FourierTransform
2.1INTRODUCTIONANDOVERVIEW
Amongalltransformsusedinsignalandimageprocessing,Fouriertransform(FT)isprobablythemostcommonlyusedtransform.Inthischapter,wefirstdescribethedefinitionaswellastheconceptsofFTandthendiscusssomeofthepropertiesofFTthatarecommonlyusedinsignalprocessing.Theemphasisofthischapterisontheconceptualinterpretationsaswellastheapplicationsofone-dimensional(1-D)andtwo-dimensional(2-D)continuousanddiscreteFTasopposedtomathematicalformulation.
2.2ONE-DIMENSIONALCONTINUOUSFOURIERTRANSFORM
AsmentionedinChapter1,asignalcanbeexpressedinmanydifferentdomainsamongwhichtimeisprobablythemostintuitivedomain.Timesignalscananswerquestionsregarding"when"eventshappen,whereasFTdomainaddressesquestionsstartingwith"howoften"(thisiswhyFTdomainisalsocalledfrequencydomain).Asanexample,assumethatyouaretostudytheshoppinghabitsofmembersinacommunitybypreparingaquestionnaire.Youwillobtainsomeusefulinformationwhenyouaskquestionssuchas"Whatdaysdoyounormallygoshopping?"or"Whattimeofthedayyounevergoshopping?"Thisinformationhelpsyouunderstandandvisualizethe"time"elementsofpeople'sshoppinghabits.Also,ifyouprepareatimesignalthatshowsthenumberofpeopleshoppingateveryinstanceoftime(i.e.,agraphofnumberofpeopleshoppingvs.time),youcanacquireanswerstoalltheaforementionedquestions.Now,consideradifferentsetofquestionssuchas"Howoftendoyougoshopping?"or"Whatpercentagesofpeoplegoshoppingtwiceaweek?"Answerstothesequestionsformthefrequencydomain,whichinsignalprocessingisformedbyFT.Letusremindourselvesthattheinformationintimeandfrequencyareexactlythesame,i.e.,neitherofthedomainsaremoreinformativethantheotherandonecanacquireallinformationononedomainfromtheother.However,consideringthecomputationsizeandthevisibilityofcertaininformationtohumans,onedomaincanbepreferredovertheother,asdiscussedinChapterI.Now,wegiveaformaldefinitionforl-DcontinuousFT.Considerg{t)asacontinuoussignalintime.TheFTofthissignal,shownasG(f),isdefinedasfollows:
+00
G(f)=FT[git)}=Jgit)e-'dt00
(2.1)
where/isthefrequencyvariable(whichisoftenexpressedinunitssuchasHz,kHz,MHz,andsoon)jistheimaginarynumber(i.e.,/=-1)
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BiomedicalSignalandImageProcessing
NotethatinEquation2.1(whichisalsoknownastheanalysisequation),theintegrationistakingplaceovertimeandthereforetheresultingfunction,i.e.,G(/),isnolongerafunctionoftime.Also,notethatG{f)isacomplexfunctionof/ThismeansthatonecandescribeG{f)asfollows:
G(/)=|G(/)|e''*'*(2.2)
where||G(/)||isthemagnitudeofG(f)G{f)representsthephaseofG(/)Acloserlookatthevalueof||G(/)||atagivenfrequencyrevealsthemainadvantagesofexpressingasignalintheFourierdomain.Continuingourexampleonpeople'sshoppinghabits,letusassumethatg{f)isthenumberofpeopleshoppingattimeUwheretimeismeasuredinseconds.Then,inordertoseehowmanypeoplewouldgoshoppingonceaday(i.e.,onceevery86,400s),allweneedtodoistocalculate||G(/)||for/=1/86,400Hz.Notethatonecouldhaveobtainedthesameinformationfromthetimesignal,buttheFTprovidesamucheasierapproachtofrequency-relatedquestionssuchastheoneweexploredearlier.IfonecancalculatetheFTforatimesignal,heorsheshouldalsobeabletocalculatethetimesignalfromafrequencysignalintheFTdomain.SuchatransformationiscalledtheinverseFT(orthesynthesistransform)thatacceptsG{f)asinputandcalculatesg{t)asfollows:
+g{t)=IFT{G(/)}=JGine''df(2.3)cc
Next,wepracticecalculatingtheFTusingsomeusefulfunctionsthatareheavilyusedinsignalprocessing.
