signal 信号与系统 周建华 光电学院教学课件

1SignalandSystem Emphasesinthischapter 1 Howtoconfirmasignal senergyandpower 2 Howtojudgeasignalisperiodicoraperiodicsignal 3 Thepropertiesofcomplexexponentialandsinusoidalsignals 4 Howtodotime shifting time reversalandtime scaling 5 Thepropertiesofunitimpulseandunitstepfunctions 6 Howtoconfirmwhetherasystemismemory invertible causal stable time invariantorlinearornot 1SignalandSystem 1 1Continuous timeanddiscrete timesignals 1 1 1ExamplesandMathematicalRepresentation A Examples 1 ASpeechSignal continuous timesignal 1SignalandSystem Shouldwechase 2 Continuous timeSignal 1SignalandSystem 3 Weeklystockmarketindex discrete timeSignal 1SignalandSystem B Representation Contiuous timeSignalExample x t cos 0tDiscrete timesignalExample x n 2n 2 GraphicalRepresentation 1SignalandSystem 1 FunctionRepresentation 1SignalandSystem Thesourceofdiscrete timesignal Itsindependentvariableisinherentlydiscrete suchasdemographicdata andyearlymobilestaledata 2 Thesamplingofcontinuous timesignals 1 1 2SignalEnergyandPower A Energy Continuous timesignal LetR 1 so Example resisterR sInstantaneouspower foranysignalx t theinstantaneouspower 1SignalandSystem Foracomplexsignalx t Energyovertheinterval t1 t t2 TotalEnergyoveraninfinitetimeinterval AveragePoweroveraninfinitetimeinterval 1SignalandSystem B Energy Discrete timesignal foranysignalx n instantaneouspower Energyovertheintervaln1 n n2 1SignalandSystem 1SignalandSystem TotalEnergyoveraninfinitetimeinterval AveragePoweroveraninfinitetimeinterval C FiniteEnergySignalandFinitePowerSignal Finite energySignal itsenergyE isfinite Finite powerSignal itspowerP isfinite butP 0 butE Definition 1SignalandSystem Infinite powerandinfinite energysignal bothitspowerandenergyareinfinite P E 1SignalandSystem Example DeterminethevaluesofP andE ofthefollowingsignals x t sin3tx n cos nx t tx t ej3t 1SignalandSystem 1 2TransformationsoftheIndependentVariable 1 2 1ExamplesofTransformations A TimeShift Rightshift t0 0 andn0 0 adelayedversionofx t orx n Leftshift t0 0 andn0 0 aadvancedversionofx t orx n x t x t t0 x n x n n0 1SignalandSystem Examples n0 0 t0 0 1SignalandSystem B TimeReversal Reflectionofx t orx n x t x t x n x n 1SignalandSystem C TimeScaling Stretchif01 x t x at a 0 1SignalandSystem Inordertox t x t 1 firstly timeshift leftorright sox t x t 2 Then timescaling stretchorcompress sox t x t 1SignalandSystem Example1 1a x t 1 x t 1SignalandSystem Leftshift compress Example1 1b x t Leftshift Reversalandcompress 1SignalandSystem Discretetimesignal example x n x 2 n 1SignalandSystem x n x 3n n 1SignalandSystem Example x n 1SignalandSystem 当 的值是整数m时 当 的值是分数时 x n x 3n 1 n 1SignalandSystem Example 1 2 2PeriodicSignals Definition ThereisapositivevalueofTwhich x t x t T foralltx t isperiodicwithperiodT 1SignalandSystem 1SignalandSystem Ifx t isperiodicwithperiodT thenx t x t mT foralltandforanyintegerm Thus x t isalsoperiodwithperiod2T 3T 4T ThesmallestpositiveTisreferedtoasthefundamentalperiodT0 ForDiscrete timeperiodsignal x n x n N foralln Inthesameway x n isalsoperiodicwithperiod2N 3N 4N ThesmallestpositivevalueofNiscalledasthefundamentalperiodN0 1SignalandSystem Examplesofperiodicsignal 1SignalandSystem Example1 4 Wetherornotthefollowingsignalisperiodic 1SignalandSystem 后面再举例介绍 1 2 3EvenandOddSignals Evensignal x t x t orx n x n Oddsignal x t x t orx n x n Anoddsignalmustbe0att 0orn 0 1SignalandSystem 3 2 10123 n x n 3 2 10123 n x n 1SignalandSystem Even OddDecomposition Anysignalcanbebrokenintoasumofevenpartandoddpart