Convolution - Rutgers University

1 Systematrest(zero initialconditions)duetoanyinputistheconv olutionofthatinputandthesystemimpulseres ponse. Wehavealreadyseenandderivedthisresultint hefrequencydomaininChapters3,4,and5,henc e,themainconvolutiontheoremis applicableto, anddomains,thatis,it , , PrenticeHall, and isdefinedby Inthisintegralisa dummyvariableofintegration,andisa ,wefirstintroducethenotionofthesignaldur ation. Thedurationofa signal isdefinedbythetimeinstants and forwhichforeveryoutsidetheinterval thesignalis equaltozero,thatis, , . Signalsthathavefinitedurationareoftencal ledtime-limitedsignals. Forexample,rectangularandtriangularpulse saretime-limitedsignals, :Theslidescontainthecopyrightedmaterialf romLinearDynamicSystemsandSignals, PrenticeHall, 21)Commutativity 2)Distributivity 3)Associativity 4)DurationLetthesignals and havethedurations,respectively,definedbyt hetimeintervals and then Theslidescontainthecopyrightedmaterialfr omLinearDynamicSystemsandSignals, PrenticeHall, 35)TimeShiftingLet.

2 Then,convolutionsofshiftedsignalsaregive nby 6)ContinuityThispropertysimplystatesthat theconvolutionisa continuousfunctionoftheparameter. , , PrenticeHall, 4 Property1)canbeprovedbyintroducingthecha ngeofvariablesintheconvo-lutionintegrala s. Thisleadsto Whichofthetwoformsoftheconvolutionintegr alshouldwechoose?Definitely, ,whileconvolving andwemayuseeitheroftheintegrals , , PrenticeHall, 5 TheproofofProperty2)followsfromthewellkn ownintegraladditionproperty!"#$%$!"#$%$! "$%$!#!"!#Property4) !and"overlaponlyintheinterval!"!", ) )and6) , PrenticeHall, :Considertheconvolutionofthedeltaimpulse (singular)signalandanyotherregularsignal &'& :We havealreadyseeninthecontextoftheintegral propertyoftheFouriertransformthattheconv olutionoftheunitstepsignalwitha regularfunction( signal )producesfunction sintegralinthespecifiedlimits,thatis&'&( '& , PrenticeHall, :Considertheconvolutionof) *and) *+)+).

3 */) 021+)+) 0We ) *) **)+0) **Theslidescontainthecopyrightedmaterial fromLinearDynamicSystemsandSignals, PrenticeHall, 8 Theevaluationofthesecondintegralrequires firstanexpansionofterm,thatis345 6345 6whichgives345 6345 65 7 Thus,bothconvolutionintegralsproducethes ameresult, , PrenticeHall, ,theconvolutionprocedureinvolvesthefollo wingsteps:Step1 :Flipabouttheverticalaxisoneofthesignals (theonethathasa simplerform(shape)sincethecommutativityh olds),thatis, :Varytheparameterfromto, thatis,slidetheflippedsignalfromtheleftt otheright,lookfortheintervalswhereit overlapswiththeothersignal, , PrenticeHall, 10 Intheabovestepsonecanalsoincorporate(ifa pplicable) case,afterthefinalconvolutionresultis ,theconvolutioncontinuitypropertymaybeus edtochecktheobtainedconvolutionresult,wh ichrequiresthatattheboundariesofadjacent intervalstheconvolutionremainsa (t)f2(t).

4 TworectangularsignalsTheslidescontainthe copyrightedmaterialfromLinearDynamicSyst emsandSignals, PrenticeHall, 11 Sincethedurationsofthesignals8and9areres pectivelygivenby88and99, weconcludethattheconvolutionofthesetwosi gnalsiszerointhefollowingintervals(Step1 )89898989 Thus, ,weflipabouttheverticalaxisthesignalwhic hhasa Notethattheconvolutionisperformedintheti mescale. , PrenticeHall, 12 InStep3,weshiftthesignal:totheleftandtot heright,thatis,weformthesignal:forand. Ashiftofthesignal:totheleftproducesnoove rlappingbetweenthesignals;and:, thustheconvolutionintegralis equaltozerofor( ). ( )031f1( ) 0 :Signals< =?>A@CBand<EDF>HGI@CBTheslidescontainthecopyrightedmate rialfromLinearDynamicSystemsandSignals, PrenticeHall, 13f1( )-1+t031 0 f2t(t- ) :SignalsJ KMLANPOandJEQRLTS UNVOXWYS[Z\Letusstartshiftingthesignal]t otheright().

5 Considerfirsttheinterval( ).Theslidescontainthecopyrightedmaterial fromLinearDynamicSystemsandSignals, PrenticeHall, 14f1( )f2(t- )-1+t031 :Signals^ _M`AaPband^dc ` ,hencetheirproductisdifferentfromzeroint hisinterval,whichimpliesthattheconvoluti onintegralisgivenbyqrstTheslidescontaint hecopyrightedmaterialfromLinearDynamicSy stemsandSignals, PrenticeHall, 15 Byshiftingthesignalufurthertotheright,we getthesame kindofoverlap for, ( )f (t- )2t-1t-1031 :Signalsv wMxAyPzandvd{ xf|h}yPzj~ |o Fromthisfigureweseethattheactualconvolut ionintegrationlimitsarefromto, thatis - Theslidescontainthecopyrightedmaterialfr omLinearDynamicSystemsandSignals, PrenticeHall, 16 Byshifting furthertotheright,for, ( )44f2(t- )031 0 :Signals M A P and d f h P j l n o Inthisinterval,theconvolutionintegralisg ivenby / - For, and donotoverlapforTheslidescontainthecopyri ghtedmaterialfromLinearDynamicSystemsand Signals, PrenticeHall, 17, thatis,theirproductisequaltozerofor, whichimpliesthatthecorrespondingintegral isequaltozerointhesameinterval, ( )f (t- )2t-1t031203.

