2015年集课件1 27给定系统是否满足下列性质

(1)无 (b)y(t)=（1）y(t)只取决于当前的输入，所以是 （2）y1(t)[cos(3t)]x1(t),令x2t)x1(tt0y2t[cos(3t)]x1(tt0，而y1(tt0[cos(3(tt0x1(tt0所以该系统是时变令x3(t令x3(t)ax1(tbx2t)y3(t)=bx2(t))=所以该系统是线性（5）因为-1£cos(3t£1有界，所以系统对于有界输入（3）y1(t)=[cos(3t)]x1(t)，y2(t)=[cos(3t)]x2

y(t)

。 x1(t)fiy1(t)=x1(t)dt,x2(t)fiy2(t)=

2(t-t0y2(t)=x1(t-t0)dt=

x1(t)fiy1(t)=x1(t)dt,x2(t)fiy2(t)=x2 x3(t)fiy3(t)=x3(t)dt=(ax1(t)+bx2 -

(1)无 y[n]=x[n-2]-2x[n- y3[n]=x3[n-2]-2x3[n-8]=ax1[n-2]+bx2[n-2]-2ax1[n-3y2[n]=x2[n-2]-2x2[n-8]=x1[n-2-n0]-2x1[n-8-n0 (g)g[n]=x[4n+ x1[n]→g1[n]=x1[4n+1],x2[n]→g2[n]=x2[4n+则当x2[nx1[n-n0]g2[n]=x2[4n+1]=x1[4n+1-n0]≠g1[n-n0]=x1[4(n-n0)+业业x1[n]→g1[n]=x1[4n+1],x2[n]→g2[n]=x2[4n+非因果，因为当nn00时，输出y[n0]与n0之后的输，也必有|g[n]||x[4n1]|非因果，因为当nn00时，输出y[n0]与n0之后的输，也必有|g[n]||x[4n1]|A业业 21

t

x2 x1(t)-x1(t-2)fiy2(t)=y1(t)-y1(t-21 t

2.21计算卷积y[nx[n*(a)x[n]=αnu[n],h[n]=βnu[n],α≠ x[k]h[n-k]=¥

aku[k]bn-ku[n-k =k akbn-k,n=k1- =bn 1-ab=bn+1-an+1b-

CausalLTISystemsDescribedbyDescribeCTsystemsintermsoflinearconstant-coefficientdifferentialequationsDescribeDTsystemsintermsoflinearconstant-coefficientdifferenceequationsyzeLTIsystemsinIntroducesomeoftheimportantideasconcerningsystemsspecifiedbylinearconstant-coefficientdifferentialequations.coefficientdifferentialequationswhichhavebeenintroducedinmathematicscourses.Pleaserecallcoefficientdifferentialequationswhichhavebeenintroducedinmathematicscourses.dvc(t)+1v(t)=1v

dv(t)+rv(t)=1fcsmmdy(t)+ay(t)=csmmdydy(t)+2y(t)=functionoftheInordertoobtainanexplicitexpression,wemustsolvethedifferentialequation findfunctionoftheInordertoobtainanexplicitexpression,wemustsolvethedifferentialequation findasolution,weneedmoreinformationthanthatprovidedbythedifferentialequationalonedydy(t)+2y(t)=Itssolutioniscomposedoftwoy(t)=yh(ty(t)=yh(t)+yp(tThehomogeneousThehomogeneoussolution,oftenreferredtoasthe

yh(t+2y(t)=Theformofthehomogeneoussolution+2y(t)=TheformofthehomogeneoussolutionisalinearcompositionsomeexponentialslikeCeλt,whereλisanThecorrespondingeigenvalueequationisl+2=Thecorrespondinghomogeneoussolution(naturalresponse)isyyh(t)=Ce2Particular Theformoftheparticularsolution,oftenreferredtoasdresponseofthesystem,arerelatedtotheformofinputsignals.Substitutetheformofthe articularsolutioniconcerneddifferentialequationanddecidependingdy(t)+2y(t)=x(t)Supposetheinputsignalx(t)=Theparticularsolutionfort>0hasthefollowingyp(t)=Plugtheinputsignalandthesolutionintothe3Ye3Plugtheinputsignalandthesolutionintothe3Ye3t+2Ye3t=Ke3tY=•5 Theoverally(t)=y(t)+y(t)=Ae-

