信號(hào)與系統(tǒng)英文版課件Chapter 2

Contents§2.1DiscreteLTIsystems:theconvolutionsum§2.2ContinuousLTIsystems:theconvolutionintegral§2.3PropertiesofLTIsystems§2.4CausalLTIsystemsdescribedbydifferentialordifferenceequations§2.5SingularityfunctionsDiscreteLTIsystems:theconvolutionsumConsidersignalx[n]:Anarbitrarysequencecanberepresentedasalinearcombinationofshiftedunitimpulses[n-k],wheretheweightsinthislinearcombinationarex[k].Writeas:……-4-3-2-101234nx[n]DiscreteLTIsystems:theconvolutionsum

(siftingproperty—篩選性).....2]-[n

x[2]1]-[n

x[1]

][]0[]1[]1[]2[]2[....][+++++-++-+=dddddnxnxnxnxDiscreteLTIsystems:theconvolutionsumLeth[n]denotetheresponseoflinearsystemto[n].

i.e.

h[n]—theunitimpulseresponse.then,each[n]ofx[n]response:

….….DiscreteLTIsystems:theconvolutionsum

….….

Thisresultisreferredtoastheconvolutionsum,andtheoperationontheright-handsideofequationisknownastheconvolutionsumofx[n]andh[n].DiscreteLTIsystems:theconvolutionsumWerepresenttheoperationas

y[n]=x[n]h[n](2.7)Thesame,isreferredtoastheconvolutionsumofx[n]and[n].Somenotes:1)AnLTIsystemiscompletelycharacterizedbyitsh[n].DiscreteLTIsystems:theconvolutionsum2)Thegraphofconvolutionsum.Transformindependentvariable:

x[n],h[n]x[k],h[k]andh[k]

h[-k]Shifth[-k]nstepsh[n-k].Foranyn,x[k]multipliedbyh[n-k]andx[k]·h[n-k].DiscreteLTIsystems:theconvolutionsumExample2.2determiney[n]=x[n]h[n].Answer:(a)DiscreteLTIsystems:theconvolutionsum

(b)Shifth[-k]totheright(n>0)ortotheleft(n<0).n<0,y[n]=0n=0,y[0]=x[0]h[0]=0.51=0.5n=1,y[1]=x[0]h[1]+x[1]h[0]=0.5+2=2.5n=1n=2DiscreteLTIsystems:theconvolutionsum

(c)Foranyparticularvalueofn,wemultiplythesetwosignalsandsumoverallvaluesofk.n=2,y[2]=x[0]h[2]+x[1]h[1]=0.5+2=2.5n=3,y[3]=x[1]h[2]=2n4,y[n]=0n=3n=4n=2DiscreteLTIsystems:theconvolutionsumory[n]={0.5,2.5,2.5,2}n=0,1,2,3,Themainstepsofconvolutionsumgraph:

reversalshiftmultiplysum(3)Calculationofconvolutionsumcanbedonebyuprightmultiplication.DiscreteLTIsystems:theconvolutionsum

h[n]{0.52}(0)

x[n]{111}(0)0.520.520.52

y[n]{0.52.52.52}(0)DiscreteLTIsystems:theconvolutionsumExample2.3x[n]andh[n]givenbyDeterminey[n]=?Answer:forn0

(Seeproblem1.54)ContinuousLTIsystems:theconvolutionintegral

Webeginbyconsideringapulseor“staircase”approximation,x(t)toacontinuoussignalx(t).IfdefineThen,,andtheshadepulseis,isAs,writtenas……tkx(k)ContinuousLTIsystems:theconvolutionintegralContinuousLTIsystems:theconvolutionintegralConsequently—siftingpropertyofFortheexampleofx(t)=u(t),ContinuousLTIsystems:theconvolutionintegralLeth(t)astheresponseofLTIto

，thenconvolutionintegralorsuperpositionintegralEq.willberepresentedsymbolicallyas

y(t)=x(t)h(t)ContinuousLTIsystems:theconvolutionintegralProcedureforevaluatingconvolution⑴Changetimevariablesandreverse:⑵Shiftlikethis:whent>0towardright;otherwisetowardleft⑶Multiplytogether⑷IntegralContinuousLTIsystems:theconvolutionintegralCompareExample2.6:Answer:ContinuousLTIsystems:theconvolutionintegralAnswer:Fort<0:BecauseThereforeFort>00101t0t>0t10t<0h(t-)1ContinuousLTIsystems:theconvolutionintegral

foralltExample:DetermineAnswer:01e2e-(t-)t01e2te-(t-)………0y(t)t…01e2x()=h()=e-(t<0)(t>0)ContinuousLTIsystems:theconvolutionintegral01e2e-(t-)t01e2te-(t-)0y(t)tContinuousLTIsystems:theconvolutionintegralExample2.7consider:Andthendetermine

x(t)={

1,0<t<T

0,otherwiseh(t)={t,0<t<2T0,otherwise1T0x(t)tT2T2Th(t)0tContinuousLTIsystems:theconvolutionintegralAnswer:(1)t<0andt>3T

