2023學(xué)年完整公開(kāi)課版chapter

1Chapter1

DiscreteSequencesandSystems2Outline1.1

DiscreteSequences1.2SignalAmplitude,Magnitude,Power1.3SignalProcessingOperationalSymbols1.4IntroductiontoDiscreteLinearTime-InvariantSystems1.5

DiscreteLinearSystems1.6Time-InvariantSystems1.7TheCommutativePropertyofLinearTime-InvariantSystems1.8TheCausalityPropertyofLinearTime-InvariantSystems1.9TheStabilityPropertyofLinearTime-InvariantSystems1.10AnalyzingLinearTime-InvariantSystems31.1.1Discrete-timeSignalsx(n)comesfrom:xa(t)=sin(2πft)Uniformedsampledas:xa(nT)=sin(2πfnT)T

denotessamplingintervalorperiod

anditsreciprocalissamplingfrequency

writtenas:samplingfrequency41.1.1Discrete-timeSignals{x(n)}:n

isintegeranddiscrete.

:

x(n)

isonesampleE.g.x(-1)=-0.1564;x(0)=0;x(1)=0.1564;x(2)=0.3090;…51.1.1Discrete-timeSignalsRelationshipbetweenxa(nT)andxa(t):Partandwhole;Manycurvesconnectingthetwopointsx(nT)andx((n+1)T),butundercertainconditions,xa(t)canbeexclusivelyreconstructedbasedonxa(nT).6(1)Unitsamplesequencediscrete-timeimpulseunitimpulseShifted:1.1.2FrequentlyUsedDiscreteSequences6()nd01()n-2d0171.1.2FrequentlyUsedDiscreteSequences7()nun01(2)Unitstepsequence81.1.2FrequentlyUsedDiscreteSequences8Relationbetweenu(n)andδ(n)δ(n)couldbeexpressedas：u(n)

couldbeexpressedas：91.1.2FrequentlyUsedDiscreteSequences9(3)Rectangularsequence()nRNn01-N101.1.2FrequentlyUsedDiscreteSequences10RelationofRN(n),u(n)

andδ(n)RN(n)isexpressedas:RN(n)isexpressedas:111.1.2FrequentlyUsedDiscreteSequences11(4)Realexponentialsequence when0<a<1()nxn0121.2SignalAmplitude,Magnitude,Power

12or131.2SignalAmplitude,Magnitude,Power

13Frequency-domainamplitudeandfrequency-domainpoweroftheXsum(n)timewaveformXsum(n)=X1(n)+X2(n)141.3SignalProcessingOperationalSymbols

OperationonSequencesSingleInput-SingleOutput:Input:corruptedsignalsOutput:puresignalsMInput-NOutput:SeveralbranchesofsignalsarecombinedtooutputButabovesystemcoulddecomposedintosimpleoperations.Including:modulator,scalarmultiplicationaddition,unitadvance151.3SignalProcessingOperationalSymbols

Addition(Adder):a(n)=b(n)+c(n)sumofsamplesatthesameinstant.15DSPWeareThinkingandInnovating…15161.3SignalProcessingOperationalSymbols

Subtract:a(n)=b(n)-c(n)differenceofsamplesatthesameinstant.16DSPWeareThinkingandInnovating…16171.3SignalProcessingOperationalSymbols

Addition(Adder):a(n)summationofsamples.17DSPWeareThinkingandInnovating…17181.3SignalProcessingOperationalSymbols

Multiplication:a(n)=b(n)·c(n)productofsamplesatthesameinstant.Like:Windowingoperation.18DSPWeareThinkingandInnovating…18191.3SignalProcessingOperationalSymbols

UnitDelay:aAdelayedversionofsample19201.4IntroductiontoDiscreteLinearTime-InvariantSystems

Becauselinearityandtimeinvariancearetwoimportantsystemcharacteristicshavingveryspecialproperties,we’lldiscussthemnow.WeneedtorecognizeandunderstandthenotionsoflinearityandtimeinvariancenotjustbecausethevastmajorityofdiscretesystemsusedinpracticeareLTIsystems,butbecauseLTIsystemsareveryaccommodatingwhenitcomestotheiranalysis20211.5DiscreteLinearSystemsThetermlineardefinesaspecialclassofsystemswheretheoutputisthesuperposition,orsum,oftheindividualoutputshadtheindividualinputsbeenappliedseparatelytothesystem.21221.5.1ExampleofaLinearSystem22231.6Time-InvariantSystemsAtime-invariantsystemisonewhereatimedelay(orshift)intheinputsequencecausesaequivalenttimedelayinthesystem’soutputsequence.23241.6.1ExampleofaTime-InvariantSystem2425E.g：Given:Isthissequencelinear?Time-shiftInvariant?Example25DSPWeareThinkingandInnovating…2526E.g：Given:(1)Linear?Example2627Example(2)Time-shiftInvariant?So,it’sTime-shiftInvariant.->ThisisnotaLTIsystem.27281.7TheCommutativePropertyofLinearTime-InvariantSystemsSwappingtheorderoftwocascadedsystemsdoesnotalterthefinaloutput.28291.8TheCausalityPropertyofLinearTime-InvariantSystemsCausalitydefinition:Thisimpliesthat:Forthecausalsystem,ifx1(n)=x2(n)forn<n0,theny1(n)=y2(n)forn<n0.29301.8TheCausalityPropertyofLinearTime-InvariantSystems30Causality:IfaLTIsystemisacausalsystem,itsatisfies:——RealizableSystem,itisusedtoprovethecausalityofthesystem(Important).311.9TheStabilityPropertyofLinearTime-InvariantSystems31Stability:Ifandonlyifforeveryboundedinput,theoutputisalsobounded(BIBO).i.e.ThenInthelaterdiscussion,theinvolveddiscretesystemistheLTIsystem.Generalpracticalsystemsarecausalandstable.321.9TheStabilityPropertyofLinearTime-InvariantSystems32SignificantConclusion:ForaLTIsystem,thesufficientbutnotnecessaryconditiontostabilityis:331.9TheStabilityPropertyofLinearTime-InvariantSystems33Proof:Because,assuming|x(n)|≤M.341.10AnalyzingLinearTime-InvariantSystemsKnowingthe(unit)impulseresponseofanLTIsyste