+KwhichisasumwhichisasumofthehomogeneoussolutionandtheSomependingcoefficientsintheoverallsolutionshouldbedecidedaccordingtotheauxiliaryconditions.Formostpartinthisbook,wewillfocusontheuseofdifferentialequationstodescribecausalLTIsystemsoftheconditionofinitialInthiscase,oftheconditionofinitialForacasualLTIifx(t)=0fort<t0,theny(t)=0fort<y(t)=Ae-2t+Ky(t)=Ae-2t+Ke3t t>5Settingy(0)=0whent=0A=-5Becauseoftheconditionofinitial55y(t)=(-Ke-2t+Ke3t55Generaly(y(n-1)(t)++ay(1)(t)+a10=bx(m)(t)+ Itcanbeexressedsim nma i=b (jjjinwhichaiandbjareconstants，an=Itssolutioniscomposedoftwoparts:thehomogeneoussolutionandtheparticularsolution.y(t)=yh(t)+ypHomogeneousHomogeneousThehomogeneoussolutionisthesolutionofy(n)(t)+y(n-1)(t)++ay(1)(t)+ay(t)=combinationofsomeexponentialslikeCeλt,whereλcombinationofsomeexponentialslikeCeλt,whereλisanThecorrespondingeigenvalueequation

++al+ RelationshipbetweeneigenvaluesandhomogeneouseigenvaluesHomogenoussinglerealCemultiplerealelt tr-1+ tr-2++C1t+C0 apairofcomplexrootsl, =p–jqept[Ccos(qt)+Dsin(qtrootsl, =p–jqept tr-1+ tr-2++C1t+C0)cos tr-1+ tr-2++D1t+D0)sin RelationshipRelationshipbetweeninputandparticularinputParticular tmPtm+ 0isnotantr[Ptm+ 0isaneigenvaluewithamultiplicityofrisnotanPe isasingleeat(Ptr+Ptr-1++Pt+P isaneigenvaluewithamultiplicityofrPcos(bt)+QNoeigenvaluefollowingExamples.Determinethehomogeneoussolutionoffollowing

y(t)+7 y(t)+16 y(t)+12y(t)=x(t) dt2 Theeigenvalueequation3+7l2+16l+12=3+7l2+16l+12=Eigenvaluesarel1=-2(doubleroot),l2=-Thehomogenoussolution3 whereA1,A2,A3are

Example:Determinetheparticularsolutionscorrespondingtothegiveninputsignals. y(t)+2 y(t)+3y(t)= +x(t)dt2 (2)x(t)=etwhenx(t)=tSubstitutingtheinputsignalintotherightsideoftheequation,weget:t2+2t( Forthebalanceoftheequation,since0isnotaneigenvalue,weassumethefollowingparticularsolution:y(t)=Bt2+Bt+ whereB1,B2andB3areSubstitutingtheassumedparticularsolutionintothedifferentialequation,wehave3Bt2+(4B+3B)t+(2B+2B+3B)=t2+BycomparingthecoefficientsontheBycomparingthecoefficientsonthetwosides,itisobvious3B1=14B+3B= +2B2+3B3=B=1,B=2,

=- Theparticularsolutionyy(t)=12p3+t92whenx(t)=

d

y(t)+2

y(t)+3y(t)

Since1isnotaneienvalueweassumethesolutionyp(t)=Be,whereBispending.tSubstitutingtheparticularsolutionintotheequation,weobtain:Bet+2Bet+3Bet=et+3B=3

y(t)=1 Example:AcausalLTIsystemisdescribedbythefollowingconstant-coefficientdifferentialequation y(t)+5 y(t)+6y(t)=x(t)dt2 Pleasedeterminethefullsolutionunderthefollowing(1)(1)x(t)=2et,t‡0;y(0)=2,dy(0)=-(2)x(t)=e-2t,t‡0;y(0)=1,dy(0)=y(y(t)+5dy(t)+6y(t)=x(t)ddt(1)Theeigenvalueequationofthegivendifferentialequationis:

Sothehomogeneoussolutionisy(t)=Ce-

+Ce-2Considertheinput2e-2Lettheparticularsolutionbey(t)=Pe-SubstitutingSubstitutingtheassumedparticularsolutionintothedifferentialequation,wehaveObviously,P=