Example2.1
Consideranexponentiallydecayingsignalg{t)=e~\(>0.Then,
C(0=Jg{t)e-'"dt
+00=I0
1-0+j2)t
1+/2pty
t'O
,,(2.4)1+jlpf
FourierTransform
Also,wecancalculatethemagnitudeandphaseofG{f)asfollows:
1
17
Vl+(2pO'
(2.5)
and
AC(0=-(i+y2p0
=-tan"\2p/')
(2.6)
Example2.2
ImpulsefunctionS{t)(alsoknownasDiracfunction)playsanimportantroleinmanyareasofscienceandengineeringsuchasphysics,differentialequations,andsignalprocessing.BeforedescribingthemathematicaldefinitionoftheimpulsefunctionandcalculatingtheFTofthisfunction,wefocusontheconceptoftheimpulsefunctionandtheneedforsuchamathematicalentity.Impulsefunctionisamathematicalfunctionthatdescribesveryfastburstsofenergythatareobservedinsomephysicalphenomena.Asanexampleofsuchburstpulses,considertheeffectofsmashingtheballinavolleyballgame.Volleyballplayersarenotallowedtoholdtheballintheirhands;rather,theyaresupposedtohittheball.Whensmashingthebail,playersapplyasignificantamountofenergyinaveryshorttime.Suchanimpulseforceappliedtotheballcreatesthemosteffectivemoveinavolleyballgame.Theeffectsofsuchanactioncanbemodeledusinganimpulsefunction.Impulsefunctionplaysanimportantroleintheidentificationofunknownsystems.Thisrolecanbedescribedthroughasimpleexample.Supposeyouaregivenablackboxandyouareaskedtodiscoverwhatisinthebox.Onequickwayofguessingthecontentsoftheboxistappingonitandlisteningtotheechoes.Inamorescientificworld,youwouldapplysomefastpressureimpulsesonthesurfaceoftheboxandobservetheresponseofthecontentsintheboxwithrespecttotheimpulsefunctionsyouapplied.Iftheboxcontainscoins,youwillhearajinglingsound,andiftheboxisfullofwater,anentirelydifferentsoundandechowillbesensed.Thistypeofidentifyingunknownsystemsisafundamentaltechniqueinafieldofsciencecalled"systemidentification,"whichfocusesonmodelinganddescribingunknownsystems.Now,weslowlyapproachamathematicalformulationoftheimpulsefunction.Theburst-likeconceptoftheimpulsefunctionimpliesthatthemathematicalmodelmustbezeroforallpointsintimeexceptforaninfinitelysmalltimeinterval(asdiscussedearlier).Inamoremathematicalmanner,assumingthattheimpulseisappliedattimet=0,themathematicalrepresentationoftheimpulsefunctionmustbezeroeverywhereexceptforasmallneighborhoodaroundtheorigin.Iftheimpulseisassumedtobenonzerosforaveryshortperiodoftime,thentheamplitudeofimpulseduringthisveryshortintervaloftimemustbeinfinitelylarge;otherwise,thetotalenergyofthesignalwouldbecomezero.Inordertoseethismoreclearly,wefocusonamathematicalmodeloftheimpulsefunction.ConsiderfunctionS{t)showninFigure2.1.Notethattheenergyofthissignal
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BiomedicalSignalandImageProcessing
a()
1/A
-A/2A/2
t
FIGURE2.15{t)function.
