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6.003(Fall2009)FinalExamination
December17,2009Name:KerberosUsername:
Pleasecircleyoursectionnumber:Section Instructor Time1 MarcBaldo
10am2 MarcBaldo 11am3 ElfarAdalsteinsson 1pm4 ElfarAdalsteinsson
2pm
Partial creditwillbe given for answers thatdemonstrate
somebutnot allofthe importantconceptual
issues.Explanationsarenotrequiredandwillnotaffectyourgrade.
Youhavethreehours.Pleaseputyourinitialsonallsubsequentsheets.Enteryouranswersintheboxes.Thisquizisclosedbook,butyoumayusefour8.511sheetsofpaper(eightsidestotal).Nocalculators,computers,cellphones,musicplayers,orotheraids.
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FinalExamination / 6.003: SignalsandSystems(Fall2009) 21.
CTSystemwithFeedback [15points]
LetGrepresentacausalsystemthatisdescribedbythefollowingdifferentialequationdy(t)
+y(t) = dx(t)x(t)dt dt
wherex(t)representstheinputsignalandy(t)representstheoutputsignal.a.
Determinetheoutputy1(t)ofGwhentheinputis
e t0x1(t) =
t;0; otherwise
Enteryourresultintheboxbelow.ty1(t) = (12t)e u(t)
(s+1)Y =(s1)XY = s1
s+1XY1= s1
s+1X1; X1=1
s+1; Res>1Y1= s1
(s+1)2 =1
s+12
(s+1)2y1(t)=etu(t)2tetu(t)
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FinalExamination / 6.003: SignalsandSystems(Fall2009)
3Nowconsiderafeedback
loopthatcontainstheGsystemdescribedonthepreviouspage.1
+ K Gw(t)x(t)
y(t)
b.
Determineadifferentialequationthatrelatesw(t)toy(t)whenK=10.Thedifferentialequationshouldnotcontainreferencestox(t).Enterthedifferentialequationintheboxbelow.
11dy(t)9y(t) = 10dw(t)10w(t)dt dt
YW =
Ks1s+1
1+Ks1s+1
=K s1s+1+KsK =K
s1(K+1)s(K1)=K
s111s9
Theminussignneartheadderindicatesthattheoutputoftheadderisw(t)y(t)1
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FinalExamination / 6.003: SignalsandSystems(Fall2009) 4c.
Determine the values of K forwhich the feedback system on
thepreviouspage isstable.Entertherange(orranges)intheboxbelow.
1 < K
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3]+w[n]+2w[n1]+3w[n2].Adirectrealizationofthedifferenceequationisshownbelow.
+ +
+
R
R
R
R
R
2
3
W
FinalExamination / 6.003: SignalsandSystems(Fall2009) 52.
SteppingUpandDown [10points]
Useasmallnumberofdelays,gains,and2-inputadders(andnoothertypesofelements)toimplementasystemwhoseresponse(startingatrest)toaunit-stepsignal
1 n0x[n] =u[n] =
0 otherwiseis
y[n] =
0 n
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FinalExamination / 6.003: SignalsandSystems(Fall2009) 63.
DTsystems [14points]
Thepole-zerodiagramforaDTsystemisshownbelow,wherethecirclehasradius1.z-plane
12
Itisknownthatwhentheinputis1foralln,theoutputisalso1foralln.Sketchtheunit-sampleresponseh[n]ofthesystemontheaxesbelow.Labeltheimportantfeaturesofyoursketch.
1
3
1n
h[n]
YX =K
z+1
2j32
z+12 +j32
z =K
z+1
2
2+3
4z =K
z2+z+1z =K(z+1+z1)
H(1)=3K=1Therefore,K= 1
3.h[n]= 13([n1]+[n]+[n+1])
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FinalExamination / 6.003: SignalsandSystems(Fall2009)
7AsecondDTsystemhasthefollowingpole-zerodiagram:
z-plane
1/2
ItisknownthatthesystemfunctionH(z)is1whenz=1.Sketchthemagnitudeofthefrequencyresponseofthissystemontheaxesbelow.Labeltheimportantfeaturesofyoursketch(includingtheaxes).
