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8TheDiscreteFourierTransform8.0INTRODUCTIONInChapters2and3,wediscussedtherepresentationofsequencesandLTIsystemsintermsofthediscrete-timeFourierandz-transforms,respectively.Forfinite-durationse­quences,thereisanalternativediscrete-timeFourierrepresentation,referredtoasthediscreteFouriertransform(DFT).TheDFTisitselfasequenceratherthanafunctionofacontinuousvariable,anditcorrespondstosamples,equallyspacedinfrequency,oftheDTFTofthesignal.InadditiontoitstheoreticalimportanceasaFourierrepre­sentationofsequences,theDFTplaysacentralroleintheimplementationofavarietyofdigitalsignal-processingalgorithms.ThisisbecauseefficientalgorithmsexistforthecomputationoftheDFT.ThesealgorithmswillbediscussedindetailinChapter9.TheapplicationoftheDFTtospectrumanalysiswillbedescribedinChapter10.Althoughseveralpointsofviewcanbetakentowardthederivationandinter­pretationoftheDFTrepresentationofafinite-durationsequence,wehavechosentobaseourpresentationontherelationshipbetweenperiodicsequencesandfinite-lengthsequences.WebeginbyconsideringtheFourierseriesrepresentationofperiodicse­quences.Althoughthisrepresentationisimportantinitsownright,wearemostofteninterestedintheapplicationofFourierseriesresultstotherepresentationoffinite­lengthsequences.Weaccomplishthisbyconstructingaperiodicsequenceforwhicheachperiodisidenticaltothefinite-lengthsequence.TheFourierseriesrepresentationoftheperiodicsequencethencorrespondstotheDFTofthefinite-lengthsequence.Thus,ourapproachistodefinetheFourierseriesrepresentationforperiodicsequencesandtostudythepropertiesofsuchrepresentations.Then,werepeatessentiallythesamederivations,assumingthatthesequencetoberepresentedisafinite-lengthsequence.623624 Chapter8TheDiscreteFourierTransformThisapproachtotheDFTemphasizesthefundamentalinherentperiodicityoftheDFTrepresentationandensuresthatthisperiodicityisnotoverlookedinapplicationsoftheDFT.8.1 REPRESENTATIONOFPERIODICSEQUENCES:THEDISCRETEFOURIERSERIESConsiderasequencex[nJthatisperiodic1withperiodN,sothati[n]x[n+rN]foranyintegervaluesofnandr.Aswithcontinuous-timeperiodicsignals,suchasequencecanberepresentedbyaFourierseriescorrespondingtoasumofharmonicallyrelatedcomplexexponentialsequences,i.e.,complexexponentialswithfrequenciesthatareintegermultiplesofthefundamentalfrequency(2rr/N)associatedwiththeperiodicsequencex[n].Theseperiodiccomplexexponentialsareoftheformek[n]=eJ(2rr/N)kn=ek[n+rN], (8.1)wherekisanyinteger,andtheFourierseriesrepresentationthenhastheform2x[n]=~LX[k]eJ(2rr/N)kn.(8.2)kTheFourierseriesrepresentationofacontinuous-timeperiodicsignalgener­allyrequiresinfinitelymanyharmonicallyrelatedcomplexexponentials,whereastheFourierseriesforanydiscrete-timesignalwithperiodNrequiresonlyNharmoni­callyrelatedcomplexexponentials.Toseethis,notethattheharmonicallyrelatedcom­plexexponentialsek[n]inEq.(8.1)areidenticalforvaluesofkseparatedbyN;i.e.,eo[n]=eNln],edn]=eN+dn],and,ingeneral,ekHN[n]=eJ(2rr:/N)(k+eN)neJ(2rr/N)kllej21r:enej(2rr/N)knek[n],(8.3)wherefisanyinteger.Consequently,thesetofNperiodiccomplexexponentialseo[n],el[n],...,eN-l[n]definesallthedistinctperiodiccomplexexponentialswithfrequen­ciesthatareintegermultiplesof(2rr/N).Thus,theFourierseriesrepresentationofaperiodicsequencex[n]needcontainonlyNofthesecomplexexponentials.Fornota­tionalconvenience,wechoosekintherangeof0toN1;hence,Eq.(8.2)hastheformN-Ix[n]=~LX[k]eJ(2rrjN)kn. (8.4)k=OHowever,choosingktorangeoveranyfullperiodofX[k]wouldbeequallyvalid.ToobtainthesequenceofFourierseriescoefficientsX[k]fromtheperiodicse­quencex[n],weexploittheorthogonalityofthesetofcomplexexponentialsequences.1Henceforth.wewillusethetildeC)todenoteperiodicsequenceswheneveritisimportanttoclearlydistinguishbetweenperiodicandaperiodicsequences.2Themultiplicativeconstant1/NisincludedinEq.(8.2)forconvenience.ItcouldalsobeabsorbedintothedefinitionofX[kJ.Section8.1RepresentationofPeriodicSequences:TheDiscreteFourierSeries625AftermUltiplyingbothsidesofEq.(8.4)bye-J(2n/N)rnandsummingfromn0ton=N-1,weobtainN-lN-l1N-lLi[n]e-J(2n/N)rn=LNLX[k]eJ(2n/NHk-r)n.(8.5)n=On=Ok=OAfterinterchangingtheorderofsummationontheright-handside,Eq.(8.5)becomesI:i[n]e-j(2n/Nlrn=I:X[k][~I:eJ(2nIN)(k-r)n].(8.6)n=Ok=On=OThefollowingidentityexpressestheorthogonalityofthecomplexexponentials:1I:ej(2n/N)(k-r)n={I,kr=mN,maninteger,(8.7)N0,otherwise.n=OThisidentitycaneasilybeproved(seeProblem8.54),andwhenitisappliedtothesummationinbracketsinEq.(8.6),theresultisN-lLi[n]e-j(2n/N)rn=X[r].(8.8)Thus,theFourierseriescoefficientsX[k]inEq.(8.4)areobtainedfromi[n]bytherelationN-lX[k]=Li[n]e-J(2rr/N)kn.(8.9)n=ONotethatthesequenceX[k]definedinEq.(8.9)isalsoperiodicwithperiodNifEq.(8.9)isevaluatedoutsidetherange0:'Sk.:'SN1;i.e.,X[0]=X[N],X[1]=X[N+1],and,moregenerally,N-lX[k+N]Li[n]e-J(2rr/N)(k+N)nn=ON-l)=Li[n]e-)(2rr/N)kne-j2:n:n=X[k],(n=Oforanyintegerk.TheFourierseriescoefficientscanbeinterpretedtobeasequenceoffinitelength,givenbyEq.(8.9)fork=0,...,(N-1),andzerootherwise,orasaperiodicsequencedefinedforallkbyEq.(8.9).Clearly,bothoftheseinterpretationsareacceptable,sinceinEq.(8.4)weuseonlythevalu