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
Download Pdf The PPT/PDF document "EECS IMPULSES AND IMPULSE RESPONSE CONT..." is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.
EECS216IMPULSESANDIMPULSERESPONSE CONTINUOUS-TIMEIMPULSES Def:Animpulseisthelimitingcaseofaconstant-areahighandfastpulse.Animpulsehaszerowidth,in niteheight,and niteareaunderit.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)fort�0using(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?Becausethecurrentisin nite! PROPERTIESOFIMPULSEANDIMPULSERESPONSE 1.Impulseresponse=h(t)=ds dt=d dt(stepresponse).Forabovecircuit,h(t)=d dt(1 e t=(RC))=1 RCe t=(RC)fort�0.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) tareunde ned. 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( )d dh dtx(t)=h(t)dx dt=dy dt;similarlyforintegrals.#1-#3greatlysimpli esmanyconvolutionsinEECS216. 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 )d soyoudon'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 d u(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!).