backexam_fall04

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1MASSACHUSETTSINSTITUTEOFTECHNOLOGY

DepartmentofElectricalEngineeringandComputerScience

6.341DiscreteTimeSignalProcessingFall2004

BACKGROUND EXAMSeptember30,2004.

FullName:

Note:Thisexamisclosedbook. Nonotesorcalculatorspermitted.The exam will be graded on the basis of the answers only. Please dontputanyexplanationsorworkintheanswerbooklet.Five pages of scratch paper are attached at the end of the exam. Thesearenottobehandedin. Letusknowifyouneedadditionalscratchpaper.


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2Problem 1 [8%]Considerthediscrete-timesystemdescribedbythefollowingequation:

y[n] = x[n]2%(a) Isthissystem linear?

2%(b) Isthissystemtime-invariant?2%(c) Isthissystemcausal?2%(d) Isthissystemstable?

Problem 2[4%]ConsideranLTIdiscretetimesystemforwhichtheinputandoutputsatisfythefollowingdifferenceequation:

1y[n] + y[n1]=x[n]2

2%(a) Isthissystemcausal?2%(b) Isthissystemstable?

Problem 3 [4%]ConsiderthediscretetimeLTIsystemdescribedbythe following frequencyresponse:

12ej H(ej) =

1 12ej (13ej)

2%(a) Isthissystemcausal?2%(b) Isthissystemstable?

Problem 4 [6%] Determine the transfer function Hxy(z) from x[n] to y[n] and the transferfunctionHey(z) from e[n] to y[n]forthefollowingsystem:


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3e[n]

?

x[n] +

-

y[n]

-+6+

1z

ZZ Z

Z -3 + ++

0.5ZZZ

Z

Figure4-1:Problem 5 [5%] Determine the frequency response H(ej ) of the stable LTI system whoseinputandoutputsatisfythedifferenceequation:

1 1y[n] y[n1]=x[n] + x[n1]

3 2Problem 6 [9%] We generate a discrete time random process x[n] by drawing a sequence ofi.i.d. numbers from a random number generator with uniform distribution in [1,1]. x[n] is thenprocessedthroughanLTIsystemwith impulseresponseh[n] = [n] + [n1]toobtainy[n],i.e. y[n] = x[n] + x[n1].

3%(a) What isthemeanmy[n]oftheprocess?3%(b) What istheautocorrelationyy [m]oftheprocess?3%(c) What isthepowerspectraldensityPyy(ej)oftheprocess?

Problem 7 [9%] We process a zero-mean, unit variance, white, discrete-time process x[n]throughthestableLTIsystemwithtransferfunction

111zH1(z) = 2

11z14togetanoutputy1[n].


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43%(a) Findastable(possiblynon-causal)systemH2(z)suchthatifwepassy1[n]throughH2(z),

theoutput iswhite.3%(b) Isyouranswerto(a)uniquewithinascalarfactor?3%(c) Assume we pass y1[n] through H2(z) to obtain y2[n]. Are x[n] and y2[n] uncorrelated?

(i.e. isxy2[m]equalto0?)

Problem 8 [8%] In the system shown below, two functions of time, x1(t) and x2(t), aremultiplied together, and the product w(t) is sampled by a periodic impulse train. x1(t) is bandlimitedto1,andx2(t) isbandlimitedto2;that is,

X1(j)=0, || 1X2(j)=0, || 2

Determinethemaximum samplingintervalTsuchthatw(t)isrecoverablefromwp(t)throughtheuseofan ideal lowpassfilter.

p(t) = (t nT)x1(t) n=

? ?w(t) - - wp(t)

6

x2(t)

X1(j) X2(j)$%e

% e% e

% e%% ee

'

1 1 2 2

Figure8-1:


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5Problem9[9%]Assumethatthecontinuous-timesignalxc(t)inthefigure9-1hasfiniteenergyand isexactlybandlimitedsothat,

Xc(j)=0 for || 2 104The continuous-time signal xc(t) is sampled as indicated in the figure below to obtain thesequencex[n],whichwewanttoprocesstoestimatethetotalareaAunderxc(t)aspreciselyaspossible. Specifically,wedefine

A= xc(t)dt

Thediscrete-timesignalcalculates

A=T x[n]n=

5%(a) FindthelargestpossiblevalueofT whichwillresultinasaccurateanestimateaspossibleofAforanyxc(t)consistentwiththestatedassumptions.

