Lecture2 discrete-time signals and systems

iilS[e-mail]-timesignalsandsystemsOutlineDiscrete-timesignalsDiscretetimesystemsDiscrete-timesystemsAnalysisofdiscrete-timelineartime-invariant(LTI,线性时不变)systemsiiiiffDiscrete-timesystemsdescribedbydifferenceequations(差分方程)q()Correlation(相关性)ofdiscrete-timesignalsSEEofSWJTU2Autumn,2011G.X.Zhang2.1Discrete-timesignalsAdiscrete-timesignalAfunctionofanindependentvariablethatisanAfunctionofanindependentvariablethatisanintegerNote:thesignalx(n)isNOTdefinedfornonintegervaluesofn.SEEofSWJTU3Autumn,2011G.X.Zhangg2.1Discrete-timesignalsRepresentationsofadiscrete-timesignal1for13nGraphicalrepresentation1,for1,3()4,for2nxnnFunctionalrepresentationi0,elsewhere21012345nTabularrepresentation()00014100xnSequencerepresentationqp()0,0,1,4,1,0,0,xn()0,1,4,1,0,0,xn()0,1,4,1xnSEEofSWJTU4Autumn,2011G.X.ZhangTimeoriginAfour-pointsequence2.1Discrete-timesignalsBasicdiscrete-timesignalsUnitsamplesequence(unitimpulse)单位采样(单位冲激)Unitsamplesequence(unitimpulse)—单位采样(单位冲激)1,for0()0f0nn0,for0nMATLABSEEofSWJTU5Autumn,2011G.X.ZhangMATLAB2.1Discrete-timesignalsUnitstepsignal(单位阶跃)1for0n1,for0()0,for0nunnMATLABSEEofSWJTU6Autumn,2011G.X.ZhangMATLAB2.1Discrete-timesignalsUnitrampsignal(单位斜坡)for0nn,for0()0,for0rnnunnSEEofSWJTU7Autumn,2011G.X.Zhang2.1Discrete-timesignalsRectangularsequence(矩形序列)1,01()0otherNnNRnn0,othernRN(n)1RN(n)0123…N-1n0123N1-1SEEofSWJTU8Autumn,2011G.X.Zhang2.1Discrete-timesignalsExponentialsignalifaisreal,x(n)isreal(),forallnxnanSEEofSWJTU9Autumn,2011G.X.Zhang2.1Discrete-timesignalsifaiscomplexvalued,,thenx(n)jare()(cossin)njnnxnrernjn()(cossin)jxnrernjnDenotedbyrealandimaginaryparts(seenextslide)()cos;()sinnnRIxnrnxnrnygyp()Representedbyamplitudeandphasefunctions()();()()nxnAnrxnnnThephasefunctionislinearwithnSEEofSWJTU10Autumn,2011G.X.Zhang2.1Discrete-timesignalsSEEofSWJTU11Autumn,2011G.X.Zhang2.1Discrete-timesignals(a)Graphof(),0.9nAnrr(b)Graphof,moduloplottedintherange()(/10)nn2(,)SEEofSWJTU12Autumn,2011G.X.Zhang(b)Graphof,moduloplottedintherange()(/10)nn2(,)Fig.2.1.7Graphofamplitudeandphasefunctionofacomplex-valuedexponentialsignal.2.1Discrete-timesignalsClassificationofdiscrete-timesignalsEnergysignalsandpowersignals(能量,功率)egysigalsadpowesigals(能量,功率)Energysignals:theenergyEisfinite,i.e.,0E2()Ee.g.,unitsamplesignal()xnExng,pgPowersignals:theaveragepowerPisfiniteandnonzero1N21lim;()21NNNnNPEExnNPowersignalspossessinfiniteenergyWhichtypesofsignals,unitstep,rectangular,unitrampSEEofSWJTU13Autumn,2011G.X.ZhangWhichtypesofsignals,unitstep,rectangular,unitrampandexponential,areenergyorpowersignals?2.1Discrete-timesignalsEXAMPLE2.1.1Determinethepowerandenergyoftheunitstepsequence.21111/1lililiNNNPTheaveragepoweroftheunitstepsignalisConsequently,theunitstepsequenceisapowersignal.Its20limlimlim212121/2NNNnPuNNNcomplexexponentialsequence0()jnxnAeConsequently,theunitstepsequenceisapowersignal.Itsenergyisinfinite.complexexponentialsequence()xnAe—powersignal(P=A2)unitrampsignal?neitheranenergynorapowersignalSEEofSWJTU14Autumn,2011G.X.Zhang----neitheranenergynorapowersignal2.1Discrete-timesignalsPeriodicandaperiodicsignals(周期,非周期)Asignalx(n)isperiodicwithperiodN(N0)ifandonlyifg()pp()y()()forallxnNxnn(fundamental)period:thesmallestvalueofNOtherwise,x(n)isaperiodic(nonperiodic)Otherwise,x(n)isaperiodic(nonperiodic)Asinusoidalsignalisperiodicwhenf0isarationalnumber,i.e.,0()sin2xnAfn,,0kfNwherekandNareintegersPeriodicsignals—energyorpowersignals?NSEEofSWJTU15Autumn,2011G.X.ZhangPeriodicsignals—energyorpowersignals?2.1Discrete-timesignalsSymmetric(even)andantisymmetric(odd)signals对称偶反对称奇对称偶反对称奇Areal-valuedsignalx(n)issymmetric(even)ifx(-n)=x(n)ifx(-n)=-x(n)––antisymmetric(odd)(seenextslide)ifx(n)x(n)antisymmetric(odd)(seenextslide)Note:ifx(n)isodd,thenx(0)=0AnyarbitrarysignalcanbeexpressedasAnyarbitrarysignalcanbeexpressedas()()()eoxnxnxnwhere1()()()xnxnxn()()()21()()()exnxnxnxnxnxnMATLABSEEofSWJTU16Autumn,2011G.X.Zhang()()()2oxnxnxnMATLAB2.1Discrete-timesignalsSEEofSWJTU17Autumn,2011G.X.Zhang2.1Discrete-timesignalsSimplemanipulationsofdiscrete-timesignalsTransformationoftheindependentvariable(time)ShiftitiShiftintime:Delay:x’(n)=x(n-k),wherekisapositiveintegerAdvance:x’(n)=x(n-k),wherekisanegativeintegerMATLABSEEofSWJTU18Autumn,2011G.X.ZhangMATLAB2.1Discrete-timesignalsEXAMPLE2.1.2AsignalisgraphicallyillustratedinFig.2.1.9(a).Showa()xnggpyggraphicalrepresentationofthesignalsand.(3)xn(2)xnSolutionThesignalisobtainedbydelayingbythreeunitsintime.TheresultisillustratedinFig.2.1.9(b).Solution.(3)xn()xng()Ontheotherhand,thesignalisobtainedbyadvancingbytwounitsintime.Theresultisillustrated(2)xn()xnSEEofSWJTU19Autumn,2011G.X.ZhanginFig.2.9(c).