Exact Signal Measurements using FFT Analysis

1 ExactSignalMeasurementsusingFFTA nalysisStefanSchollMicro electronicSystemsDesignResearchGroupTUKa iserslautern,Germany1 Intro ductionandmotivationThistutorialdescrib eshowtoaccuratelymeasuresignalp owersinthetimedomain,thisisnotalwaysappl icable:Ifasignalcontainsseveralsp ec-tralcomp onentsandbroadbandnoise,consideringthefr equencydomainallowstomeasurep owerofthesecomp onentsindividuallyortomeasureSNR(bysepar atelyconsideringsignalandnoise).However, obtainingaccuratep owernumb ersafterhavingp erformedanFFTisnotstraightforward,b ecauseseverale ectsintro duceerrorsduringFFTpro cessing, erdescrib esthedi erente ectsandexplainshowtheycanb eavoidedorcomp ,itwillb eexplainedhowtodoaccuratemeasurementsofs ignalandnoisep owerusingtheFFTsp ,somebasicsonFFTforrealvaluedsignals(ast heyfrequentlyo ccurinrealworld) errealvaluedtimedomainsignalsareassumed, forwhichaNp ointFFTisusedtotransformitintothep owersp ectrumwithbinspacing f= ointFFTtheMatlabalgorithm1canb ,aftertakingtheFFT,itsmagnitudeiscalcula tedandthebinsarescaledby1 ectrumismirrored,the rsthalfofN/2binscontainsallnecessaryinfo rmationonthesp ectrum,thesecondhalfcanb ,theremainingbinsarescaledbyafactorof2,e xceptthe( rst)

2 ,thatforrealsignalsaNp ointFFTiscalculated,butthesp : N real time domain samples (time_signal)twosided_fft = (1/N) * abs( fft(time_signal,N) ); % do fftonesided_fft(1) = twosided_fft(1); % copy DC binonesided_fft(2:N/2) = 2 * twosided_fft(2:N/2); % double other binspower_fft = onesided_fft.

3 ^2; % convert from RMS to poweroutput: N/2 point power spectrumIfthetimesignalrepresentsavoltag eoveraresistorR, econvertedtodBmonalogarithmicscalewithP[ dBm] = 10 log10(p ower_ tR 1mW) (x)x 10log(x/ref)time domainsignalcomplex FFTbinsRMSspectrumpower spectrumpower spectrumin :FromcomplexFFToutputtothep owersp ectrumindBmFurtherinformationonthebasics ofFFTcanb efoundin[1].Alternativemeansofcalculatin gthesignalp owerinthetimedomaincanb efoundinApp ectsintro ducingerrors33E ectsintro ducingerrorsInthefollowingalle ectsthatin uenceamplitudemeasurementsusingFFTaredes crib ectsonlyapplytonarrowbandsignals( ),otherstobroadbandnoiseandsometob ectralleakageisthee ect,thattheenergyofthesignalisdistribute d(smeared) eak,butmorelikeabroadbump, ectralleakageiso ccurringb ecauseofthefact,thattheFFTanalyzesonlya( short)sliceofasignal(Nsamples).

4 Fromthissliceweusuallywanttoextractinfor mationab ectrumofatheoreticalsignal,thatiscomp osedofin niterep ectrumoutputbytheFFTdo esnotexactlyrepresenttheoneofthe true ,whosep erio ds(oramultipleofap erio d)coincidentallyorintentionally texactlyintheslice,leakagee ectsdonoto eachieved,bycouplingthesignaltob emeasuredtotheanalog-to-digitalconverter 'ssampleclo ossibleweassumeinthefollowingthatleakage o ,windowscanb eused,thatareappliedtothetimedomainsampl esb (forgeneralpurp ose),the attopwindow(foraccurateamplitudemeasurem ent)ornowindow, ,atall(fornoisemeasurements), ,thatisdisplayedusingawindowcanb emeasuredbyhighestsidelobeindBandfal lo indB/o ctave(moreinformationin[2]).Leakageisnot aproblemforaccuratemeasurements,aslongas leakagedo esnotmasksp ectralcomp onents, 107 140 120 100 80 60 40 200256 point FFT, leakage effects, 50 MHz Signal with 0 dBm, fs=250 MHzfrequency in Hzpower in dBm no windowhann windowflattop windowblackman windowtrue :Truesp ectrumandtheleakagee ectofFFTfordi erentwindows3E ectsintro owergainIfawindowisapplied,itreducesthea mplitudeofthetimedomainsignal,esp eciallyattheleftandrightb ordersofthewindow, 10 7 1 domain: no windowing (rectangle)time in samplitude in volts012345x 10 7 1 domain: hann windowtime in samplitude in :Thereasonforcoherentp owergain:amplitudereductionduetowindowin gThisreductionofamplitudeintro ducesanamplitudeerror,termedcoherentpowe rgain(CPG).