or 1SignalandSystem Examples Determineandsketchtheevenandoddpartsofthesignal 1SignalandSystem 1SignalandSystem Problem1 231 24 1SignalandSystem 1 3ExponentialandSinusoidalsignal 1 3 1Continuous timeComplexExponentialandSinusoidalSignals 1SignalandSystem RealExponentialSignals a 0 a 0 Growingexponential Decayingexponential 1SignalandSystem C aarerealvalue B PeriodicComplexExponentialandSinusoidalSignals Cisreal aispurelyimaginary x t satisfiesforx t x t T andSox t isperiodic Consider 0isreferredtoasthefundamentalfrequency 1SignalandSystem Thesinusoidalsignal Thefundamentalperiodis andhavethesamefundermentalperiod 1SignalandSystem Euler sRelation Wealsohave and 1SignalandSystem Aconstantsignal Itsfundamentalfrequencyiszero Anditsfundamentalperiodisundefined 1SignalandSystem 1SignalandSystem Conclusion Periodicsignalisfinite powersignal P E Example Thesetsofharmonicallyrelatedcomplexexponential thatis setsofperiodicexponentials allofwhichareperiodicwithacommonperiodT0 Thekthharmonic k t itsfundamentalfrequencyis k 0 anditsfundamentalperiodis 1SignalandSystem Example1 5Plotthemagnitudeofthesignal 1SignalandSystem TheapplicationofEuler sRelation C GeneralComplexExponentialSignals inwhich So 1SignalandSystem Signalwaves r 0 r 0 1SignalandSystem 1 3 2Discrete timeComplexExponentialandSinusoidalSignals ComplexExponentialSignal sequence 1SignalandSystem A RealExponentialSignal 1 0 1 1SignalandSystem RealExponentialSignal 1 0 1 1SignalandSystem B SinusoidalSignals Complexexponential Sinusoidalsignal 1SignalandSystem x n cos 2 n 12 1SignalandSystem C GeneralComplexExponentialSignals ComplexExponentialSignal inwhich then 1SignalandSystem RealorImaginaryofSignal 1SignalandSystem 1SignalandSystem 1 3 3PeriodicityPropertiesofDiscrete timeComplexExponentials Continuous time N Discrete time 1SignalandSystem Bydefinition thusSo Conditionofperiodicity 2 0isrational Fundamentalperiod Nandmhavenofactorsincommon 1SignalandSystem Calculateperiod If2 0isnotrational isaperiodicsignal TheperiodicitypropertyofDiscretetimesinusoidalsignalisinthesamewayasComplexexponentialsignal 1SignalandSystem Theperiodofthelinearcombinationsofperiodicsignals Period T1 T2 IfT1 T2isrational x t isperiodic ItsperiodistheminimalcommonmultipleofT1andT2 orelse x t isnotperiodic 1SignalandSystem Period T1 T2 Similaritily Period N1 N2 IfN1 N2isrational x n isperiodic ItsperiodistheminimalcommonmultipleofN1andN2 orelse x n isnotperiodic 1SignalandSystem Period T1 T2 N 12 T 12 N 31 T 31 4 aperiodic T 12 Example Determinetheperiodofthefollowingsignals 1SignalandSystem N 24 Example1 6Determinetheperiodofthefollowingsignal 1SignalandSystem a b c aperiodic Solution T Example Determinethefollowingsignals period 1SignalandSystem a b 1SignalandSystem Solution a 1SignalandSystem Solution b Thesetsofharmonicallyrelatedcomplexexponential thatis setsofperiodicexponentials allofwhichareperiodicwithacommonperiodN 1SignalandSystem ThereareonlyNdistinctperiodicexponentialsintheset Problem1 251 26 1SignalandSystem 1 4TheUnitImpulseandUnitStepFunctions 1 4 1TheDiscrete timeUnitImpulseandUnitStepSequences 1 UnitSample Impulse 1SignalandSystem UnitStepFunction 1SignalandSystem 2 RelationBetweenUnitSampleandUnitStep or Thefirstdifference Therunningsum 1SignalandSystem 3 SamplingPropertyofUnitSample 1SignalandSystem IllustrationofSampling 1SignalandSystem 1 4 2TheContinuous timeUnitStepandUnitImpulseFunctions 1 UnitStepFunction 1SignalandSystem UnitImpulseFunction 1SignalandSystem t 0 1SignalandSystem Note t 0 u t