM.Holleretal.:A6–12GHzAnalogueLag-CorrelatorforRadioInterferometry
(a) Realcorrelatorschematiclayout (b) Complexcorrelatorschematiclayout
Fig.6.Thefullschematicofthereal(a)andthecomplex(b)correlatorsforasinglebaseline.Therealcorrelatorsamplesat16
lagsinstepsofδτ ,whilethecomplexcorrelatorsamplesat8lagsinstepsof2δτ .Ateachlagofthecomplexcorrelator,the
N N
signalissplitandonearmisphase-shiftedby90◦.Inbothcases,thereisa180◦ boardwithonearmphase-shiftedby180◦.0◦
phase shiftersof similar slotline designare used in correspondingpositionsto match the response with thatof the 180◦ phase
shifters.
ulatedmodel.Themeasuredvideobandwidthwasintherange so the impedance is complex as well as depending on input
of800–1100kHz. frequency.Theminimumabsolutevalueofthe impedancefor
ourfrequencybandcanbecalculatedtoabout100Ω.Tokeep
reflections and signal scatterings between channels to a min-
RF C R V imum, we required S to be low across the whole band. We
in b L out 11
used two element matching with a series microstrip line and
a shunt line. In addition, the high impedance of the diode is
Fig.7.Thepowerdetectorcircuit.Theforward(direct)current reduced through a 100Ω parallel lumped resistor at the cost
through the diode produces a voltage across the load resistor oflosingsensitivity.Theresistorwillbypasspartofthesignal
RL.ThebypasscapacitorCblookslikeashortfortheRFsignal. awayfromthediodewhichwillnotbedetected.Thecharacter-
isticsofthefinaldetectorcircuitareshowninFig.9.
(a) Detectorresponse (b) DetectorPassband
Fig.8.(a)Thedetectorresponseagainstpowerfora9GHzsig-
nal. The square law region extends to about −11dBm input
power.(b)Thepassbandofthedetector(includingthematch-
ingcircuit)from5to13GHz.Inbothplots,thesolidlinesare
Fig.9.A photographofthefinaldetectorcircuit(left) andthe
themeasurementsandthedashedlinesaresimulations.
result of the matching (right). The plots show the measured
inputreflection(blackcurve)togetherwithasimulation(gray
curve).
Matching the diode to the 50Ω microstrip circuit was an
importanttask, both to maximise power transfer to the detec-
torandtoavoidsettingupstandingwavesinthecorrelatorsig-
naldistributioncircuit.Thediodeincludesreactivecomponents
ChristianM.Holleretal.:A6–12GHzAnalogueLag-CorrelatorforRadioInterferometry 7
4. CorrelatorHardware 4.2.RFmeasurements
Thetransmissioncharacteristicsofthecorrelatorthroughtothe
4.1.Design
input to the diode detector have been plotted in Fig. 11. The
signalhascascadedthroughthe180◦slotlinephaseshifter,five
Inthefinaldesign,0◦ and180◦ correlationsforasinglebase- signal splitters and one signal combiner (and one arm of the
line (see Fig. 6) were implemented on two separate boards. 90◦ phaseshifterinthecaseofthecomplexboard).Alossless
Figures10aand10bshowthephysicallayoutofthe0◦realand systemwouldshowaflatresponseatabout−15dBm.Butmi-
180◦ complex boards. The designs show all the components; crostripintroducesloss,whichisproportionaltofrequencyand
splitters, phase shifters and the detector circuits. To combine causes the slope across the band. The slope can be corrected
thesignalsbeforethedetectors,weusedWilkinsondividersin for by inserting equalisers before the correlator input and is
reverse.Eachlaghasdifferentdelaylengths.Thelayoutswere thereforeofsmallconcerntous.Largeripplesacrosstheband
designedtoavoidrightanglebendsandtokeepcomponentsat wouldbeaproblem,sincetheyreducethesignal-to-noiseper-
a distancetotry tominimisepossible coupling.Thelayoutof formanceofthecorrelator.
the complex correlator was very similar to the real correlator.
Themostsignificantdifferencesarethe90◦ phaseshiftersand
differentdelaylengths.
Fig.11. Left: signal transmission through the correlator lags.
Right:correlatorinputreflection. The results for the Complex
andRealboardsareverysimilar.