(2)0<t<T(3)T<t<2T

1T0x()t-2T0tT1T<t<2TT2T2Th()0t-2T0tT10<t<TtContinuousLTIsystems:theconvolutionintegral(4)2T<t<3Ty(t)=0,t<0andt>3TAssignments(P139)2.10,2.11t-2T0tT1tT2T3T..…………y(t)0PropertiesofconvolutionThecommutativepropertyx(t)y(t)h(t)x(t)y(t)h(t)PropertiesofconvolutionThedistributivepropertyAlso,x(t)＋y(t)＋x(t)y(t)PropertiesofconvolutionTheassociativeproperty(c)y[n]x[n]=*(a)x[n]y[n](d)x[n]y[n](b)y[n]x[n]=*PropertiesofLTIsystemsLTIsystemswithandwithoutmemoryFordiscretesystemwithoutmemory:

h[n]=0forn≠0Inthiscasewherek=h[0]isaconstant,andconvolutionsumisPropertiesofLTIsystemsWithmemory:h[n]≠0forn≠0.Forcontinuoussystemwithoutmemory:

h(t)=0fort≠0

k—aconstantIfk=1,systemsbecomeidentitysystems,andPropertiesofLTIsystemsInvertibilityofLTIsystemIfthenthesystemwithistheinverseofthesystemwith.x(t)h0(t)w(t)=x(t)(a)h1(t)x(t)identitysystemh(t)=(t)x(t)(b)PropertiesofLTIsystemsSimilarly,ifthensystemofistheinversesystemofExample:considerdeterminePropertiesofLTIsystemsCausalityforLTIsystemsIfh[n]=0,forn<0orh(t)=0,fort<0thenthesystemiscausal.InthiscasePropertiesofLTIsystemsStabilityforLTIsystems

absolutelysummable.

absolutelyintegrable.Example:ifthenisstableifisn’tstable

Assignments:2.28(a)(d),2.29(a)(d)PropertiesofLTIsystemsTheunitstepresponseofanLTIsystemTheunitstepresponse,s(t)ors[n],correspondingtotheoutputwhen

x[n]=u[n]orx(t)=u(t)PropertiesofLTIsystems

Assignments:P1392.12,2.23CausalLTIsystemsdescribedbyequationsLinearconstant-coefficientdifferentialequationsAfirst-order

ageneralNth-order

(2.109)CausalLTIsystemsdescribedbyequations(1)theresponsetoaninputx(t)willgenerallyconsistofthesumofaparticularsolutionandahomogeneoussolution,i.e.(2)Inordertodeterminey(t),wemustspecifyauxiliaryconditions.(3)Wewillusetheconditionofinitialrestasauxiliarycondition.Thatis,ifx(t)=0for,weassumethat(4)Undertheconditionofinitialrest,thesystemdescribedbyEq.(2.109)iscausalandLTI.[determineh(t)fromeq(2.109),seeproblem2.56.]CausalLTIsystemsdescribedbyequationsLinearconstantcoefficientdifferenceequationsThenth-order(1)Inamannerexactlyanalogoustothatfordifferential(2)Auxiliaryconditionalsoisinitialrest,

i.eifx[n]=0forn<n0,theny[n]=0for

n<.ThissystemisLTIandcausal.CausalLTIsystemsdescribedbyequations(3)Eq.(2.113)canberearrangedintheformiscalledarecursiveequation(N≧1).Determineh[n]andy[n]ofEq.(2.115)bymeansofrecursion.CausalLTIsystemsdescribedbyequationsExample2.15considerdetermineh[n]=?Solution:Letandsystemisinitialrest,i.e.,forn≦-1.forn≧0,……infiniteimpulseresponsesystem，Assignment:2.18(P140)CausalLTIsystemsdescribedbyequationsBlockdiagramrepresentationsoffirst-ordersystemsdescribedbydifferentialanddifferenceequationsConsiderequation(2.126)Threebasicoperations:＋(a)anadderx[n]aax[n](b)multiplicationbyacoefficientDx[n-1](c)aunitdelayx[n]CausalLTIsystemsdescribedbyequationsRewriteequation(2.126)asarecursive(2.127)BlockdiagramdescribedbyEq.(2.126)＋Dy[n]-ay[n-1]x[n]bCausalLTIsystemsdescribedbyequationsAfirst-orderdifferentialequationrewriteitasThreebasicoperations:Additionmultiplicationbyacoefficientdifferentiationy(t)＋Dx(t)CausalLTIsystemsdescribedbyequationsOursystemcanbeimplementedusinganintegrator.Assignment:p(148)2.38(a)2.39(b)x(t)＋y(t)-abSingularityfunctionsTheunitimpulseasanidealizedshortpulseTheish(t)ofidentitysystem.Equation(2.134)isabasicpropertyof.again0t2t0SingularityfunctionsSimilarly,thelimitsasoformustbeunitimpulses,andsoonDefiningtheunitimpulsethroughconvolutionUptothepresent,bedefinedas

(or)Thelimitsofshortpulsewithdurationandthearea=1.

SingularityfunctionsSampling(orsifting)property:Definingtheunitimpulsethroughconvolution:1.letx(t)=1(forallt)Then2.Supposeanyfunctionx(t)andf(t)boundedeverywhereandcontinuousatt=0ort=,then

SingularityfunctionsInferenceAnd3.isevenfunction:SingularityfunctionsProve:Consi