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ConvolutionintegralrepresentationofCTLTIsystemsRepresentationofCTSignalsintermsofshiftedunitimpulsesPropertiesandExamplesTheunitimpulseasanidealizedpulsethatis“shortenough”:Theoperationaldefinitionofδ(t)SignalsandSystemsFall2003Lecture#416September2003
RepresentationofCTSignalsApproximateanyinputx(t)asasumofshifted,scaledpulses
hasunitareaTheSiftingPropertyoftheUnitImpulse
ResponseofaCTLTISystemImpulseresponse:TakinglimitsConvolutionIntegral
OperationofCTConvolutionExample:CTconvolution
TimeIntervalOutput
PROPERTIESANDEXAMPLESCommutativity:Anintegraor:Siftingproperty:SoifinputoutputSteprespose:
DISTRIBUTIVITY
ASSOCIATIVITYCommutativity
Causality:CTLTIsystemiscausalStability:CTLTIsystemisstable
Theimpulseasanidealized“short”pulseConsiderresponsefrominitialresttopulsesofdifferentshapesanddurations,butwithunitarea.Asthedurationdecreases,theresponsesbecomesimilarfordifferentpulseshapes.
TheOperationalDefinitionoftheUnitImpulseδ(t)δ(t)—idealizationofaunit-areapulsethatissoshortthat,foranyphysicalsystemsofinteresttous,thesystemrespondsonlytotheareaofthepulseandisinsensitivetoitsdurationOperationally:TheunitimpulseisthesignalwhichwhenappliedtoanyLTIsystemresultsinanoutputequaltotheimpulseresponseofthesystem.Thatis,δ(t)isdefinedbywhatitdoesunderconvolution.
TheUnitDoublet—DifferentiatorImpulseresponse=unitdoubletTheoperationaldefinitionoftheunitdoublet:
Tripletsandbeyond!nisnumberofdifferentiationsOperationaldefinitionsntimes
“-1derivatives=integral?I.R.=unitstepIntegratorsImpulseresponse:Operationaldefinition:Cascadeofnintegrators:ntimes
Integrators(continued)theunitrampMoregenerally,forn
NotationThenE.g.Defineandcanbe
SometimesUsefulTricksDifferentiatefirst,thenconvolve,thenintegrate
Example
Example(continued)
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