安捷伦射频脉冲的频谱与信号分析.docx

安捷伦射频脉冲的频谱与信号分析.docx

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安捷伦射频脉冲的频谱与信号分析

PropertiesofTransformsTimeoperationMultiplicationf（tg（tFrequencyoperationConvolution1F（sg（ω−sds2π−∫∞∞SignificanceThespectrumoftheproductoftwotimefunctionsistheconvolutionoftheirspectra.Thisisthemoregeneralstatementofthemodulationproperty.Forexample,samplingasignalisequivalenttomultiplyingitbyaregulartrainofunitareaimpulses.Thespectrumofthesampledsignalconsistsoftheoriginalsignalspectrumrepeatedabouteachcomponentofthe（linespectrumofthetrainofimpulses（seepair6S,page24.Fornooverlap,thehighestfrequencyinthesignaltobesampledmustbelessthanhalfthesamplingfrequency.Ifthisistrue,originalsignalspectrum（hencesignalcanberecoveredwithalowpassfilter（samplingtheorem.Thespectrumofthenthderivativeofafunctionis（iωntimesthespectrumofthefunction.A“differentiatingnetwork”has（overtheappropriatefrequencyrangeatransmissionKpDifferentiationdnf（tdtnMultiplicationbyppnF（pThus,theoutputwaveisproportionaltothederivativeoftheinput.IntegrationMultiplicationby1pω0whereKisdimensionlessorhasthedimensionsofimpedanceoradmittance.∫∫∫−∞f（τ（dτn123ntThespectrumofthenthintegralofafunctionis（iω−ntimesthespectrumofthefunction.Thus,theresponseofanyfiltertoastepfunctionistheintegralofitsimpulseresponse.An“integratingnetwork”has（overtheappropriatefrequencyrangeatransmissionKω0p1F（ppn,whereKisdimensionlessorhasthedimensionsofimpedanceoradmittance.Thus,theoutputisproportionaltotheintegralofthepastoftheinput.26

AppendixB.IFAmplifierResponseandDistortionIFamplifierresponseMentionwasmadeinthetestofthephenomenonofdecreasedsensitivityandresolutionthatresultswhenaCWsignalissweptbytheIFamplifieratahighratecomparedtothebandwidthsquared.AssumingaGaussianresponsefortheamplifier,theresultingtransientcanbedeterminedasfollows:

AsweepfrequencysignalasillustratedinFigureB-1canberepresentedbySubstitutingbackforτandsimplifyingTδ21−js12πFs1=y（tδ2t2exp−22Tδ22πFs1−j1+s2TsδFπ2sFs=（texp（jπTst2s（B-1usingpair10ofAppendixA（B-61S（ω=τ2πexp−（τω22Theenvelopeofy（tisthen14jTsτ=where（B-2y（t=2πFsIfweassumeaGaussianresponse,2πFs1+Tδ2s21δ2t2exp−22Tδ2s1+2πFs（B-71ω2ωexp−H（=2δtheproductofS（ωH（ωgivesNotethatforlowsweeprates（B-3Ts1?

22πFsδ12πFs−y=（texp2Tδs22t11Y（ω=S（ωH（ω=τ2πexp−τ2+2ω22δ（B-4（B-8Theoutputtransientistheinversetransformofthisfunction,againusingpair101τ2y（texp−=2τ2+11τ2+2δ2δτThis,aswasstatedearlier,isaplotofthefrequencyresponseoftheIFamplifier.（B-5fFsweepwidthslope=s=TssweeptimetFigureB-1.Asweepfrequencysignal27

DistortionIftheconditionon（B-8isnotsatisfied,theresultingtransientwillbealteredinbothwidth（timedurationandamplitude.Thereductioninamplitudewillbe1α=22πFs41+（B-9Tδ2sπBwhereBisthe3dBbandwidth,Notingthatδ=ln（2Inalikemanner,the3dBbandwidthofthefunction（B-7isδT2ln（21+s2πFδπs=∆t'Theratioofthesetimesis2（B-122πFs∆t'=1+Tδ2∆ts2（B-13α=1222ln（2Fs41+（B-102πTsBAplotofthisfunctionindBversus–Fs/（TsB2isincludedasFigureB-2.Ifwesolveforthe3dBtimedurationΔtfromequation（B-8bysettingαto1/√2andsolvingfortheappropriateΔt,wegetThisistheratiooftheeffectiveresolvingbandwidthofaspectrumanalyzertothebandwidthoftheIFamplifierasafunctionofsweeprate.Rewrittenintermsof3dBbandwidthB.Beff2ln（2=1+Bπ2FsTB2s2（B-14ThisfunctionisplottedinFigureB-2.Fs=SweepwidthTs=SweeptimeB=3dBIFbandwidthBeff=Effectivebandwidthln（2δTs∆t=πFs（B-111000Lossinamplitudeandsensitivitya（dB0510a202530350.11.010NormalizedsweeprateBeffB101Fs1001000TsB2FigureB-2.Sensitivitylossandnormalizedeffectivebandwidthvs.normalizedsweeprate28Normalizedeffectivebandwidth100BeffB

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January6,2012*0.125€/minuteProductspecificationsanddescriptionsinthisdocumentsubjecttochangewithoutnotice.©AgilentTechnologies,Inc.2012PublishedinUSA,July5,20125952-1039