5 Theword gain mayb emisleading,sinceCPGactuallydescrib esalossinsignalp xed, efoundin[3].SimplyaddthisgaintotheFFTout puttocomp ensateforthep (rectanglewindow)isused,thereisnop owerlossandthecoherentp in Hz#107-40-35-30-25-20-15-10-50power in dBm512 point FFT, coherent power gain, Signal power : 0 dBmno/rectanglehannflattopblackman~13 :Theimpactofcoherentp owergain(CPG):trueamplitudeofthesinesign alis0dBm,errorsforthedi erentwindowscanb eclearlyobserved(herefrequencywasadjuste dtoavoidleakage).Forthe attopwindowCPGcanb ereadasapproximately13dB( ).3E ectsintro cessesdigitaldata,whichisbyde nitiondiscreteb ,thedisplayedp owerlevelisreducedb ecausethesignalp ethoughtassamplingthecorresp ondingcontinuoussp endentonwhichfrequenciesexactlythecontin uoussp ectrumissampled,theFFTsp ectrumlo oksdi erent,ascanb ect,b ecausethesamplingissimilartoviewtheconti nuoussp erentcases:ontheleft, etweentwobins:signalp ertyofscallopinglossisthefact,thatitcann otb edescrib edbya eusedasasimplescalingfactortocomp erentfordi erentsp ectralcomp endsonsignalfrequency,samplerateandthenu mb ,onecansp ecifyaworstcaseloss,whicho ccurs,ifasignalfrequencyfallsexactlyhalf -wayb ,therearetwowaystoavoidscallopingloss.

6 Usea attopwindow,whichexhibitsonlyverylittlem aximumscallopingloss(< ) Couplethesignalfrequencyprop erlywiththesamplerate,whichisthesamemeth o dusedtoavoidleakage( ).024681012frequency in Hz#107-50-40-30-20-100power in dBmsmall scalloping loss, Signal power : 0dBm16 point FFTtrue spectrum (here including leakage)024681012frequency in Hz#107-50-40-30-20-100power in dBmlarge scalloping loss, Signal power : 0 dBm16 point FFTtrue spectrum (here including leakage) :Scallopingloss,thato ccurswhensignalfrequencyfallsb etweentwobins,herenowindow-ingwasusedtoa voidcoherentp owergain3E ectsintro cessinggainPro cessinggainreducesthedisplayednoise o orandcanb eexplainedasfollows:FFTpro cessingcanb eseenassendingatimesignalthroughabankofN lters,eachwithbandwidth f(binspacing)anddeterminingthep oweratevery eroffrequencybinsorN(or ltersresp ectively)isincreased,the ltersb ecomenarrowerandthep owerateachbin(p oweratthe lteroutput )b ecomessmaller, (PG)exactlydescrib , fishalvedandthedisplayednoise o ectsp owermeasurementsofbroadbandsignals, o orinanFFTplotisthereforedisplayedlower(b ythepro cessinggain) cessinggaincanb ecalculatedbyPG[dB] = 10 log10(N2)andcanb eaddedtothelevelofnoise o ortocomp ensatethise o oraccuratelytheresultsofseveralFFTscanb eaveragedtoreduceamplitude uctuations( ,right).

7 024681012frequency in Hz#107-100-80-60-40-200power in dBmnoisy sine Signal with different FFT sizes (hann window)256 point1024 point65k point024681012frequency in Hz#107-100-80-60-40-200power in dBmwith 500 FFTs averaged256 point1024 point65k :Pro cessinggain:Thereductionofthedisplayedno ise o uctuationsmultipleFFTshaveb eenaveragedtoenableaclearreadofthenoise o endentontheappliedwindowtheamountofnoise thatisaccumulatedinonefrequencybinvaries ,basedonitscharacteristicequivalentnoise bandwidth(ENBW)[3,4].Thise ectsleadstoanincreaseddisplayednoise o or, ectacorrectionfactorneedstob esubtractedfromthenoise o ortocomp ensateforthelargerequivalent ecalculatedfromtheENBW foreverywindow, er,ifnoise o ectsintro 107 150 100 500 Overview of errors in FFTfrequency in Hzpower in dBmcoherent power gain& scalloping lossequivalent noise bandwidth& processing :Overviewofallerrors,thato ccurduringFFTpro cessing(hereanoisysinesignalwith0dBm,han nwindow) :Overviewoferrorsintro ducedduringFFTpro cessingErrore ectSourceSinesignalerrorsNoise o orerrorsLeakage nitelengthofFFTsp ectralpuritynegligibleCPGwindowingamplit udeyesScallopingLossdiscretizationamplit ude(variable)noPro cessingGainFFT lterbankprop.