isdiscontinuous theu 0 isundefinedor n 0 u 0 1 t 0 t 0 n 0 n 0 1 1SignalandSystem 2 RelationBetweenUnitImpulseandUnitStep 1SignalandSystem Sinceu t isdiscontinuousatt 0 u t shouldbenotdifferentiable Why Howtointerpretit 1SignalandSystem t 0 t 0 1 Defining So 1SignalandSystem 1SignalandSystem 3 SamplingPropertyof t 1SignalandSystem Example1 7 1 21 1234 t x t x t 1SignalandSystem 1 5Continuous timeandDiscrete timeSystem Definitionofsystem 1 InterconnectionofComponent device subsystem Broadestsense 2 Aprocessinwhichsignalscanbetransformed Narrowsense RepresentationofSystem 1 Relationbythenotation 1SignalandSystem 2 PictorialRepresentation model Continuous timesystem x t y t Discrete timesystem x n y n 1SignalandSystem 1 5 1SimpleExampleofsystems Example1 8 RCCircuitinFigure1 1 Vc t Vs t RCCircuit system vs t vc t 1SignalandSystem Example1 10 Balanceinabankaccountfrommonthtomonth balance y n netdeposit x n interest 1 soy n y n 1 1 y n 1 x n ory n 1 01y n 1 x n Balanceinbank system x n y n 1SignalandSystem 1 5 2InterconnectionsofSystem 1 Series cascade interconnection 1SignalandSystem 2 Parallelinterconnection Series Parallelinterconnection 1SignalandSystem 3 Feed backinterconnection 1SignalandSystem 1 6BasicSystemProperties 1 6 1SystemswithandwithoutMemory Memorylesssystem Itsoutputisdependentonlyontheinputatthesametime 1SignalandSystem Contrarily ifitsoutputisrelatedwithitspastinputoritsfutureinput thesystemismemorysystem 1SignalandSystem Examplesofmemorylesssystem y t Cx t ory n Cx n ory n 0 5y n 1 2x n Examplesofmemorysystem Aresistorisamemorylesssystem Acapacitorisamemorysystem 1SignalandSystem Aaccumulatorisamemorysystem Adelayisamemorysystem 1 6 2InvertibilityandInverseSystems Invertiblesystemdefinition Ifdistinctinputsleadtodistinctoutputs thesystemisinvertiblesystem 1SignalandSystem 1 Ifsystemisinvertible thenaninversesystemexists 2 Aninversesystemcascadedwiththeoriginalsystem yieldsanoutputequaltotheinput Features 1SignalandSystem Examples 1 6 3Causality Causalsystemdefinition AsystemiscausalIftheoutputatanytimedependsonlyonvaluesoftheinputatthepresenttimeandinthepast 1SignalandSystem noncausalsystem Iftheoutputatsometimedependsonvaluesoftheinputinthefuture thesystemisnoncausal 1SignalandSystem Example Causalsystem noncausalsystem conclusion Allmemorylesssystemsarecausal 1SignalandSystem Example1 12Determinewhetherthefollowingsystemsarecausalornot 1 2 Noncausalsystem Causalsystem 1 6 4Stability Stablesystemdefinition Finiteinputleadtofiniteoutput 1SignalandSystem 1SignalandSystem Smallinputsleadtoresponsesthatdonotdiverge Examples StablependulumMotionofautomobile Example1 13 1SignalandSystem Examples1 13Showthefollowingsystemsarestablesystems 1 2 unstablesystem stablesystem 1SignalandSystem Examples Showthefollowingsystemsarestablesystems Solution 1 If 2 If 1 6 5TimeInvariance Time invariantsystemdefinition Ifatimeshiftintheinputsignalresultsinanidenticaltimeshiftintheoutputsignal Time invariantsystem Ifx t y t thenx t t0 y t t0 Ifx n y n thenx n n0 y n n0 1SignalandSystem Example1 141 151 16 Determinewhetherornoteachofthefollowingsystemsaretime invariant 1SignalandSystem 1 6 6Linearity Linearsystemdefinition Thesystempossessestheimportantpropertyofsuperposition 1 Additivityproperty Theresponsetox1 t x2 t isy1 t y2 t 2 Scalingorhomogeneityproperty Theresponsetoax1 t isay1 t whereaisanycomplexconstant a 0 1SignalandSystem Thatis linearsystemsatisfies Theresponsetoax1 t bx2 t isay1 t by2 t L x1 t x2 t y1 t y2 t ax1 t x1 t x2 t ax1 t bx2 t ay1 t y1 t y2