信号与系统英文版课件Chapter 3

Contents§3.1TheresponseofLTIsystemsto complexexponentials.§3.2Fourierseriesrepresentationof continuousperiodicsignals§3.3ConvergenceoftheFourierseries§3.4Propertiesofcontinuous-time Fourierseries.§3.5Fourierseriesrepresentationofdiscrete–timeperiodicsignals§3.6PropertiesofdiscreteFS(omit)§3.7FourierseriesandLTIsystems§3.8Filtering1768-1830TheresponseofLTIsystemstocomplexexponentialsH(s)andH(z)–systemfunction.Whenthedominanceconditionbesatisfied,theresponseofanLTIsystemtoestandznisthesamecomplexexponential.es1t—Eigenfunction,H(s1)—Eigenvaluex(t)H(s)y(t)x[n]H(z)y[n]TheresponseofLTIsystemstocomplexexponentials[Show]:ConsiderLTIsystemwithh(t).whenAssumingthattheintegralconverges,thenTheresponseofLTIsystemstocomplexexponentialsWhereH(s)isacomplexconstantwhosevaluedependsonsandwhichisrelatedtoh(t)byExample:determineTheresponseofLTIsystemstocomplexexponentialsMoregenerallyAcontinuousperiodicsignalrepresentaslinearcombinationsofEq.(3.25)isreferredtoastheFourierseriesrepresentationintermofcomplexexponentialsofx(t).Thesetof{ak}iscalledtheFourierseriescoefficientsorthespectralcoefficientsofx(t).Eq.(3.31).(3.32)aretheFourierseriesintermsoftrigonometricfunction.Acontinuousperiodicsignalrepresentaslinearcombinationsof(3.32)Determinationofcoefficientsak

Deduction(omit)involvetheEq:Thea0istheDCorconstantcomponentofx(t)Determinationofcoefficientsakwhichistheaveragevalueofx(t)overoneperiod.WriteEq(3.25)and(3.39)onceagain:(3.25)DeterminationofcoefficientsakTheyalsoisreferredtoasFSpair,Eq(3.25)representfrequencycompositionofx(t),ak

alsoisreferredtoasspectralamplitudeofeachfrequencycompositions.Assignments(P251):3.33.4DeterminationofcoefficientsakExample3.5determinetheFourierseriescoefficientsofx(t).-TT-T/20T/21x(t)t…………Solution:0=2/TDeterminationofcoefficientsak-8-4482T1/TIfT=8T1k0(k0)DeterminationofcoefficientsakExample:CalculatetheFScoefficientsforimpulsetrain.Thisisperiodicunitimpulsessignal,andisdenotedbyasymbolδT(t).……0-TT2TtDeterminationofcoefficientsak……-2-1012k……0-TT2Tt0=2/TDeterminationofcoefficientsakak=jCkDeterminationofcoefficientsak2.Spectralpropertyofperiodicsignal:(a)discrete,(b)harmonic3.FrequencybandwidthofsignalForsincfunctionspectral,fromzerofrequencyoffirstpasszeropointas.of1/10akas.Determinationofcoefficientsak4.Therelationbetween

andtimeproperty(a)samewaveformsignal,durationT1,.(b)differentwaveformsignal,varianceT,(Tak).Assignments：(p251)3.22(a)Figurep3.22(a)(d)ConvergenceoftheFourierseriesTherearetwosomewhatdifferentclassesofconvergencecondition.1)Finiteenergycondition2)TheDirichletconditionG.L.Dirichlet

1805-1859ConvergenceoftheFourierseriesExample:aperiodicsignal(2)Inanyfiniteintervaloftime,x(t)isofboundedvariation.Anexamplethatmeets(1)butnot(2)is-1012x(t)……t(3)Inanyfiniteintervaloftime,thereareonlyafinitenumberofdiscontinuities.Furthermore,eachofthesediscontinuitiesisfinite.Seefigure3.8(b)1………-1-112t0…ConvergenceoftheFourierseriesConvergenceoftheFourierseries1601x(t)t………8…meets(1)and(2)butnot(3)Propertiesofcontinuous-timeFourierseries.Wewillusethenotationtosignifythepairingofaperiodicx(t)withitsFScoefficients.thenz(t)=Ax(t)+By(t)1)LinearityTimeshifting-3-2-1012345……tak=?-TT01t……Fig.p2563.22(f)Timeshifting-3-2-1012345……t-TT01t……TimereversalAconsequenceoftimereversalpropertyisthatTimereversalExample:1-1-10123x(t)t1-1-10123y(t)tTimescalingIfx(t)isperiodicwithperiodT,0=2/T,Thenx(at),periodT/a(a,real>0),fundamentalfrequencya0,akremainthesame.Differentiation(orfromobservation)DifferentiationExample:determineakofx(t)illustratedinFigure.1-1-10123x(t)tSolution:T=2,0=,a0=01-30-232t-11-2(t-1)ParsevaltheoremAveragepowerofx(t)overoneperiodAssignments(p251):3.5,3.24x[n]representaslinearcombinationsofharmonicallyrelatedcomplexexponentials.Thesetoffunctionareharmonicallyrelated.N–fundamentalperiod.0=2/N–fundamentalfrequency.∵thereareonlyNdistinctsignalsinEq.(3.85)k=0,1,2…(3.85)(3.86)i.e0[n]=N[n],1[n]=N+1[n],x[n]representaslinearcombinationsofharmonicallyrelatedcomplexexponentials.TheFSrepresentationofperiodicx[n]iswherek=<N>isdenotationwhichneedonlyincludeNitems,beginningwithanyvaluesofk.Forexample,kcouldtakeonthevaluesk=0,1,….,N-1,ork=3,4,…..,N+2.Eq.(3.88)–discreteFourierSeriesak–FScoefficients(3.88)TheEq.similartointegralofcontinuousfunctions:(Seeproblem3.54)DeterminationofcoefficientsakNowconsiderEq.(3.88),multiplyingbothsidesbye-jr(2/N)n

andsummingoverNitems:(3.95)orDeterminationofcoefficientsakDeterminationofcoefficientsakEq.(3.88)and(3.95)arediscreteFSpair,ak(onlyNvalues)isreferredtoasspectralcoefficientofx[n].ReferringtoEq.(3.88)Iftakekintherangefrom0toN-1,thenDeterminationofcoefficientsakIfkrangefrom1toN,theni.e.thevaluesakrepeatperiodicallywithperiodN.SodiscreteFSisafiniteserieswithNitems.DeterminationofcoefficientsakExample3.12:evaluatecoefficientsak

ofFSfortheperiodicwaveshowninFigure.……-N0Nn1Determinationofcoefficientsak(seeproblem1.54)Determinationofcoefficientsak∵2N1+1=5,andletN=10,thenplotofak〜kasFigure.0.50.323-0.123-0.1230.10.5-0.123-0.1230.10.3230.3230.5-10-8-6-4-226810k0•••••••••4•FourierseriesandLTIsystems1.Recollection

Ifx(t)=est(or

x[n]=zn)istheinputtoLTIsystem,andwhenthedominanceconditionbesatisfied,thentheoutputisgivenbyy(t)=H(s)est

(ory[n]=H(z)zn)where

Inwhichh(t)(orh[n])istheimpulseresponseoftheLTIsystem.

FourierseriesandLTIsystems

Whensandzaregeneralcomplexnumbers,H(s)andH(z)arereferredtoasthesystemfunctionsofthecorrespondingsystems.isreferredtoasthefrequencyresponseofthecontinuoussystem.Letx(t)bethenoutputisFourierseriesandLTIsystems—frequencyresponseofthe