(i.e.,theareaunderthecurve)Istheduration(i.e.,A)multipliedbytheheightofthepulse(i.e.,1/A).ThisshowsthattheenergyofthesignalisalwaysoneregardlessofthevalueofA,i.e.,thearea(energy)independentofA.Nowwecandefinetheimpulsefunctionasfollows:
d(t)=limdA(t)
(2.7)
Thepreviousdefinitionimpliesthateventhoughtheimpulsefunctionisnonzeroonlybetween0"and0%theenergyofthesignalisstillone,i.e.,
(2.8)0"
Inordertohaveameaningfulvisualrepresentationfortheimpulsefunctionemphasizingthefactthattheamplitudeoftheimpulsefunctionis0everywhereexceptattheorigininwhichtheamplitudeapproachesinfinity,anarrowpointedtowardinfinityisusedtoshowtheimpulsefunction(Figure2.2).
8{t)
t
FIGURE2.2Visualrepresentationofanimpulsefunction.
FourierTransform
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Otherusefulpropertiesoftheimpulsefunctionincludethesamplingcapabilityoftheimpulsefunction,i.e.,foranysignalg{t),
J*d(t)g(t)dt=j'1,wecaneasilycalculatetheFTusingG(f)asfollows:
(2.22)
Thepreviousequationassertsthatonceafunctioniscompressedintime,thefunctioninfrequencydomainexpandswiththesamerate.Thismeansthatoncethewidthofasignalintimedomainapproacheszero,itswidthinfrequencydomainapproachesinfinity.ThisobservationfurtherexplainswhytheFTofanimpulsefunctionmustbeinfinitelyflat.
2.3SAMPLINGANDNYQUISTRATEThetechnologicaladvancementsoftheInternetandotherdigitalmedia,digitalcomputers,digitalcommunicationsystems,andotherdigitalmachinesandsystemsmakestheprocessingofdigitalsignalsandimagesavaluedtechnique.Inaddition,theexistenceofveryfastdigitalsignalprocessorsthataretailoredtoprocessdigitalsignalswithamazinghighspeeds,supportstheprocessingofsignalsinadigitalform.However,knowingthatalmostallsignalscollectedfromnature(includingbiomedicalsignals)arecontinuousinnature,wewouldneedto"digitize"continuoussignalstoformdigital(ordiscrete)signalstobeprocessedwithdigitalsignalprocessors.Next,letusdiscusstwoimportantquestionsthatrequireanswersbeforeanyattemptstosamplethecontinuoussignalscanbemade:"Isitpossibletoformadigitalsignalfromacontinuoussignalwhilemaintainingallinformationinthecontinuoussignal?"Andiftheanswertothefirstquestionisyes,then"Howfastarewesupposedtosampleacontinuoussignalsuchthatallinformationofthecontinuoussignalispreservedintheresultingsampled(discrete)signal?"Theanswertothefirstquestionis"Yes!"Thisanswermaybetosomedegreecounterintuitive.Thereasonwhyitmaybecounterintuitiveisbecauseoncethecontinuoussignalissampled,apparentlythereisnoguaranteethatonecanrecovertheexactvaluesofthesignalbetweenthesamples.Inotherwords,ifwecanreconstructthecontinuoussignal
FourierTransform
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fromthediscretesignal,thenwecouldclaimthatnoinformationhasbeenlostfromthecontinuoussignal.However,itseemsimpossibletoreconstructtheexactvaluesbetweenthesampledvaluessincetheremightbeseveralpossibleoptionsforeachintermediatepoint.Thekeyissuetoaddresslieson"Howfastcanacontinuoussignalbesampled?"AtheoremcalledNyquistorShannontheoremaddressesourproblem.Themathematicaldetailsandproofofthetheoremwillnotbegivenhere.However,thepracticalprocedureintroducedbythetheoremforsamplingacontinuoussignalwhilemaintainingallinformationintheresultingdiscretesignalisdescribedinthefollowingsteps:
Step1:CalculateFTofthecontinuoussignal.Step2:Findmaximumfrequencyofthesignal,i.e.,maximumfrequencyatwhichtheFTofthesignalisnonzero.Callthisfrequency/.Step3:Samplethecontinuoussignalwithasamplingfrequencywhichisatleasttwiceof/,i.e.,/>2/.Inotherwords,takesamplesofthecontinuoussignalevery
