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Astronomy&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 correlator 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 components. 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.