An impulse has zero width in64257nite height and 64257nite area under it Math Mathematicians Impulses are distributions or generalized functions Dont call them Dirac delta functionsDirac would sue for de famation Def The impulse response of a syste ID: 24901
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EECS216IMPULSESANDIMPULSERESPONSE CONTINUOUS-TIMEIMPULSES Def:Animpulseisthelimitingcaseofaconstant-areahighandfastpulse.Animpulsehaszerowidth,inniteheight,andniteareaunderit.Math:Mathematicians:Impulsesaredistributionsorgeneralizedfunctions.Don'tcallthemDiracdeltafunctions{Diracwouldsuefordefamation!Def:Theimpulseresponseofasystemissimplyitsresponsetoanimpulse:Impulse=(t)! SYSTEM !h(t)=impulseresponse(makessense).Def:Thestepresponseofasystemissimplyitsresponsetoastepfunction:step=1(t)! SYSTEM !s(t)=stepresponse(iseasiertocompute). CONTINUOUS-TIMEIMPULSERESPONSE Q:Howtocomputetheresponsetoaninputthatdoesn'texistphysically?A:ConsideraseriesRCcircuitdrivenbyahighandfastvoltagepulse:1.Source=step=x(t)=1(t)!y(t)=s(t)=[1 e t=(RC)]1(t)(familiar).2.Stepup,thendown:x(t)=1(t) 1(t )!y(t)=s(t) s(t ).3.Scaleprevious:x(t)=1 [1(t) 1(t )]!y(t)=1 [s(t) s(t )].4.Fort,thisbecomesy(t)=1 [1 e t=(RC)] 1 [1 e (t )=(RC)]=[e=(RC) 1]1 e t=(RC) RC1 e t=(RC)=1 RCe t=(RC)fort0using(ex 1)=(1+x+x2 2!+::: 1)xforx==(RC)1.5.Responsey(t)toapulsex(t)ofwidth,height1 andarea =1isindependentofand1 ,aslongasRC(notetheunits). PHYSICALINTERPRETATIONOFIMPULSEANDIMPULSERESPONSE 1.TakeNortonequivalent:Impulsivecurrentsource1 R.Duration=.2.Shotofcharge1 Rchargescapacitorupto1 C1 R=1 RC.(q=Cv)3.Capacitorvoltagedecays.Impulsivesourcelikeaninitialcondition!4.Current=1 Randduration=don'tmatter{onlycharge=product.5.Howcanthecapacitorvoltagejump?Becausethecurrentisinnite! PROPERTIESOFIMPULSEANDIMPULSERESPONSE 1.Impulseresponse=h(t)=ds dt=d dt(stepresponse).Forabovecircuit,h(t)=d dt(1 e t=(RC))=1 RCe t=(RC)fort0.Forabovecircuit, 2.h(t)=L 1fH(s)g=L 1f1=(sC) R+1=(sC)g=L 1f1=(RC) s+1=(RC)g=1 RCe t=(RC).3.Rba(t c)dt=areaunder(t)=1ifacb:R1 1(t a)dt=1.4.R1 1x(t)(t a)dt=x(a)(siftingproperty).x(t)(t a)=x(a)(t a).5.(at)=1 jaj(t)(bothhavearea1 jaj).(t)1(t)and(t) tareundened. EECS216COMPUTINGCONVOLUTIONS TENRULESFORCOMPUTINGCONVOLUTIONS 1.h(t)[ax(t)+by(t))=a[h(t)x(t)]+b[h(t)y(t)]Breakupconvolutionsusinglinearcombinations. 2.Ifh(t)x(t)=y(t)thenh(t a)x(t+b)=y(t a+b)Thisisveryusefulifthegivenfunctionshavedelays. 3.x(t)(t a)=x(t a)andx(t)u(t)=Rt 1x()ddh dtx(t)=h(t)dx dt=dy dt;similarlyforintegrals.#1-#3greatlysimpliesmanyconvolutionsinEECS216. 4.Ifbothh(t)andx(t)arecausal,theny(t)=h(t)x(t)=[Rt0h()x(t )d]u(t)isalsocausal. 5.Ifx(t)isamorecomplicatedfunctionthanh(t),usey(t)=R1 1x()h(t )dsoyoudon'tsubstitutet . 6.Ifh(t)=0outside0tLandx(t)=0outside0tM,theny(t)=h(t)x(t)=0outside0t(L+M).Anduse#2! 7.Even*even=even;Even*odd=odd;Odd*odd=evenfunctions.Use#2toshiftsymmetricandantisymmetricfunctions. 8.(t)(t)=(t)andu(t)*u(t)=r(t)=tu(t)andrect(t)*rect(t)=triangle(t). 9.Ify(t)=h(t)x(t)thenRy(t)dt=[Rh(t)dt][Rx(t)dt].Goodcheck. 10.Discretizeh(t)andx(t)anduseMatlab'sconvtocheckform.Youstillneedtosetthescalefactorproperly(use#9). EX#1:Computey(t)=e tu(t)[u(t) u(t a)]. Sol'n:Using#1-#5,y(t)=e tu(t)u(t) e tu(t)u(t a)1stterm=Rt0e du(t)=[1 e t]u(t):2ndterm=[1 e (t a)]u(t a).y(t)=[1 e t]u(t) [1 e (t a)]u(t a)=RCcircuitpulseresponse. EX#2:Computey(t)=e tu(t)t T[u(t) u(t T)].Hint:Notet T[u(t) u(t T)]=1 TRt0[u(t) u(t T) T(t T)]dt. Sol'n:Abusingnotationforclarityandusing#3andEx#1above,wehavey(t)=Rt0[e tu(t)1 T[u(t) u(t T) T(t T)]]dty(t)=1 TRt0[1 e t]dtu(t) 1 TRtT[1 e (t T)]dtu(t T) RtTe (t T)dtu(t T)y(t)=1 T[t+e t 1]u(t) 1 T[(t T)+e (t T) 1]u(t T)+[e (t T) 1]u(t T)ComparetoSolimanandSrinathp.60{yes,theyDOagree(tryit!).