1 The DFT The Discrete ourier ransform DFT is the equi alent of the continuous ourier ransform for signals kno wn only at instants separated by sample times ie 64257nite sequence of data Let be the continuous signal which is the source of the d ID: 23810
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Lecture7-TheDiscreteFourierTransform7.1TheDFTTheDiscreteFourierTransform(DFT)istheequivalentofthecontinuousFourierTransformforsignalsknownonlyat instantsseparatedbysampletimes(i.e.anitesequenceofdata).Letbethecontinuoussignalwhichisthesourceofthedata.Let samplesbedenoted\n \r \r \r \r \n \r.TheFourierTransformoftheoriginalsignal,,wouldbe "!$#%'&(*)+),.-+/102,3Wecouldregardeachsample \rasanimpulsehavingarea 45\r.Then,sincetheintegrandexistsonlyatthesamplepoints: 6!$#%7&(98;:�+=@-+/A02B3& C \rD-+/EGF \rD-+/10H?IFF \rD-+/10J?IF H-+/A08;:�+=ie. "!$#%K&:+=LJNMOE \rD-+/10J?Wecouldinprincipleevaluatethisforany#,butwithonly datapointstostartwith,only naloutputswillbesignicant.YoumayrememberthatthecontinuousFouriertransformcouldbeevaluatedoveraniteinterval(usuallythefundamentalperiod@)ratherthanfrom QPto82 FPifthewaveformwasperiodic.Similarly,sincethereareonlyanitenumberofinputdatapoints,theDFTtreatsthedataasifitwereperiodic(i.e. to Histhesameas Rto S H.)HencethesequenceshownbelowinFig.7.1(a)isconsideredtobeoneperiodoftheperiodicsequenceinplot(b).0123456789101100.20.40.60.81(a)05101520253000.20.40.60.81(b)Figure7.1:(a)Sequenceof &TU samples.(b)implicitperiodicityinDFT.Sincetheoperationtreatsthedataasifitwereperiodic,weevaluatetheDFTequationforthefundamentalfrequency(onecyclepersequence,:?Hz,VXW:?rad/sec.)anditsharmonics(notforgettingthed.c.component(oraverage)at#Y&Z ).i.e.set#Y&Z [H\ \ S]^\ ]`_\ S] Uor,ingeneral _\r&:+=LJaMOE \rD-+/$bdcegfJ_&Z ih S H83 _\ristheDiscreteFourierTransformofthesequence \r.Wemaywritethisequationinmatrixformas:jkkkkkl C \r \r \r... S \rmnnnnno&jkkkkkkklppVprqTp:+=pVptspruTp:+VpvqpuprwTp:+q...p:+=p:+Vp:+qpmnnnnnnnojkkkkkl C \r \r \r... \rmnnnnnowherep&Zxy[z{ !\}| andp&ZpV:etc.&~.DFTexampleLetthecontinuoussignalbeK&dcF $U H\ @1HzF O\ 2Hz012345678910-4-20246810Figure7.2:ExamplesignalforDFT.Letussampleat4timespersecond(ie.=4Hz)from&r to&qs.Thevaluesofthediscretesamplesaregivenby: \r&tF$5WV @F$\{byputting&r&Js84 i.e.\n \r&t, \r&, \r&t, \r&Z , &ZRTherefore _\r&qLE \rd-+/cbfJ&qLJNMOE \rA !fJjkkl C \r \r \r \rmnno&jkkl ! ! ! !mnnojkkl C \r \r \r \rmnno&jkkl$ !RH!RmnnoThemagnitudeoftheDFTcoefcientsisshownbelowinFig.7.3.012305101520f (Hz)|F[n]|Figure7.3:DFToffourpointsequence.InverseDiscreteFourierTransformTheinversetransformof _\r&:+=LJaMOE \rd-+/bdcegfJ85 is\n 4\r& :+=LfMOE _\rD-U/bdcefJi.e.theinversematrixis:timesthecomplexconjugateoftheoriginal(symmet-ric)matrix.Notethatthe _\rcoefcientsarecomplex.Wecanassumethatthe\n 4\rvaluesarereal(thisisthesimplestcase;therearesituations(e.g.radar)inwhichtwoinputs,ateach,aretreatedasacomplexpair,sincetheyaretheoutputsfrom oand odemodulators).Intheprocessoftakingtheinversetransformtheterms _\rand _\r(re-memberthatthespectrumissymmetricalabout:V)combinetoproducefre-quencycomponents,onlyoneofwhichisconsideredtobevalid(theoneatthelowerofthetwofrequencies,_*]VXW?Hzwhere_ :V;thehigherfrequencycomponentisatanaliasingfrequency(_¡:V)).Fromtheinversetransformformula,thecontributionto \rof _\rand ¢ _\ris:f \r& £ _\rD-/bdcegfJ¤F _\rD-/bdce8;:+f�"J¥(7.2)Forall \rreal S _\r&:+=LJaMOE \rD-+/bdce8:+f�¦JBut-+/bdce8:+f�¦J&-+/VXWJ1forall-/$bdc¨§eJ&Z-/$bdcegfJi.e. S _\r& ª©_(i.e.thecomplexconjugate)86 SubstitutingintotheEquationforf \rabovegives,f 4\r& £ _\rd-/bdcegfJ¤F ©_.-+/bdcegfJH¥since-/VXWJ&Tie.f \r& £«¬£ _\r¥ \ _ ¦®£ _\r¥¨¯6°\ _¥orf \r& T± _\r± £\ _F²$³¨´ _\rd¥i.e.asampledsinewaveatVXWf:?Hz,ofmagnitudeV:± _\r±Forthespecialcaseof_&µ , \r&S¶·\n 4\r(i.e.sumofallsamples)andthecontributionof C \rto\n 4\risE \r¤&: \r&averageof \r¤&d.c.compo-nent.Interpretationofexample1. C \r&r$ impliesad.c.valueof: C \r&VEs&r(asexpected)2. \r{& !R¸& ª© \rimpliesafundamentalcomponentofpeakamplitudeV:± ¦\r±&Vs]ª&rwithphasegivenby²³¨´ \r& ¹ oi.e. \ o¤&tK U\ Y o(asexpected)3. \r&ºH_&:Vnoother S _componenthere)andthisimpliesacomponentV \r& \rD-/bdce¼»VJ& 4H\rd-/WJ&\{(asexpected)since¨¯D°\{i&t forall87 01230123456f (Hz)|F[n]|sqrt(2)3/sqrt(2)Figure7.4:DFToffourpointsignal.Thus,theconventionalwayofdisplayingaspectrumisnotasshowninFig.7.3butasshowninFig.7.4(obviously,theinformationcontentisthesame):Intypicalapplications, ismuchgreaterthan;forexample,for &½U R, _\rhasU components,but^ U Rarethecomplexconjugatesof^$ ,leaving¾ ¿E ÀVsasthed.c.component,VVs{Á¾ ¿ÁÂVtoVVsGÁ¾ ¿ÃÁÂVascompletea.c.com-ponentsandVs¾ ¿ÃVÀÂVasthecosine-onlycomponentatthehighestdistinguishablefrequency_&:V.MostcomputerprogrammesevaluateÁ¾ ¿fÀÁ:(orÁ¾ ¿fÀÁb:forthepowerspectralden-sity)whichgivesthecorrectshapeforthespectrum,exceptforthevaluesat_&Z and:V.7.2DiscreteFourierTransformErrorsTowhatdegreedoestheDFTapproximatetheFouriertransformofthefunctionunderlyingthedata?ClearlytheDFTisonlyanapproximationsinceitprovidesonlyforanitesetoffrequencies.Buthowcorrectarethesediscretevaluesthemselves?TherearetwomaintypesofDFTerrors:aliasingandleakage:88 7.2.1AliasingThisisanothermanifestationofthephenomenonwhichwehavenowencounteredseveraltimes.Iftheinitialsamplesarenotsufcientlycloselyspacedtorepresenthigh-frequencycomponentspresentintheunderlyingfunction,thentheDFTval-ueswillbecorruptedbyaliasing.Asbefore,thesolutioniseithertoincreasethesamplingrate(ifpossible)ortopre-lterthesignalinordertominimiseitshigh-frequencyspectralcontent.7.2.2LeakageRecallthatthecontinuousFouriertransformofaperiodicwaveformrequirestheintegrationtobeperformedovertheinterval-PtoFPoroveranintegernumberofcyclesofthewaveform.IfweattempttocompletetheDFToveranon-integernumberofcyclesoftheinputsignal,thenwemightexpectthetransformtobecorruptedinsomeway.Thisisindeedthecase,aswillnowbeshown.Considerthecaseofaninputsignalwhichisasinusoidwithafractionalnum-berofcyclesinthe datasamples.TheDFTforthiscase(for_&Ä