信号与系统课件chapter1.ppt

SignalsandSystemsAlanV OppenheimAlanS WillskyS HamidNawabTeacher 湛柏明E mail zhanbm Introduction Status SignalsandSystemsisaveryimportantfundamentalcourse thefundamentaltheories conceptsandmethodsestablishedinthiscoursearethefoundationofourspecialty Introduction Objective DiscussandstudythefundamentaltheoriesandmethodsforwhichthedeterministicsignalspassthroughtheLTIsystems Preview Chapter1 SignalsandSystemsConstructthebasicconceptsaboutsignalsandsystemsChapter2 LTISystemsIntroducethetheoriesandmethodsforthetime domainanalysisofLTIsystems Theemphasisistheconvolutionmethod x t y t Preview Chapter3and4 Continuous TimeFourierSeriesandTransformAnimportanttoolforsignalanalysisChapter6 TimeandFrequencyCharacterizationofSignalsandSystemsChapter7 SamplingSamplingisabridgeconnectingtheanalogsignalsandthedigitalsignals Preview Chapter8 CommunicationSystemsIntroducetheapplicationsoftheFourierTransformChapter9 TheLaplaceTransformTheLaplacetransformisalsoanimportanttoolinsignalandsystemanalysis Goback Chapter1SignalsandSystems Contents 1 1Continuous timeanddiscrete timesignals1 Whatisasignal 2 Whatisasystem 3 Continuous timeanddiscrete timesignals4 Howtorepresentasignal 5 Examplesofsignals6 Summary Contents 1 2TransformationoftheIndependentVariable1 Timeshifting2 Timereversal3 Timescaling4 Examples5 Periodicsignals6 Evenandoddsignals Contents 1 3 1 4Somebasicbuildingblocksofsignals1 SinusoidalSignals2 ExponentialSignals3 UnitImpulseandUnitStepFunctions Contents 1 5Continuous timeanddiscrete timesystems1 Continuous timeanddiscrete timesystems2 Examplesofsystems3 Interconnectionsofsystems Contents 1 6Thesystemproperties 1 SystemswithandwithoutMemory 2 InvertibilityandInverseSystems 3 Causality 4 Stability 5 TimeInvariance 6 Linearity Examples Summary Whatisasignal Thesignals whicharefunctionsofoneormoreindependentvariables containinformationaboutthebehaviorornatureofsomephysicalphenomena Thecurrentandvoltageinacircuit thespeechsignal etc Goback Whatisasystem Asystemisaninterconnectionofsomeunits devicesandsubsystemsaccomplishingacertainfunction Thecommunicationsystems thecontrolsystems etc Goback Transmitorprocess x t y t 1 1Continuous timeandDiscrete timeSignals DefinitionAsignalx t issaidtobeacontinuous timesignalorananalogsignaliftheindependentvariableiscontinuous Otherwise ifthetimevariableisdiscrete thesignalissaidtobediscrete Acontinuous timesignal Goback Adiscrete timesignal PeriodicSignals wherekisinteger Tiscalledtheperiod 1 2 2PeriodicsignalsAcontinuous timesignalissaidtobeperiodicifitsatisfies PeriodicSignals wherekandNareintegers Niscalledtheperiod Adiscrete timesignalissaidtobeperiodicifitsatisfies PeriodicSignals Thesmallestperiodofaperiodicsignaliscalledthefundamentalperiod Thereciprocalofthefundamentalperiodiscalledthefundamentalfrequency FundamentalPeriodandFundamentalFrequency Goback EvenandOddSignals 1 2 3EvenandOddsignalsArealsignalcanalwaysbeexpressedasasumoftwoparts evenandoddparts Goback 1 3ExponentialandSinusoidalSignals SinusoidalSignals A magnitude phasemeasuredinradians 0 angularfrequencymeasuredinradians s Goback BasedontheparametersCanda thesignalexhibitsdifferentcharacteristics 1 3ExponentialandSinusoidalSignals Continuous timeexponentialsignal Periodiccomplexexponentialsignal Realexponentialsignal Harmonicrelation Examples Generalcomplexexponentialsignal Likethecontinuous timeexponentialsignal alsobasedontheparametersCand thesignalexhibitsdifferentcharacteristics 1 