奥本海姆版信号与系统ppt

Chapter1SignalsAndSystems崔琳莉ContentsDescriptionofsignalsTransformationsoftheindependentvariableSomebasicsignalsSystemsandtheirmathematicalmodelsBasicsystemsproperties1.1Continuous-TimeandDiscrete-TimeSignals1.1.1ExamplesandMathematicalRepresentation(1)AsimpleRCcircuitSourcevoltageVsandCapacitorvoltageVcA.Examples(2)AnautomobileForceffromengineRetardingfrictionalforceρVVelocityV(3)ASpeechSignal(4)APicture(5)vitalstatistics(人口统计)NoteInthisbook,wefocusonourattentiononsignalsinvolvingasingleindependentvariable.Forconvenience,wewillgenerallyrefertotheindependentvariableastime,althoughitmaynotinfactrepresenttimeinspecificapplications.0tA)(tx10t)(tx0,00,)(ttatetxRttAtx),sin()(B.Twobasictypesofsignalst:continuoustimex(t):continuumofvalue1.Continuous-Timesignal2.Discrete-Timesignaln:discretetimex[n]:adiscretesetofvalues(sequence)Example1:1990-2002年的某村农民的年平均收入][nx)(txsk81SamplingExample2:x[n]issampledfromx(t)WhyDT?(1)FunctionRepresentationExample:x(t)=cos0tx[n]=cos0nx(t)=ej0tx[n]=ej0n(2)GraphicalRepresentationExample:(Seepagebefore)(3)Sequence-representationfordiscrete-timesignals:x[n]={-21321–1}orx[n]=(-21321–1)C.Representation3Note:Sincemanyoftheconceptsassociatedwithcontinuousanddiscretesignalsaresimilar(butnotidentical),wedeveloptheconceptsandtechniquesinparallel.Therearemanyothersignalsclassification:Analogvs.DigitalPeriodicvs.AperiodicEvenvs.OddDeterministicvs.Random……1.1.2SignalEnergyandPowerInstantaneouspower:)()(1)()()(22tiRtvRtitvtpLetR=1Ω,so)()()()(222txtvtitp+R_)(tv)(tiEnergy:t1tt2212121)()()(22ttttttdttxdttvdttpAveragePower:2121)(1)(121212ttttdttxttdttpttTotalEnergyAveragePower21212][11nnnnxnnDefinition:212)(ttdttx21212)(1ttdttxtt212][nnnnxContinuous-Time:(t1tt2)Discrete-Time:(n1nn2)Wewillfrequentlyfinditconvenienttoconsidersignalsthattakeoncomplexvalues.whenNNnNnxNP2][121limdttxdttxETTT22)()(limntTotalEnergyAveragePowernNNnNnxnxE22][][limTTTdttxTP2)(21limNote:Itisimportanttorememberthattheterms“Power”and“energy”areusedhereindependentlyofthequantitiesactuallyarerelatedtophysicalenergy.Withthesedefinitions,wecanidentifythreeimportantclassofsignals——a.finitetotalenergyb.finiteaveragepowerc.infinitetotalenergy,infiniteaveragepowerE02limTEPTP),0(limTTTPEthenPifPE,ReadtextbookP71:MATHEMATICALREVIEWHomework：P57--1.21.2.1ExamplesofTransformations1.TimeShiftx(t-t0),x[n-n0]t00Advance1.2TransformationoftheIndependentVariablen00DelayTimeShiftx(t)andx(t-t0),orx[n]andx[n-n0]:TheyareidenticalinshapeIft00,x(t-t0)representsadelayn00,x[n-n0]representsadelayIft00,x(t-t0)representsanadvancen00,x[n-n0]representsanadvance2.TimeReversalx(-t),x[-n]——Reflectionofx(t)orx[n]2.TimeReversalx(-t),x[-n]——Reflectionofx(t)orx[n]amirrorTimeReversalx[n]x[-n]LookingformistakesNote:thedifferencebetweenx(-t)and–x(t)x(t)x(-t)???-x(t)0tt4248421x(t)t4428421x(t/2)t4428421x(2t)3.TimeScalingcompressedstretchx(at)(a0)TimeScalingx(at)(a0)Stretchifa1Compressedifa1Howaboutthediscrete-timesignal?x[n]Generally,timescalingonlyforcontinuoustimesignalsx[2n]x[n]x[2n]0123456nThisisalsocalleddecimationofsignals.（信号的抽取）x[n/2]x[n]222Example011tx(t)Solution1：Solution2：Solution1：Solution2：011tx(t)01tx(t-1/2)1/23/201tx(3t-1/2)1/61/2011tx(t)01/31tx(3t)01tx(3t-1/2)1/61/2()ft12121t0t1221t0(1)ft)1(tf01122t)31(tft0131232)(tft011212t112012)1(tf)3(tf12t03132)31(tf012t32－31shiftreversalScalingreversalshiftScalingreversalshiftScalingExamplef(t)f(1-3t)1.2.2PeriodicSignalsAperiodicsignalx(t)(orx[n])hasthepropertythatthereisapositivevalueofT(orintegerN)forwhich:x(t)=x(t+T),foralltx[n]=x[n+N],forallnIfasignalisnotperiodic,itiscalledaperiodicsignal.ExamplesofperiodicsignalsCT:x(t)=x(t+T)DT:x[n]=x[n+N]PeriodicSignalsThefundamentalperiodT0(N0)ofx(t)(x[n])isthesmallestpositivevalueofT(orN)forwhichtheequationholds.Note:x(t)=Cisaperiodicsignal,butitsfundamentalperiodisundefined.Examplesofperiodicsignals1.tAtx83sin)(Itisperiodicsignal.ItsperiodisT=16/3.2.0,00,cos)(ttttxItisnotperiodic.3.tBtAtx41sin31cos)(8,621TTx(t)isperiodic.Itsperiodis24TThesmallestmultiplesofT1andT2incommon4.tttx2coscos)(21,2TTItisaperiodic,too.ThereisnothesmallestmultiplesofT1andT2incommon5.nnx4cos][x(t)isaperiodic.6.nnx83cos][ItisperiodicwithperiodN=16.-20-15-10-505101520-101-20-15-10-505101520-101-20-15-10-505101520-202CosπtCos2tcosπt+cos2t1.2.3EvenandOddSignalsNote:Anoddsignalmustnecessarilybe0att=0,orn=0.ie.x(t)=0,orx[n]=0.Evensignal:x(-t)=x(t)orx[-n]=x[n]Oddsignal:x(-t)=-x(t)orx[-n]=-x[n]Even-OddDecomposition——AnysignalcanbeexpressedasasumofEvenandOddsignals.)]()([21)()}({txtxtxtxEve)]()([21)()}({txtxtxtxOdo]}[][{21][]}[{nxnxnxnxEve]}[][{21][]}[{nxnxnxnxOdox(t)=xeven(t)+xodd(t)x[n]=xeven[n]+xodd[n]Exampleoftheeven-odddecompositonExampleoftheeven-odddecompositonHomework：P57--1.91.101.21(a)(b)(c)(d)1.22(a)(b)(c)(g)1.231.241.3ExponentialandSinusoidalSignals1.3.1Continuous-timeComplexExp