6 Signals M A P and E R T V X Y [ Insummary,theconvolutionoftheconsidereds ignalsisgivenby Theslidescontainthecopyrightedmaterialfr omLinearDynamicSystemsandSignals, PrenticeHall, 18 Notethatfromtheconvolutioncontinuityprop erty,theconvolutionsignalobtainedis a continuousfunctionof. ,alsoforwehave, andfinallyforweget. Thusthefunctionobtained, , (t)f (t)2tt01222102-t+ :Twosignals:rectangularandtriangularpuls esSincebothsignalshavethedurationinterva lsfromzerototwo, , PrenticeHall, 19 Inthenextstepweflipabouttheverticalaxist herectangularsignalsinceitapparentlyhasa simplershape, ,weslidetherectangularsignaltotherightfo r, ,andfor, (t )f1(t )t=0<0<2t<2<4t f-21( )2t00-2024t(a)(b)(c)-2+ , PrenticeHall, 20 Theconvolutionintegralinthesetwointerval s, ,isrespectivelygivenby Insummary,wehaveobtained It canbeeasilycheckedthattheobtainedconvolu tionresultrepresentsa , PrenticeHall.]

7 We nowconsidertheslightlymoredifficultprobl emofconvolvingtwosignalswithtriangularsh apes, Theproblemis ( )f1( ) 10101 +1 :TwotriangularlyshapedsignalsLetusflipab outtheverticalaxisthesignal . , PrenticeHall, 22f2(t )f2(t )f2( ) +1 t-1t<1<0t<2<1tt=0 101t-1-10 +1t 02(a)(b)(c)-1+ ,nowgivenby . Fromtheconvolutiondurationproperty,wecon cludethattheconvolutionisequaltozerofora nd. Thus, , PrenticeHall, 23 Considertheinterval. Inthisinterval,thesignal is givenby , overlapswiththesignal intheintervalfromzeroto, theconvolutionisgivenby For, thesignal , ,overlapswiththesignal intheintervalfromto. Here,theconvolutionis givenby / Theslidescontainthecopyrightedmaterialfr omLinearDynamicSystemsandSignals, PrenticeHall, (t)ttf1(t) :Convolutionwitha shiftedsignalAccordingtotheconvolutionti meshiftingproperty,wecanshiftthesignal totheoriginandfindtheconvolutionoftheshi ftedsignal andthesignal.

8 Letrepresenttheconvolutionof and .Inordertofindtherequiredoriginalconvolu tionresult,theconvolutionobtainedthrough theregularconvolutionprocedurewith and , PrenticeHall, 25 However, and , weconcludethatthecorrespondingconvolutio nis equaltozerofor and .Letusflipabouttheverticalaxistherectang ularsignal , (t )f1(t )f1( ) 1-3-10<3<1tt=0<3<5t -11-1-201230tt(c)(a)(b) , PrenticeHall, 26 Intheinterval, theconvolutionisgivenby( ) / - Intheinterval, / Inthenextsectionweapplytheconvolutionfor mulatolinearcontinuous-timeinvariantsyst emsandshowthatthesystemresponsetoanyinpu tis thatend,wewillusetheconceptsofsystemtran sferfunctionandsystemimpulseresponseintr oducedinChapters3 , PrenticeHall, systematrest(zeroinitialconditions)dueto anyinput,say.

9 Inthepreviouschaptersonthefrequencydomai ntechniques, (t)Linear SystemH(s), h(t)y(t)=? :SystemresponseduetoanarbitraryinputReca llthat. We assumethatthesysteminitialconditionsarez ero(systematrest)andthatiscausal().Hence , , PrenticeHall, 28 Sincethesystemimpulseresponseisknown,wed oknowtheanswertothesystemimpulseresponse problem,whichis (systemoperation)ontheknowninputwewillha veorsimilarly,byassumingtimeinvariance )o)(t-t h(t-tLinear Systemh(t), :Systemresponseduetotheimpulsedeltafunct ionTheslidescontainthecopyrightedmateria lfromLinearDynamicSystemsandSignals, PrenticeHall, 29We canpresentanyinputsignalintermsofthedelt aimpulsesignalas( ) , weget R Thisformulaestablishesinthetimedomainthe mostfundamentalresultoflinearsystemtheor y, , PrenticeHall, continuous-timelinearsystematrest(zero-s tateresponse) Systemy(t)=f(t)*h(t)f(t)zsh(t).

10 Zero-statesystemresponseistheconvolution ofthesysteminputandthesystemimpulserespo nseNotethattheobtainedzero-stateresponse convolutionformula R canberepresentedasa sumofthreeintegralsTheslidescontaintheco pyrightedmaterialfromLinearDynamicSystem sandSignals, PrenticeHall, 31 R Sincefor(causalinputsignal), ,theintegrationisperformedintheregionwhe re(causallinearsystem),hence, ,weareleftwith R whichproducesthezero-statesystemresponse at changeofvariableas, it canbeeasilyshownthat R Theslidescontainthecopyrightedmaterialfr omLinearDynamicSystemsandSignals, PrenticeHall, :Givena lineardynamicsystemrepresentedby H Itstransferfunctionandtheimpulseresponse aregivenby Thesystemzero-stateresponsedueto H is F M.