3ExponentialandSinusoidalSignals Adiscrete timeexponentialsignalhasthegeneralform 1 3ExponentialandSinusoidalSignals Adiscrete timeexponentialsignalhasthegeneralform Generalcomplexexponentialsignal Realexponentialsignal Periodicity Sinusoidalsignal Goback harmonicrelation Example 1 4TheUnitImpulseandUnitStepFunction Thediscrete timeunitimpulseandunitstepfunctions1 Definition2 Therelationshipbetweentheunitimpulseandunitstepfunctions3 Samplingpropertyoftheimpulse 1 4TheUnitImpulseandUnitStepFunction Thecontinuous timeunitimpulseandunitstepfunctions1 Definition2 Therelationshipbetweentheunitimpulseandunitstepfunctions3 SamplingpropertyoftheimpulseSignalsexpressedintermsoftheunitstep Goback Summary Inthischapter wehavedevelopedanumberofbasicconceptsrelatedtocontinuous timeanddiscrete timesignalsandsystems SignalscarryinformationSignalscanbemathematicallyexpressedasfunctionsofoneormoreindependentvariables Inourbook wefocusontheone dimensionalsignalwhichinvolvesasingletimevariable Summary Sometimes weoftenrepresentasignalingraph Therearesometypicalsignalsandtheircharacteristicsthatwemustlearnbyheart Theunitstepandunitimpulse therealandcomplexexponentialsignals andsin cossignals Usingthesesignalswecanbuildothercomplicatedsignals Summary Systemsareaninterconnectionofsubsystems Thephysicalmeaningsofsystemsareverybroad Inthischapter werepresentsystemsusingthesebasicways BlockdiagramsMathematicalequationsWehavealsodiscussedsomebasicpropertiesofsystemsandthewaysthathowtoverifytheseproperties Summary Theemphasesare linearity time invariant causalityandstabilityproperties Thesystemsthatsatisfybothlinearityandtime invariantpropertiesarecalledtoastheLTIsystems TheLTIsystemsaretheprimaryfocusinourbook becausealargeclassofnaturesystemscanbecharacterizedbyLTIsystems Homework Readthetextbookfromp1top56 9 14 15 20 21 31Note yourexercisesmustbewritteninEnglishexceptyourname ExamplesofSignals 1 VoltagesandCurrentsinacircuit RCcircuit vs t andvc t voltagesofsourceandcapacitor i t currentinthecircuit ExamplesofSignals 2 Speechsignal ExamplesofSignals 3 Thestockmarketindex Goback RepresentationsofSignals Therearethreewaystorepresentasignal 1 Themathematicalfunction 2 Graphicrepresentation RepresentationsofSignals 3 Foradiscrete timesignal wecanrepresentthesignalasasequenceofnumbers x n 0 0 1 0 23 1 2 1 2 Goback Summaryfortheconceptofsignal Signalscontaininformationingeneral Wecanuseamathematicfunction agraphorasequenceofnumberstorepresentasignal Boththefunctionvalueandtheindependentvariableofcontinuous timesignalsarecontinuous Summaryfortheconceptofsignal 4 Theindependentvariableofadiscrete timesignalisdiscrete 5 Forsomediscrete timesignals theindependentvariableisinherentlydiscrete butotherdiscrete timesignalsaregeneratedbysamplingcontinuous timesignals Goback Timeshifting TimeshiftingOriginalsignalx t Timeshiftedversionofx t Timeshiftedversionsofx t t0ispositive Goback Timereversal TimereversalOriginalsignalx t Timereversedversionofx t Goback Timescaling TimescalingOriginalsignalx t Timescaledversionofx t aisanarbitraryrealvalue When a 1 x t iscompressedtox1 t When a 1 x t isexpandedtox1 t asillustratedinthefollowingfigure Timescaling Goback ExamplesofTransformationsoftheindependentvariable Example1 1 1 2 1 3Givenasignalx t findsignalx 3t 1 Solution Steps x t Timeshifting x t 1 x t 1 Timereversal x t 1 x t 1 Timescaling x 3t 1 IntegrationforTransformationoftheIndependentVariable Givenacontinuous timesignalx t thestepstodeterminethesignalx