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Lecture7-TheDiscreteFourier

Transform

7.1TheDFT

Transformforsignalsknownonlyat

?instantsseparatedbysampletimes?(i.e. afinitesequenceofdata). Let ?samples bedenoted

TheFourierTransformoftheoriginalsignal,

???????,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 83

Wemaywritethisequationinmatrixformas:

??and???? ?etc.???.

DFT-example

Letthecontinuoussignalbe

dc 1Hz 2Hz

012345678910-4

-2 0 2 4 6 8 10

Figure7.2:ExamplesignalforDFT.

Letussample

?.The valuesofthediscretesamplesaregivenby: 84

Therefore

01230
5 10 15 20 f (Hz) |F[n]|

Figure7.3:DFToffourpointsequence.

InverseDiscreteFourierTransform

Theinversetransformof

85
is 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) 86

SubstitutingintotheEquationfor??

?????abovegives, ???since? ie.?? or?? i.e.asampledsinewaveat ??Hz,ofmagnitude

Forthespecialcaseof

contributionof ??????to???????is? nent.

Interpretationofexample

1. ???(asexpected) 2. ?????withphasegivenby ????o i.e. ????o o ?(asexpected) 3. ?-noother ????componenthere)andthisimpliesa component since 87
01230
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 frequency

Mostcomputerprogrammesevaluate

?(or ?forthepowerspectralden- ???and

7.2DiscreteFourierTransformErrors

88

7.2.1Aliasing

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.

89
components.

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) 90

7.3TheFastFourierTransform

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:

91

Also,if

eg.if

7.3.1Decimation-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 ?withperiod

Forexample,for

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