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Lecture7-TheDiscreteFourier
Transform
7.1TheDFT
Transformforsignalsknownonlyat
?instantsseparatedbysampletimes?(i.e. afinitesequenceofdata). Let ?samples bedenotedTheFourierTransformoftheoriginalsignal,
???????,wouldbe integrandexistsonlyatthesamplepoints: ie. ?datapointstostart with,only ?finaloutputswillbesignificant. ?)ratherthanfrom ???to 82??to ??????isthesameas???????to??? theperiodicsequenceinplot(b).
012345678910110
0.2 0.4 0.6 0.8 1 (a)0510152025300
0.2 0.4 0.6 0.8 1 (b)Figure7.1:(a)Sequenceof
??Hz, i.e.set or,ingeneral 83Wemaywritethisequationinmatrixformas:
??and???? ?etc.???.DFT-example
Letthecontinuoussignalbe
dc 1Hz 2Hz012345678910-4
-2 0 2 4 6 8 10Figure7.2:ExamplesignalforDFT.
Letussample
?.The valuesofthediscretesamplesaregivenby: 84Therefore
012305 10 15 20 f (Hz) |F[n]|
Figure7.3:DFToffourpointsequence.
InverseDiscreteFourierTransform
Theinversetransformof
85is i.e.theinversematrixis ric)matrix.
Notethatthe
inputs,ateach and ??odemodulators). ?and ?(re- memberthatthespectrumissymmetricalabout ?)combinetoproduce?fre- lowerofthetwofrequencies, ?Hzwhere ?;thehigherfrequency componentisatan"aliasingfrequency"( ???????of ?and is: ?????(7.2)Forall
???????real? But?1forall?
i.e. ?(i.e.thecomplexconjugate) 86SubstitutingintotheEquationfor??
?????abovegives, ???since? ie.?? or?? i.e.asampledsinewaveat ??Hz,ofmagnitudeForthespecialcaseof
contributionof ??????to???????is? nent.Interpretationofexample
1. ???(asexpected) 2. ?????withphasegivenby ????o i.e. ????o o ?(asexpected) 3. ?-noother ????componenthere)andthisimpliesa component since 8701230
1 2 3 4 5 6 f (Hz) |F[n]| sqrt(2)3/sqrt(2)
Figure7.4:DFToffourpointsignal.
Intypicalapplications,
?ismuchgreaterthan?;forexample,for ?has???????components,but??? ?arethecomplexconjugatesof????? leaving ??asthed.c.component, ?to ?ascompletea.c.com- ponentsand frequencyMostcomputerprogrammesevaluate
?(or ?forthepowerspectralden- ???and7.2DiscreteFourierTransformErrors
887.2.1Aliasing
frequencyspectralcontent.7.2.2Leakage
integrationtobeperformedovertheinterval- ?to ?oroveranintegernumber berofcyclesinthe ?datasamples.TheDFTforthiscase(for ???to isshownbelowin7.5.024680
2 4 6 8 freq |F[n]|Figure7.5:Leakage.
89components.
05101520253035404550-1
-0.5 0 0.5 1 inordertocalculatetheDFT. correctlocationismuchreduced,asinFig7.7.024680
1 2 3 4 5 6 7 (a)024680
1 2 3 4 5 (b) 907.3TheFastFourierTransform
theDFT,thisnumberisdirectlyrelatedto ?(matrixmultiplicationofavector), where ?ischosentobe sideration. volvesalotofredundantcalculations:Re-writing
?as itiseasytorealisethatthesamevaluesof? ??arecalculatedmanytimesasthe ?repeatsfordifferentcom- binationsof ?and ?;secondly,? ??isaperiodicfunctionwithonly ?distinct values.Forexample,consider
????(theFFTissimplestbyfarif ?isanintegralpower of2) ?????say? Then?Fromtheabove,itcanbeseenthat:
91Also,if
eg.if7.3.1Decimation-in-timealgorithm
?samplesinto2summations, eachwith ?samples,onefor?evenandtheotherfor?odd.Substitute
?for?evenand??? ?for?oddandwrite:Notethat?
Therefore
ie.Thusthe
?-pointDFT ?canbeobtainedfromtwo ?-pointtransforms, oneoneveninputdata, ?,andoneonoddinputdata,??? ?.Althoughthefre- quencyindex ?rangesover ?values,only ?valuesof???? ?and??? ?needtobe computedsince ?and??? ?areperiodicin ?withperiodForexample,for
92N/2 point
DFTN/2
point DFT f[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:FFTflowgraph1.
Assumingthan
?-pointtransforms,breakingthemdownto ?-pointtransforms,etc?????,untilwe comedownto ?-pointtransforms.For ????,onlyonefurtherstageisneeded (i.e.thereare ?stages,where ??????),asshownbelowinFig7.9. tionisoftheformofFig7.10 93quotesdbs_dbs20.pdfusesText_26