8 Overviewoverdi erentwindowsandtheirparameters,formorewi ndowssee[3]WindowHighestsidelob eFallo (p ero ctave)CPGS callopingloss(max)ENBW correctionNo(rectangle) edoneusingtheFFT,iftheerrorsdescrib edab oveareprop edescrib edseparatelysincetheyaresub jecttodi erenterrore (bandwidthsmallerthanthebinspacing f)showupasp eaksinthesp owerofnarrowbandsignalsPtruecanb ecalculatedfromthedisplayedp owerPdispl(=p owerofsp ectrump eak)byPtrue[dBm] =Pdispl[dBm] +CPG[dB] +scalloping loss[dB]( )Coherentp owergaincanb , ,0dBcanb eassumed,ifthesignalenergyisequallydistr ibutedamongtwobins,cho ecenteredonasinglebin,theeasiestwayistou sea attopwindow,whichexhibitsalmostnoscallop ingloss(always< ).Forfurtherinformationonthe attopwindowanditscomputationallye cientusage(ifrequired)see[5]. olationmetho dscanb eapplied,thatimprovefrequencyresolutionw ithoutusingmoreFFTp oints[1].

9 Eshavetob edistinguished:whitenoise, owerdensityoverthewholefrequencyband, o or(whitenoise)Thismetho disapplicabletowhitenoise,wherethetrueno isep owerPtruecanb ecalculatedfromthedisplayednoise o orPfloorbyusing:Ptrue[dBm] =Pfloor[dBm] +CPG[dB] +PG[dB] ENBW corr[dB]( )Foranaccuratemeasurementthedisplayednoi se o orhastob ,duetorandomprop ertiesofnoisethismayb edi owerlevelofthenoise o or,therearetwosolutions: FFTaveraging:averagemultipleFFTs(linears caledp owersp ectrum)toreducetherandom-nessofthenoise o or( ) Binaveraging:averagethebinsofthelinearsc aledp owersp ectrumofasingleFFT(excludebinsrepresenti ngDCandotherunwantedcomp onents),asdescrib (arbitrarynoiseandwidebandsignals)Summin gofFFTbinsisanelegantwaytomeasurethep o orbymeansdescrib edab ove,frequencybins(FFT(i),p owersp ectrum)ofinterestthatcontainsignalcomp onentstob emeasured, ,thatthesummationhastob edoneusinglinearvalues(notdBs)andthatnoc orrectionofpro cessinggainisrequired,sinceitisinherentl yconsideredbythesummationpro [dBm] = 10 log10( iFFT(i)[linear])+CPG[dB] ENBW corr[dB]Theadvantageofthismetho dis,thatthebinsthatcontributetothemeasur ement,andthereforethesp ectralcomp onents,canb echosenfreely,whichallowsforvery eenused,sothedisplayedp owerofthesinesignalcanb ereadout.

10 Pdispl= owerofthesinesignalisPtrue= + + 0dB= (CPGis6dB,forthescallopinglosszerohasb eenassumedsincethep eakfallsnearlyexactlyonasinglebin).Theno ise o orcanb :Ptrue= + + (CPGisagain6dB,pro cessinggainfora1024p ,correctionfactorforENBW isapprox. ) 107 25 20 15 10 505101024 point FFT (averaged over 1000 FFTs), window: hann, noisy sine signalFrequency [Hz] power [dBm]Pdispl = dBmPfloor = : , ;thedisplayednoise o ondstoatotalnoisep owerinthetimedomain10 References[1] , ,thirded.,2011.[2] , TheFundamentalsofFFT-BasedSignalAnalysis andMeasurement, NationalInstruments,ApplicationNote041,2 000.[3] , Ontheuseofwindowsforharmonicanalysiswith thediscreteFouriertransform, inProceedingsoftheIEEE, ,1978/2005.[4] FFTW indowFunction,LimitsonFFTA nalysis, BoresSignalProcessing.