The performance of the phase shifters inside the correla-
tor isshownin Fig. 12. The 180◦ slotline phaseshiftershows
an exceptional behaviour with a maximum error of ±6◦ over
(a) Prototype0◦RealCorrelator the bandof5–13GHz.The 90◦ microstripphaseshifter hasa
maximumerrorofabout±15◦.
(a) 180◦shifter (b) 90◦Phaseshifter
Fig.12.(a)Performanceofthe180◦phaseshifterinthesystem.
Fortheperformanceofthe90◦phaseshifter(b),thedifference
betweenthesetwocurveshastobetaken.
(b) Prototype180◦ComplexCorrelator
The frequency response, including all microstrip compo-
Fig.10.Thephysicallayoutofthe0◦ realcorrelatorboard(a)
nents,thedetectorcircuitandthevideobandbuffersofthepro-
andthe180◦ complexcorrelatorboard(b).Thecorresponding
totypesystemisshowninFig.13(a).Therearesomevariations
180◦ realand0◦ complexcorrelatorboardsarenotshown.For
acrossthebandbutitisrelativelyflatwithagentleslope.The
theseprototypeversions,onedetectorwasswappedfora50Ω
rising passbandis similar to the detector’spassband (Fig. 8b)
output to enable measurements on the network analyser. The
but there is a noticeable dip at 6 GHz. The structures in the
numbersontherealcorrelatorcorrespondtothelagorders.
passbandsoftheotherlagsdifferslightlyandtheaveragesig-
nallevelsvarybyafactorof2to3.Thereforecalibrationofthe
correlatorwillbenecessary.
8 ChristianM.Holleretal.:A6–12GHzAnalogueLag-CorrelatorforRadioInterferometry
analogue broadband correlators (Harris&Zmuidzinas 2001;
Robertsetal.submitted).
Thecausesofthelagerrorsareunclear.Thelagerrorsare
not consistent between boards so a design error is ruled out.
By replacinglumpedcomponents,we determinedthatthe lag
spacings are insensitive to the exact positions of the compo-
nents. Inspection of the boards showed no problems with the
printing or production of the board. A solder mask layer is
printedtoprotectthelinesandrestrictsolderflowingalongthe
tracks.Avariationinthislayercouldinfluencetheeffectivedi-
electricconstant.We manufacturedacorrelatorboardwithout
(a) Passband (b) Dispersion asoldermaskbutthelagerrorspersisted.Beforethethingold
layer can be plated on the tracks, the surface of the copper is
Fig.13.(a)TypicalpassbandofafulllagincludingallRFcom- roughened.Itis conceivablethat the roughenedsurface could
ponentsplusthedetectorcircuitandthetwofirststagesofam- increase the path length at high frequencies because most of
plification.Thisshowsadipat6GHzcomparedtotheresponse the current flows along the surfaces of the lines. To test this,
ofthedetectorinFig.8b.(b)Dispersionintheoutermostlags, wealsomanufacturedacorrelatorboardwithnogoldcoating.
whichhavethelargestdifferentialpathlength.Theslopeacross Reliable measurements for this board could not be made be-
thebandis±25◦. causeofpoorcontacttotheconnector.
Wefoundthatgroupsoflagshadsimilarlagerrorstoeach
other.Thisindicatedthatthesegroupsmayhavebeenaffected
by a long arm of the delay line shared by the whole group.
Besidesbeinglossy,particularlyathigherfrequencies,mi-
Variationsintherelativepermittivityofthedielectricǫ could
crostripalsoexhibitsdispersion.Foroursubstratewithǫ =9.7 r
r beacausebuttheerrorsinneighbouringlagsaretoolargetobe
aphysicallengthofonewavelength(1.0λ)at6GHzis2.065λ
explainedbythis.Variationsinǫ of±0.5wouldbenecessary
r
at 12GHz. However, this will have less influence on the cor-
to explain the average errors and only if the dielectric varia-
relatorthanonemightexpect.Thetwoopposingsignalsinthe
tions exactly followed the geometry of the board. This level
correlator travel through similar length of microstrip line so
ofvariationinǫ ismuchhigherthanthemanufacturingtoler-
r
dispersionis compensated.It is only the differentiallength of
ance. Whatever the cause may be, the lag spacings are stable
striplinethatproducesdispersionandtheworstaffectedarethe
overtime.Thisenablesustocorrectforthelagerrorsusinga
two outer lags. Figure 13(b) showsa measurementfor one of
calibrationscheme we have developedfor the broadbandcor-
these lags. It shows a phase slope of ±25◦ across the band of
relators.