at b are Determinex t b TimeshiftingDeterminex t b Timereversalifa 03 Determinex at b Timecompressionorexpandingby a Goback ExamplesofTransformationsoftheindependentvariable Timeshiftx t totheleftby1second Goback ExamplesofTransformationsoftheindependentvariable Timereversex t 1 Goback ExamplesofTransformationsoftheindependentvariable Timecompressx t 1 by3 Goback RealExponentialSignals RealExponentialSignalsCandaarereal Fora 0 x t isgrowing Fora 0 x t isdecaying Fora 0 x t isconstant Goback PeriodicComplexExponentialSignals PeriodicComplexExponentialSignalsIfaisapurelyimaginarynumber Leta j 0 C 1 Itisperiodic LetTbetheperiod So PeriodicComplexExponentialSignals Thefundamentalfrequencyisdefinedby or Thefundamentalperiod k 0 1 2 PeriodicComplexExponentialSignals Euler sRelation or PeriodicComplexExponentialSignals Further PeriodicComplexExponentialSignals Forexample acomplexexponentialsignalgivenby PeriodicComplexExponentialSignals Anditwillplayacentralroleinmuchofourtreatmentofsignalsandsystems inpartbecausetheyserveasextremelyusefulbuildingblocksformanyothersignals Page19 Periodiccomplexexponentialisabasicperiodicsignalwhichisimportantbothintheoryandengineering Goback HarmonicRelation HarmonicRelationGivenaperiodiccomplexexponential Forsignalifitsfrequencyisanintegermultipleof 0 i e k k 0 thenwesaythatxk t isthek thharmonicofx t HarmonicRelation Usingaweightedsumofasetofharmonicallyrelatedcomplexexponentials wecanconstructmanyotherperiodicsignals Goback GeneralComplexExponentialSignals GeneralComplexExponentialSignalsThemostgeneralcaseofacomplexexponentialcanbeexpressedintermsofthetwocaseswehaveexaminedsofar therealexponentialandtheperiodiccomplexexponential IfCandaarecomplex thenx t isacomplexexponential GeneralComplexExponentialSignals WeoftenexpressCinthepolarformandexpressaintherectangularform Polarform Rectangularform Then Realpartofx t Imaginarypartofx t r 0 r 0 GeneralComplexExponentialSignals Thesegrowinganddecayingsinusoidalsignalsareplottedinthefollowingfigure Theenvelopesare Goback 1 3 1Continuous timeComplexExponentials Example1 5Givenasignal Expressx t asaproductofasingleperiodiccomplexexponentialandasinglesinusoid Theansweris Goback 1 3 1Continuous timeComplexExponentials Solution Thesignalx t canbeexpressedas FromtheEuler srelation wecanget Sothemagnitudeofx t is Goback 1 3 2Discrete TimeExponentialandSinusoidalSignals RealExponentialSignalsIfCand arereal thesignalx n iscalledtherealexponentialsignal 1 0 1 0 1 3 2Discrete TimeExponentialandSinusoidalSignals For 1 thesignalx n isdecaying 0 1 0 1 3 2Discrete TimeExponentialandSinusoidalSignals For 1 thesignalx n isaconstant 1 1 Goback 1 3 2Discrete TimeExponentialandSinusoidalSignals SinusoidalSignals 1 3 2Discrete TimeExponentialandSinusoidalSignals Ageneralsinusoidalsequence and Goback 1 3 2Discrete TimeExponentialandSinusoidalSignals GeneralComplexExponentialSignalsForthecomplexexponentialsignalx n thenx n canbeexpressedas 1 3 2Discrete TimeExponentialandSinusoidalSignals for 1and 1 Goback 1 3 3PeriodicityofDiscrete timeComplexExponentials minteger PeriodicityNowconsider AssumethatitsperiodisN then Conclusion Thesignalej 0nisperiodicifandonlyif2 0isarationalnumber 1 3 3PeriodicityofDiscrete timeComplexExponentials For 0 0 6radians 0 0 2 radians nonperiodic periodic 1 3 3PeriodicityofDiscrete timeComplexExponentials DeterminationofthefundamentalperiodFortheseperiodicsignals Note BothNandmarepositiveintegers andhavenofactorincommon Thefundamentalfrequencyisdefinedby Usetheformula 1 3 3PeriodicityofDiscrete timeComplexExponentials Averyinterestingphenomenonof Let 0bedifferentvaluesrespectively 1 3 3PeriodicityofDiscrete timeComplexExponentials Weseethatthehighestrateofoscillationofsignaloccursat 0 1 3 3PeriodicityofDiscrete timeComplexExponentials Thisimpliesthatthediscrete timecomplexexponentialsignalsarealwaysperiodicsignalsof withperiod2 Becauseoftheperiodicity thesignalexp j 0n doesnothaveacontinuallyincreasingrateofoscillationas 0isincreasedinmagnitude Goback 1 3 3PeriodicityofDiscrete timeComplexExponentials TheharmonicrelationGivenaperiodiccomplexexponential Forsignal Ifxk n hasafrequency kwhichisanintegermultipleof 0 i e k k 0 thenwesayxk n isthek thharmonicofx n 1 3 3PeriodicityofDiscrete timeComplexExponentials Defineasetofharmonicallyrelatedsequencesto k 0 1 2 ItcanbeseenthereareonlyNdistinctperiodicexponentialsinthesetgivenin k n Goback Thefundamentalperiodis 1 3 3PeriodicityofDiscrete timeComplexExponentials Example 1 6Determinethefundamentalperiodofthediscrete timesignal Solution 1 3 3PeriodicityofDiscrete timeComplexExponentials Goback 1 4TheUnitImpulseandUnitStepFunction 1 4 1TheDiscrete TimeUnitImpulseandUnitStepSequencesThediscrete timeunitimpulseandstepsequencesaredefinedby Goback 1 4TheUnitImpulseandUnitStepFunction Relationshipbetweentheimpulseandthestepfunctions 1 Thediscrete timeimpulseisthefirstdifferenceoftheunitstep thatis 1 4TheUnitImpulseandUnitStepFunction 2 Thediscrete timeunitstepistherunningsumoftheunitimpulse Forn 0 Forn 0 1 4TheUnitImpulseandUnitStepFunction or Forn 0 Forn 0 Goback 1 4TheUnitImpulseandUnitStepFunction Moregenerally SamplingPropertyofthediscrete timeunitimpulse Goback 1 4TheUnitImpulseandUnitStepFunction 1 4 2TheContinuous TimeUnitImpulseandUnitStepFunctionsThecontinuous timeunitstepfunctionisdefinedby Note theunitstepisdiscontinuousatt 0 sothevalueofu t att 0isundefined 1 4TheUnitImpulseandUnitStepFunction Theunitimpulsefunction t isdefinedby Note Theunitimpulsefunction t hasnonzeroatt 0 andhaszeroforallnonzerovaluesoft Andtheareaunder t is1 Goback 1 4TheUnitImpulseandUnitStepFunction 1 Theunitimpulse t isthefirstderivativeoftheunitstepfunctionu t 1 4TheUnitImpulseandUnitStepFunction 2 Theunitstepistherunningintegraloftheunitimpulse Goback 1 4TheUnitImpulseandUnitStepFunction Samplingproperty and Goback 1 4TheUnitImpulseandUnitStepFunction Signalsareoftendefinedintervalbyinterval Forexample supposethatx t isgivenby Signalsexpressedintermsofunit step wherex1 t x2 t x3 t arearbitrarycontinuousfunctionsoft 1 4TheUnitImpulseandUnitStepFunction Suchsignalscanbeexpressedanalyticallyintermsoftheunit stepfunctionu t andtimeshiftsofu t 1 4TheUnitImpulseandUnitStepFunction Example1 7Considerthediscontinuoussignalx t depictedinthefigure Itcanbeexpressedintermsoftheunitstep 1 4TheUnitImpulseandUnitStepFunction Fromtheexpressionofx t intermsoftheunitstep wecanreadilycalculateandgraphthederivativeofx t Goback Thecontinuous timeanddiscrete timesystems Continuous timesystems Discrete timesystems Goback 1 5 1SimpleExamplesofSystems 1 5 1SimpleExamplesofSystemsExample 1 8ConsidertheRCcircuitinthefollowingfigure Determinetherelationshipbetweenvc t andvs t Answer 1 5 1SimpleExamplesofSystems Example 1 10Considerasimplemodelforthebalanceinabankaccountfrommonthtomonth Lety n denotethebalanceattheendofthen thmonth andsupposethaty n evolvesfrommonthtomonthaccordingtothedifferenceequation x n Thenetdeposit 1 5 