to_&:V)isshownbelowin7.5.0246802468freq|F[n]|Figure7.5:Leakage.WemighthaveexpectedtheDFTtogiveanoutputatjustthequantisedfrequen-89 cieseithersideofthetruefrequency.Thiscertainlydoeshappenbutwealsondnon-zerooutputsatallotherfrequencies.Thissmearingeffect,whichisknownasleakage,arisesbecauseweareeffectivelycalculatingtheFourierseriesforthewaveforminFig.7.6,whichhasmajordiscontinuities,henceotherfrequencycomponents.05101520253035404550-1-0.500.51Figure7.6:Leakage.Therepeatingwaveformhasdiscontinuities.Mostsequencesofrealdataaremuchmorecomplicatedthanthesinusoidalse-quencesthatwehavesofarconsideredandsoitwillnotbepossibletoavoidin-troducingdiscontinuitieswhenusinganitenumberofpointsfromthesequenceinordertocalculatetheDFT.ThesolutionistouseoneofthewindowfunctionswhichweencounteredinthedesignofFIRlters(e.g.theHammingorHanningwindows).Thesewindowfunctionstaperthesamplestowardszerovaluesatbothendpoints,andsothereisnodiscontinuity(orverylittle,inthecaseoftheHanningwindow)withahypotheticalnextperiod.Hencetheleakageofspectralcontentawayfromitscorrectlocationismuchreduced,asinFig7.7.0246801234567(a)02468012345(b)Figure7.7:LeakageisreducedusingaHanningwindow.90 7.3TheFastFourierTransformThetimetakentoevaluateaDFTonadigitalcomputerdependsprincipallyonthenumberofmultiplicationsinvolved,sincethesearetheslowestoperations.WiththeDFT,thisnumberisdirectlyrelatedto V(matrixmultiplicationofavector),where isthelengthofthetransform.Formostproblems, ischosentobeatleast256inordertogetareasonableapproximationforthespectrumofthesequenceunderconsiderationhencecomputationalspeedbecomesamajorcon-sideration.HighlyefcientcomputeralgorithmsforestimatingDiscreteFourierTrans-formshavebeendevelopedsincethemid-60's.TheseareknownasFastFourierTransform(FFT)algorithmsandtheyrelyonthefactthatthestandardDFTin-volvesalotofredundantcalculations:Re-writing _\r&:+=LJaMOE \rD-+/bdcefJas _\r&:+=LJNMOE \rdpfJ:itiseasytorealisethatthesamevaluesofpfJ:arecalculatedmanytimesasthecomputationproceeds.Firstly,theintegerproduct_repeatsfordifferentcom-binationsofand_;secondly,pfJ:isaperiodicfunctionwithonly distinctvalues.Forexample,consider &9(theFFTissimplestbyfarif isanintegralpowerof2)pÅ&t-+/bdcÆ&t-+/sÃXÇ& !È&ZÉ=sayThenÉV& !ÉRq& !5Éi& É©És& ÉÃ& ÉÉu&Ê!É5Ë%&Ê!5ÉÌ&9É©ÉÅ&~Fromtheabove,itcanbeseenthat:91 ptsÅ& pEÅpÃÅ& pÅpuÅ& pVÅpvËÅ& pvqÅAlso,if_fallsoutsidetherange0-7,westillgetoneoftheabovevalues:eg.if_&tandi&rÍ^pqÃÅ&tÉqÃ&ºÉÅsÎÉq&tÉq7.3.1Decimation-in-timealgorithmLetusbeginbysplittingthesinglesummationover samplesinto2summations,eachwith:Vsamples,oneforevenandtheotherforodd.SubstituteÏS&JVforevenandÏS&J+=Vforoddandwrite: _\r&eb+=LÐMOE\n 4ϸ\rdpVÐf:Feb+=LÐMOE ÏF\rdp8VÐ�f:NotethatpVÐf:&t-+/bdce8VÐf�&9-+/bdcebÐf&rpÐfebTherefore _\r&eb+=LÐMOE\n 4ϸ\rdpÐfebFpÐ:eb+=LÐMOE ÏF\rdpÐfebie. _\r&tÑÒ _\rFpÐ:iÓ _\rThusthe -pointDFT _\rcanbeobtainedfromtwo:V-pointtransforms,oneoneveninputdata,ÑÒ _\r,andoneonoddinputdata,Ó _\r.Althoughthefre-quencyindex_rangesover values,only:VvaluesofÑÒ _\randÓ _\rneedtobecomputedsinceÑÔ _\randÓ _\rareperiodicin_withperiod:V.Forexample,for &9:92 Õeveninputdata \r \rd Ö$\r\n ×\rÕoddinputdata\n ¦\rd \rd 4H\r Í\r \r&tÑÒ \rFpEÅÓ C \r ¦\r&tÑÒ ¦\rFpÅÓ \r 4H\r&tÑÒ 4\rFpVÅÓ \r \r&tÑÒ \rFprqÅÓ \r C\r&tÑÒ \rFptsÅÓ C \r&tÑÒ C \r pEÅÓ C \r 4H\r&tÑÒ ¦\rFpÃÅÓ \r&tÑÒ \r pÅÓ \r ×\r&tÑÒ 4\rFpuÅÓ \r&tÑÒ \r pVÅÓ \r 4ÍH\r&tÑÒ \rFpvËÅÓ \r&tÑÒ \r prqÅÓ \rThisisshowngraphicallyontheowgraphofFig7.8:N/2pointDFTN/2pointDFTf[0]f[2]f[3]f[4]f[6]f[1]f[5]f[7]H[0]H[3]G[3]G[0]F[0]F[7]Figure7.8:FFTowgraph1.Assumingthan isapowerof,wecanrepeattheaboveprocessonthetwo:V-pointtransforms,breakingthemdownto:s-pointtransforms,etc,untilwecomedownto-pointtransforms.For & ,onlyonefurtherstageisneeded(i.e.thereareØstages,where &ZHÙ),asshownbelowinFig7.9.ThustheFFTiscomputedbydividingup,ordecimating,thesamplesequence \rintosub-sequencesuntilonly-pointDFT'sremain.Sinceitistheinput,ortime,sampleswhicharedividedup,thisalgorithmisknownasthedecimation-in-time(DIT)algorithm.(Anequivalentalgorithmexistsforwhichtheoutput,orfrequency,pointsaresub-dividedthedecimation-in-frequencyalgorithm.)ThebasiccomputationattheheartoftheFFTisknownasthebutterybecauseofitscriss-crossappearance.FortheDITFFTalgorithm,thebutterycomputa-tionisoftheformofFig7.1093 N/4 pointDFTN/4 pointDFTN/4 pointDFTN/4 pointDFTf[0]f[4]f[2]f[6]f[1]f[5]f[3]f[7]F[0]F[7]Figure7.9:FFTowgraph2.ÚÛÚ p*Ü:ÛÚFp*Ü:ÛÝÝÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞRßàààààààààààààààààààRá pÜ:FpÜ:Figure7.10:ButteryoperationinFFT.whereÚandÛarecomplexnumbers.Thusabutterycomputationrequiresonecomplexmultiplicationand2complexadditions.Notealso,thattheinputsamplesarebit-reversed(seetablebelow)becauseateachstageofdecimationthesequenceinputsamplesisseparatedintoeven-andodd-indexedsamples.(NB:thebit-reversalalgorithmonlyappliesif isanintegralpowerof).94 Index [BinaryBit-reversedBit-reversedrepresentationBinaryindex00000000100110042010010230111106410000115101101561100113711111177.3.2ComputationalspeedofFFTTheDFTrequires Vcomplexmultiplications.AteachstageoftheFFT(i.e.eachhalving):Vcomplexmultiplicationsarerequiredtocombinetheresultsofthepreviousstage.SincethereareBâBUãV stages,thenumberofcomplexmul-tiplicationsrequiredtoevaluatean -pointDFTwiththeFFTisapproximately |âBUãV (approximatelybecausemultiplicationsbyfactorssuchaspE:,peb:,peä:andpµåeä:arereallyjustcomplexadditionsandsubtractions). V(DFT):Vâ,UãV (FFT)saving321,0248092æ25665,5361,02498æ1,0241,048,5765,12099.5æ7.3.3PracticalconsiderationsIf isnotapowerof,thereare2strategiesavailabletocompletean -pointFFT.1.takeadvantageofsuchfactorsas possesses.Forexample,if isdivisi-bleby(e.g. &ÄR),thenaldecimationstagewouldincludea-pointtransform.95 2.packthedatawithzeroes;e.g.include16zeroeswiththe48datapoints(for &vR)andcomputea×$-pointFFT.(However,youshouldagainbewaryofabrupttransitionsbetweenthetrailing(orleading)edgeofthedataandthefollowing(orpreceding)zeroes;abetterapproachmightbetopackthedatawithmorerealisticdummyvalues).96