6–12GHz,asexpectedfromthedistancetothezerolag.This
impliesthattheelectricalpositionofthepointofcorrelationis
afunctionoffrequency.Thiseffectwillbecompensatedbythe 5. Calibration
calibrationprocessthatwillbediscussedinSec.5.1.
5.1.Method
Theperformancesoftherealandcomplexcorrelatorswere
comparable.Theabilitytodirectlymeasuretheamplitudeand Theeightcomplexfrequencychannelscanberecoveredfrom
phaseateachlaginthecomplexcorrelatorisattractiveforsome the16lagdatabytakingadirectFouriertransform(DFT).The
commissioningtests.Butcalibratingforthesmallerrorsinthe DFT kernel can be convenientlyexpressed with the 16 by 16
90◦phaseshiftersoverthepassbandaddsanextralayerofcom- DFTmatrix
plexity.Fromapracticalviewpoint,themorecompactformat
W0 W0 W0 ... W0
of the real correlator fitted better into standard 6U racks. For
W0 W1 W2 ... WN−1
these reasons, we chose the real correlators for the final sys-
tpteihcmiypO.saicnteaedls.deAitsstoafnamcceeoabnsesutewrqeuemeenenncttehs,edtlhiadegnscorvotastrusi-ercndoomrrueutlcaahstiomenxopfreuecnttchetdain.oTnahnies- F=WW...00 WWN...2−1 WW2(...N4−1) ...... WW(N2−(1...N)(−N1−)1) (13)
whereWk =exp(−2πjk/N)andN =16.Theinputdataarethe
notuniformlysampledanda standardDFT cannotbe usedto
16voltagesmeasuredatthelags,writtenasacolumnvectorv .
recoverthe true spectrum. Instead, a means of calibrating the 0
Therecoveredspectrumisgivenby
correlatorhadtobefound.Theexactpositionsofthe correla-
tionscouldbemeasuredandthisinformationcouldbeusedto
s =Fv . (14)
0 0
estimatethetruespectrum.Thelagpositionerrorsonthepro-
totype board were apparently random with values of 5–10% Only the first eight of the elements in s are used because
0
of the ideal lag spacing. However, in the final production se- the remainingeightare relatedto the first eightelements(see
ries, the lag errors were found to be smaller than in the pro- Sec. 2.2). This opens up a way for speeding up the DFT by
totype boards. Nevertheless, they will cause some reduction adoptingan8by16matrix.Tokeepthingssimple,weproceed
in the SNR. Such lag errors have also been found in other withthefull16by16matrix.
ChristianM.Holleretal.:A6–12GHzAnalogueLag-CorrelatorforRadioInterferometry 9
DFT usually assumes uniform sampling but Sec. 4.2 Fouriertransformofthedetectors’responsesoverthefullcal-
showed that the sampling may be irregular by up to 10% be- ibrationrun.Theindividualspectraofthedetectorscanbeac-
causeoferrorsinthelagspacing.Intheabsenceofnoise,the countedforbygatheringthesemeasurementsintothespectrum
true spectrum can be recovered from non-uniformly sampled matrixS,thentherecoverymatrixisgivenby
data – see, for example Bagchi&Mitra (1998). We have de-
veloped a method that is particularly well suited for interfer- R=F−1SV−1. (20)
ometersandusesanastronomicalsourceforcalibration.Itcan
beshownthatthedegreeofcorrelatednoisebetweenthelagsis
ItcanbeseenfromtheDFT matrix(Eq.13) thatthefirstele-
minimisedwhenNyquist-sampled.Inacorrelatorwithlager-
mentofthespectrumvectors inEqs.(14)and(15)straddles
rors,the noise becomesmorecorrelated.Theoveralleffectof 0
thezero-frequencyband.Becauseofthechoiceoflagspacings,
non-uniformlysampleddataistodegradetheSNRwhenthere
thissub-bandcorrespondsto12GHzIF.Theinformationcon-
isnoise–see,forexampleBracewell(1999).Thesamplinger-
tentofthiszero-frequencybandisonlyhalfofwhatitshould
rors are calibrated by defining a recovery matrix R that will
be because its two halves are redundant. At the other end of
return the true spectrum s , given the non-uniformlysampled
0
the spectrum,thehighestfrequencysub-bandissplitbetween
lagdatav;
positive and negative frequencies. This causes its centre fre-
FRv=s . (15) quency to be ill-defined. These undesirable symptoms can be
0
avoided by shifting the sub-bands by half channel width us-
R can be thought of as a transformation that interpolates the
ing the shift theorem for DFT: Multiplying the lag data by a
non-uniformlysampled data v to the uniformly sampled data
setofcomplexfactorsshiftsthesub-bandsinfrequencyspace.
v thatwouldhavebeenmeasuredhadtherebeennolagerrors.