4%(b) UnderthestatedassumptionswillyourestimateofAbeexactorapproximate?

Discrete-time- - -C/D system

x[n]

A=

estimate

of

Axc(t)

6T

Figure9-1:

Problem 10[8%]InthesystembelowH(ej)isadiscretetimeall-passfilterwithaconstantgroupdelayof3.7andcontinuousphase, i.e.,H(ej) = ej(3.7), || < .

xc(t) - C/D -xd[n] H(ej) -yd[n] D/C - yc(t)6 6T1 T2


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64%(a) Assumingnoaliasing,describe inwords(onesentence)theoutputoftheoverallsystem,

yc(t), intermsofxc(t),xd[n],T1 andT2.4%(b) Assuming no aliasing, describe in words (one sentence) the output of the discrete time

systemyd[n] intermsofxc(t),xd[n],T1 andT2.Problem 11 [4%] For the system shown below, if the discrete-time system H(ej) is LTI, is theoverallsystemHc(j)LTI?

Hc(j)

xc(t) - C/D -xd[n] H(ej) -yd[n] D/C - yc(t)6 6T T

Problem 12 [4%] Determine the output y[n] for a system with the following input sequencex[n]and impulseresponseh[n].

3x2

2x x[n] x h[n]1x 1x x1

- -

0 1 2 n 1 0 1 nProblem 13 [8%]Forthepole-zeroplotgivenbelowanswerthefollowingquestions:

4%(a) IftheROCis |z|>2:2%(i) Isthesystemstable?2%(ii) Isthesystemcausal?

4%(b) IftheROC|z|


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71

0.5

ImaginaryPart

0

0.5

1

H(z)

1 0.5 0 0.5 1 1.5 2Real Part

Figure13-1: Pole-ZeroDiagram

Problem 14 [6%] Consider a continuous-time LTI system for which the magnitude of thefrequency response is 1 and the phase is as shown below. Determine the response of thatsystemtotheinputx(t) = s(t)cos(ct),whereS(j)=0for || c.2

H(ej)0

0

Problem 15 [8%] The first plot in Figure 15-1 shows a signal x[n] that is the sum of threenarrow-bandpulseswhichdonotoverlap intime. Itstransformmagnitude |X(ej)| isshowninthesecondplot. ThegroupdelayandfrequencyresponsemagnitudefunctionsofFilterA, a discrete-timeLTIsystem,areshowninthethirdandfourthplots,respectively. Theremainingplots in Figures 15-2 and 15-3 show 8 possible output signals, yi[n] i = 1,2,...8 DeterminewhichofthepossibleoutputsignalsistheoutputofFilterAwhenthe input isx[n].

Problem 16What isyourbestestimateofyourgradeonthisexam?


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8

-1

-0.5

0

0.5

1

x[n]

Input Signal x[n]

Magnitude(dB)

GroupDelay(Samples)

|X(e^jw)|

0 50 100 150 200 250 300 350 400

n

20Fourier Transform Magnitude of Input x[n]

15

10

5

00 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Frequency Normalized by pi

250Group Delay of filter A

200

150

100

50

00 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2


Frequency Response Magnitude of filter A0

-50

-100

-150

-2000 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2


Figure16-1: InputandFilterA information


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9

-1.5

-1

-0.5

0

0.5

1

y1[n]

Possible Output y1[n]

y4[n]

y3[n]

y2[n]

0 50 100 150 200 250 300 350 400

n

1Possible Output y2[n]

0.5

0

-0.5

-1-200 -150 -100 -50 0 50 100 150 200

n


0.5

0

-0.5

-1

-1.50 50 100 150 200 250 300 350 400

n

Possible Output y4[n]1

0.5

0

-0.5

-10 50 100 150 200 250 300 350 400

n

Figure16-2: Outputsy1y4


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10

-1

-0.5

0

0.5

1

y5[n]


y7[n]

y6[n]

-200 -150 -100 -50 0 50 100 150 200

n


0.5

0

-0.5

-10 50 100 150 200 250 300 350 400

n


0.5

0

-0.5

-10 50 100 150 200 250 300 350 400

n

-1

-0.5

0

0.5

1

y8[n]


0 50 100 150 200 250 300 350 400

n

Figure16-3: Outputsy5

y8

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