So,thefullsolutionofthedifferentialequationy(t)=y(t)+y(t)=Ce-

+Ce-3t+e-22

dy(t)=-2Ce-

-3C bythegiveninitialconditions,wedy(0)=-2C-3C-1=-12C1=3,C2=-y(0)=Thesolutionundercondition(1) x x =e-2tt(2)Thehomogeneoussolutionandtheparticularsolutionof dy(t)+ y(t)+6y(t)=dgivendifferentialequationareh12y(t)=(Pt+P)e-ph12y(t)=(Pt+P)e-p10y(t)=yh(t)+ypinwhichy(t)=yh(t)+yp 2=Ce-2t+Ce-3t+te-2t+P 2=(C+P)e-

+Ce-3t+te-

y(t)=(C+P)e-2t+Ce-3t+te- dy(t)=-2(C+P)e-2t-3Ce-

yy(0)=1,dy(0)= accordingtothegiveninitialy(0)=1,dy(0)=we

dy(0)=-2(C+P)-3C+1= C1+P0=,C2=-Thefullsolutionundercondition(2)

Introducesomeoftheimportantideasconcerningsystemsspecifiedbylinearconstant-coefficientdifferencePleaserecallthesolutionoflinearconstant-coefficientdifferenceequationswhichhavebeenintroducedinmathematicscourses.Letusconsideradiscrete-timeLTIsystemdescribedbyanNth-orderconstant-coefficientdifferenceequation.OrOrinasimplifiedNM SolutionofdifferenceequationsissimilartothatofdifferentialDifferenceequationsmaybesolvedbyExample.Adiscrete-timeLTIsystemisdescribedy[n]+3y[n-1]+2y[n-2]=Giventheauxiliaryconditiony[0]=0,y[1]=2,andtheinputsignalx[n]=2nu[n],determinetheoutputy[n].Solution:Accordingtothegiveny[n]=-3y[n-1]-2y[n-Whenn=2,incorporatingtheauxiliaryconditions,wey[2]=-3y[1]-2y[0]+x[2]=-6-0+4=- y[3]=-3y[2]-2y[1]+x[3]=y[4]=-3y[3]-2y[2]+x[4]=-…….Generally,thesolutionofdifferenceequationsiscomposedofthehomogeneoussolutionandtheparticularsolution.Thehomogeneoussolutionisdeterminedbytheeigenvaluesofthedifferenceequation.ThehomogeneousequationN ay[n-k]= TheTheeigenvalueNalNkkThehomogeneoussolutionwhenalleigenvaluesλ1,λ2,…,aresingley[n]=Cln+Cln++Ch Foreigenvalueβ1ofmultiplicityk,thecorrespondingsolution(Cnk1+Cnk2++C)bn Thecorrespondinghomogeneoussolutiontakesasinusoidalformwhentheeigenvaluesarecomplex.Theparticularsolutionisdeterminedbythe locatedontherightsideofthedifferenceequation.Theformoftheparticularsolutioncanbedeterminedaccordingtotheformofthese terms(input). n

++D

1isnotana Da aisnotanExample:Givenadifferencedeterminedetermine Theeigenvaluel1=l2=1,l3=j,l4=-Thehomogeneousy[n]=(Cn+C)(1)n+C( 22

+C(-44jj 2yy[n]=Cn+C+Pcos( h1222P=C3+C4,Q=j(C3-Bytheinitial n=10=2C1+C2-1=0=2C1+C2-1=3C+C-12n=n=n=C1=0,C2=1,P=1,Q= BlockDiagramRepresentationofSystemsDescribedbyDifferentialandDifferenceEquationsBlockdiagramprovidesapictorialrepresentationwhichcanaddtoourunderstandingofthebehaviorandpropertiesofthesesystems.simulationorimplementationofsystemsdescribedbydifferentialanddifferenceequations.Suchrepresentationscansimulationorimplementationofsystemsdescribedbydifferentialanddifferenceequations.Itcanbedirectlytranslatedintoaprogramforthesimulationofsuchasystemonadigitalcomputer.

dy(t)+ay(t)=bx(t)

1 =-

dy(t)=-ay(t)+bx(t)ty(t)=[bx(t)-t d d dt2 — 1a0Payattentiontotheoutputoftheadder,we(t)=-d(t)-(t)+10Rearrangingtheequation,we

y(t)+ dy(t)+ay(t)=1 Somerelatedsignalstounitimpulsefunctionknownassingularityfunctions.HowSomerelatedsignalstounitimpulsefunctionknownassingularityfunctions.HowLTIsystemsrespondtounitimpulsesothersingularityTheUnitImpulseasanIdealizedShortd(t)=limdDDfixx(t)=x