0 We define a diagonal matrix H whose diagonal elements are
R also corrects for variations in the gain of individual detec- (ǫ0,ǫ,ǫ2,···,ǫN−1),whereǫ =e−πj/N.Thischannelshiftmatrix
torsanddispersioninthemicrostripdelaylines.Thisissimilar
needstooperatebeforetheDFT.Thefinalrecoveredspectrum
totheschemeusedfortheWASP2 lagcorrelationspectrome-
isgivenby
ter(Harris&Zmuidzinas2001),exceptthatWASP2wascali-
bratedusingaCWsignaltodeterminetherecoverymatrixR.
s =FHRv. (21)
Becauseitisalaborioustasktocalibrateeachcorrelatorindi- 0
vidually,weuseanunresolvedastronomicalsourcetocalibrate
allcorrelatorssimultaneouslyinsitu.Inanidealcorrelator,the Note that H should not be applied duringthe calibration step
cross-correlationfunctionis boundby a sinc envelope.When (Eq.20)becausethenHwillbecancelledout.
Nyquist-sampled,thecentralpeakofthesincenvelopelieson
a lag and the nulls of the envelopes coincide with the other
lags(whenτ = τ).To formthe16 by16calibrationmatrix, Lags 1 − 8 Lags 9 − 16
g i
thepeakisalignedtothefirstlag.Thiswillgiveasetofmea-
surementslikev = (1,0...0)T, whereTis thetranspose.As
1
thesourceistracked,thepeakdriftsthroughindelayspaceand
whenitisalignedwiththesecondlag,thesecondmeasurement
v = (0,1,0...0)T willbetaken(Fig.14).Repeatingthispro-
2
cess and gathering these 16 column vectors of measurements
gives
V =(v v ··· v ). (16)
0 1 2 16
Intheidealcase, suchasthis,V istheidentitymatrixI. The
0
matrixS isthecorrespondingcollectionofidealspectrafrom
0
thefullcalibrationprocess.ThismeansthatS isequaltothe
0
Fouriertransformmatrix;
−0.2 −0.1 0 0.1 0.2 −0.2 −0.1 0 0.1 0.2
FI=S =F. (17) HA /hrs HA /hrs
0
Fig.14. Lag data during a calibration run on an astronomical
Inreality,therearegainvariations,dispersionandnon-uniform
source.Tocalibratethecorrelator,asourceistrackedwiththe
sampling. For small errors, V will be close to diagonal. We
path compensatorheld fixed. Unlike earlier plots which were
introducetherecoverymatrixRtofindtheidealspectrumS ,
0
plotted in instrument delay space (τ constant, τ variable in
g i
FRV=S0. (18) Eqn2),thesetime-streamdataplottedinhouranglewhichcor-
respondstoτ constant,τ variable,andhencethefringeperiod
SinceS ≡Ffrom(17), i g
0 isappropriatetotheRFfrequency(15GHz)ratherthantheIF
R=V−1. (19) frequency(9GHz).ThematrixV0(Eq.16)canbeconstructed
fromsuchadataset.