13
|H(ej)|
2
1
Sketchtheangleofthefrequencyresponseofthissystemontheaxesbelow.
Labeltheimportantfeaturesofyoursketch(includingtheaxes).
H(ej)
2
2
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FinalExamination / 6.003: SignalsandSystems(Fall2009) 84.
CTSystems [14points]
AcausalCTsystemhasthefollowingpole-zerodiagram:s-plane
121
Lety(t)=s(t)representtheresponseofthissystemtoaunit-stepsignal1
; t0
x(t) =u(t) =0; otherwise.
Assumethattheunit-stepresponses(t)ofthissystemisknowntoapproach1ast.Determiney(t)
=s(t)andenteritintheboxbelow.
t/2)u(t)y(t) = (1+et2e
Fromthepole-zerodiagram,H(s)= K
(s+1)(s+ 12).
Since the system is stable (system is causal and poles are all
in left half-plane), the unit-stepresponsewillapproachH(0)ast.
Therefore
H(0)= K(1)(1
2)=2K=1
andK= 12. TheLaplacetransformoftheunitstepisX(s)=1
s forRe{s}>0. ThereforeY = 12
s(s+1)(s+ 12)=
1s+
1s+1
2s+ 1
2
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FinalExamination / 6.003: SignalsandSystems(Fall2009)
9AsecondCTsystemhasthefollowingpole-zerodiagram:
s-plane
1
1
Assumethattheinputsignalis11; cost >
x(t) = 0; otherwise.
Let ak and bk represent theFourier series coefficientsof the
inputandoutput signals,respectively,wherethe fundamental(lowest
frequencycomponent)ofeachsignalhasaperiodof2.Itisknownthat b0
=1.Determine b1.
a0 a1b1a1 =
21+3j
H(s)= K(s+1)(s+1j)(s+1+j)=
K(s+1)((s+1)2+1)=
K(s+1)(s2+2s+2)
b0a0
=1=H(j0)=K2
Thefundamentalfrequency is= 22 =1.
b1a1 =H(j1)=
K(1+j)(1+2j)=
21+3j
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FinalExamination / 6.003: SignalsandSystems(Fall2009) 10
ConsiderthefollowingsystemforDTprocessingofCTsignals:H1(j)
H2(e
j) H3(j)uniform
sampler
sample-to-
impulsexa(t)
xb(t) xc[n] yd[n] ye(t)yf(t)
wherexc[n]=xb(nT)and0
n=
5. DTprocessingofCTsignals [15points]
ye(t) =
yd[n](tnT).ThefrequencyresponsesH1(j)andH3(j)aregivenbelow.
2T
2T
32T
32T
H1(j)
1
T
T
H3(j)
T
a.
AssumeinthispartthatH2(ej)=1forallfrequencies.Determineyf(t)when
5
xa(t)=cos t +sin t .2T 4T
2Tt
yf(t) = cos
2Tt
1
4sin
3
4Tt
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FinalExamination / 6.003: SignalsandSystems(Fall2009) 11
b. Forthispart,assumethat1 ; ||
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FinalExamination / 6.003: SignalsandSystems(Fall2009) 126.
WhichareTrue? [16points]
ForeachoftheDTsignalsx1[n]throughx4[n](below),determinewhethertheconditionslistedinthefollowingtablearesatisfied,andanswerTfortrueorFforfalse.
X(ej0) = 0
X(e
j) d = 0
X(ej) is purely imaginary
ejkX(ej) is purely real for some integer k
x1[n] x2[n] x3[n] x4[n]
T F T T
T T T F
F F T F
F T F F
1n
x1[n]
1n
x2[n]
1n
x3[n]
1n
x4[n]=x4[n + 5]
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FinalExamination / 6.003: SignalsandSystems(Fall2009) 147.
MultipliedSampling [16points]
TheFouriertransformofasignalxa(t)isgivenbelow.
2
2
Xa(j)
1
Thissignalpassesthroughthefollowingsystem K H(j)
uniform
sampler
sample-to-
impulsexa(t)
xb(t) xc[n] xd[n] xe(t)
cos(7t)
xf(t)
wherexc[n]=xb(nT)and0
n=xe(t) = xd[n](tnT)
andT if||