1SimpleExamplesofSystems Howtorepresentasystem Fromtheaboveexamples weseethatwedescribeasystemusingamathematicalequation calledthemathematicalmodel Foracontinuous timesystem themathematicalmodelisadifferentialequation andforadiscrete timesystem themathematicalmodelisadifferenceequation Goback 1 5 2InterconnectionsofSystems 1 5 2InterconnectionsofSystemsManyrealsystemsarebuiltasinterconnectionsofseveralsubsystems Therearefourtypesofinterconnections series parallel series parallelandfeedbackinterconnections 1 5 2InterconnectionsofSystems Series cascade interconnection Parallelinterconnection 1 5 2InterconnectionsofSystems Series parallelinterconnection 1 5 2InterconnectionsofSystems Feedbackinterconnection Goback 1 6 1SystemswithandwithoutMemory 1 6 1SystemswithandwithoutMemoryDefinition Asystemissaidtobememorylessifitsoutputforeachvalueoftheindependentvariableatagiventimeisdependentonlyontheinputatthatsametime Considerthesystemcharacterizedby Itisamemorylesssystem Acontinuous timesystemcharacterizedby 1 6 1SystemswithandwithoutMemory Thissystemisamemorysystem ismemoryless Considerthesystemdescribedby 1 6 1SystemswithandwithoutMemory Acapacitorisanexampleofacontinuous timesystemwithmemory since Goback 1 6 2InvertibilityandInverseSystems 1 6 2InvertibilityandInverseSystemsDefinition Asystemissaidtobeinvertibleifdistinctinputsleadtodistinctoutputs x n 1 6 2InvertibilityandInverseSystems Anexampleofaninvertiblecontinuous timesystemis Theinversesystemisconstructedas 1 6 2InvertibilityandInverseSystems Givenasystemdescribedby Itsinversesystemis Cascadetheoriginalsystemwiththeinversesystemwegettheidentitysystem Goback 1 6 3Causality 1 6 3CausalityAsystemissaidtobecausaliftheoutputatanytimeonlydependsonvaluesoftheinputatthepresenttimeandinthepast Page46Thusinacausalsystemitisimpossibletogetanoutputbeforeaninputisappliedtothesystem assumingnoinitialenergy 1 6 3Causality TheRCcircuitiscausal sincethecapacitorvoltagerespondsonlyonthepastsourcevoltage Butthesystemsdefinedby and arenot Why Goback 1 6 4Stability 1 6 4StabilityStabilityisanotherimportantsystemproperty Informally astablesystemisoneinwhichsmallinputleadtoresponsesthatdonotdiverge Stablesystem Unstablesystem 1 6 4Stability Considerthesystemgivenby Wehave Itcanbeseenthaty n growswithoutbound Sothesystemisunstable 1 6 4Stability BIBOdefinitionAsystemissaidtobestableifaboundedinputleadstoaboundedoutput Howtocheckasystemwhetherisstableornot Goback 1 6 5TimeInvariance 1 6 5TimeInvarianceConceptually asystemistimeinvariantifthebehaviorandcharacteristicsofthesystemarefixedovertime AnexampleoftimeinvariantsystemistheRCseriescircuitifthevaluesofRandCareconstants 1 6 5TimeInvariance 1 6 5TimeInvarianceConceptually asystemistimeinvariantifthebehaviorandcharacteristicsofthesystemarefixedovertime AnexampleoftimeinvariantsystemistheRCseriescircuitifthevaluesofRandCareconstants 1 6 5TimeInvariance IfthevaluesofRandCarechangedovertime thenthecircuitistimevariant ThevalueRCchangestogreaterovertime ThevalueRCchangestosmallerovertime Goback 1 6 6Linearity 1 6 6LinearityAlinearsystemisasystemthatpossessestheimportantpropertyofsuperposition Ifaninputconsistsoftheweightedsumofseveralsignals thentheoutputisthesuperposition thatis theweightedsum oftheresponsesofthesystemtoeachofthosesignals 1 6 6Linearity Foralinearsystem itmustsatisfybothadditivityandhomogeneity Additivity 1 6 6Linearity Homogeneity Combiningthetwoproperties Goback Examples Causality Example1 12Considerthesystemdefinedby To