In the real system, the spectrum will not be rectangular. The
spectrum for each detector may be recovered by taking the
10 ChristianM.Holleretal.:A6–12GHzAnalogueLag-CorrelatorforRadioInterferometry
5.2.Simulations
Fringe function and detector positions
The calibration method was tested on simulated data with
1
analytically-generated astronomical fringes assuming a flat e
d
spectrum. The spectrum was evaluated for an ideal correlator plitu 0.5
with no sampling errors and also with up to 10 % lag errors. m 0
a
Figures15a and15b show thatthe calibratedspectraare sim- e
ilar to the true spectra, while the uncalibrated spectra deviate ng −0.5
Fri
noticeably.Thefullspectra,includingthenegativefrequencies, −1
−9−8−7−6−5−4−3−2−1 0 1 2 3 4 5 6 7 8 9
have been shown here. The two halves of the spectrum with
Lag numbers
channelshift(Fig.15b)aresymmetricandredundant. Uncalibrated Spec. (amp) Calibrated Spec. (amp)
The recovered spectra shown in Fig. 15 are not flat, de-
Uncalib Calibrated
spitetheflatband-limitedpassbandthatwentintomodelingthe 0.4 0.4
e Ideal e Ideal
cross-correlationfunction.Furthermore,theripplesacrossthe ud 0.3 ud 0.3
spectrumchangeasthesourcedriftsthroughthesky.Thisisil- mplit 0.2 mplit 0.2
lustratedinFig.16.Fig.16ashowsasimulationofaprocessed A A
0.1 0.1
data stream from a 5m east-west baseline with a single point
0 0
sourceatthecentreofthefieldofview.Afterfringe-rotation3 1 9 16 1 9 16
andstandardphasecalibration,weexpectconstantamplitudes
Uncalibrated Spec. (phase) Calibrated Spec. (phase)
andzerophaseinallsub-bands.Insteadweseeperiodiccycles 180 180
in both amplitude and phase at the fringe rate for each sub-
g g
band. The magnitude of the cycles are worse for a correlator de de
with lag errors. Calibration does reduce the magnitude of the e / 0 e / 0
s s
a a
cycles (Fig. 16b) but it remains noticeably higher than for a h h
P P
correlatorwithnolagerrors(compareFig.16awithFig.16b).
−180 −180
Thecyclesarisefromaliasing.Becausewecanonlysample 1 9 16 1 9 16
thecross-correlationfunctionoversomefiniterangeofdelays, Freq channel Freq channel
the recovered spectrum is a convolution of the true spectrum
with the spectral response of the window function. But our
(a) Nochannelshift
signal is critically sampled, so the spectrum lies shoulder-to-
shoulderwith its aliased images (see Fig. 17). The convolved
Uncalibrated Spec. (amp) Calibrated Spec. (amp)
wingsofthe imagesextendbeyondthe 6GHzbandwidthand
aliasthesignal’sspectrum.Thisiswhytheedgesub-bandsare Uncalib Calibrated
0.4 0.4
most affected. Simulations show that narrowing the passband de Ideal de Ideal
u u
bandwidth by one sub-bandreduces these alias cycles, as ex- mplit 0.2 mplit 0.2
pected.
A A
Applyinga taperedwindowfunctionslike Hammingwin-
0 0
dowbeforetheDFTmakesthealiascyclesworse.Theresponse 1 9 16 1 9 16
functionofthe Hammingwindowhasa widerlobecompared
Uncalibrated Spec. (phase) Calibrated Spec. (phase)
to a rectangularwindow(i.e.applyingnowindowing).So the 180 180
recoveredspectrum will have wider convolvedwings and the
g g
alias cycles will be worse. However the Hamming window e e
d d
willsuppressspectralleakagebetweenchannelsbecauseithas e / 0 e / 0
s s
lower sidelobes(−43dB)comparedto a rectangularwindows ha ha
P P
(sidelobesupto−13dB).ApplyingtheHammingwindowre-
−180 −180
ducesaliascyclesinthecentralsub-bandsbutthebenefitsare 1 9 16 1 9 16
marginal(compareFigs.16aand18). Freq channel Freq channel
Window functions could be beneficial in correlators that (b) Withchannelshift
oversamplethe cross-correlationfunction.Forexample,ifthe
6–12GHz signal is oversampled at 24GHz , half the sub- Fig.15.Calibrationcanimprovetherecoveredspectrumsothat
bandsbetween0–6GHzwill haveno signal.But these empty itisclosertowhatwouldhavebeenmeasuredbyanidealcor-
relator with no lag errors. Here the lag errors are up to 10%
3 In interferometers with independently-mounted antennas, the ofthelagspacings.(a)Thetopplotistheresponseofapoint
phase centre drifts as the source is tracked. This changes the phase sourceininstrumentdelayspace.Theopencirclesarethevolt-
ofeachsub-bandataconstantrate.Therateisdeterminedbythege- agesmeasuredateachlag.Thelowerplotsaretheamplitudes
ometry of the telescope and the frequency of the sub-band. As we
and phasesof the recoveredspectra withoutand with calibra-
knowthisrate,thevaryingphaseofthesub-bandscanbemadecon-
tion.In(b),thechannelswereshifted.Theeffectofthisismost
stantbyaprocesscalledfringe-rotation.Thisisequivalenttostopping
dramaticinthephase.
thefringesindelayspace.
