MIT6 011S10 Finadfgdfl Ans

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MassachusettsInstituteofTechnologyDepartmentofElectricalEngineeringandComputerScience

6.011: IntroductiontoCommunication,ControlandSignalProcessingFINALEXAM,May18,2010ANSWERBOOKLET

YourFullName:RecitationTime : oclock

This exam is closed book, but 4 sheets of notes are allowed. Calculators and otherelectronicaidswillnotbenecessaryandarenotallowed.Check that thisANSWERBOOKLET has pages numbered up to 26. The bookletcontainsspacesforallrelevantreasoningandanswers.Neatworkandclearexplanationscount;showallrelevantworkandreasoning!Youmaywanttofirstworkthingsthroughonscratchpaperandthenneatlytransfertothis booklet the work you would like us to look at. Let us know if you need additionalscratchpaper.Onlythisbookletwillbeconsidered inthegrading;noadditionalanswerorsolutionwrittenelsewherewillbeconsidered. Absolutelynoexceptions!Thereare5problems,weightedasshown, fora totalof100points. (Thepointsindicated on the following pages for the various subparts of the problems are our bestguessfornow,butmaybemodifiedslightlywhenwegettograding.)

Problem YourScore1(17points)2(18points)3(25points)4(20points)5(20points)

Total(100points)

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Problem1 (17points)Note that1(d)doesnotdependonyouranswers to1(a)-(c), and canbedone

independentlyofthem.Xc(j ) = 0 for || 2103 .

xc(t) - C/D6

-x[n]

h[n] -y[n] PAM6

-r(t)

C/D6

-q[n]=r(nT2)

T1 p(t), T2 T2

-

6H(ej)

1

2 2

1(a) (3points)Determinethe largest valueofT1 toensurethaty[n] =xc(nT1).

LargestpossibleT1 is:2


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1(b) (6points)WithT1 pickedas in1(a),determineachoiceforT2 andp(t)toensurethatr(t) =xc(t).

(Youcan leaveyourexpressionsforT2 andp(t) intermsofT1, insteadofsubstituting inthenumericalvalueyouobtained in1(a)forT1.)

T2 = p(t) =

(Continue1(b)onnextpage:)3


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1(b) (continued)AlsodetermineifthereisanotherchoiceofT2 andp(t)thatcouldensuretheequalityr(t) =xc(t). Explainyouranswercarefully.

IsthereanotherpossiblechoiceofT2? IfyouanswerYes,thenspecifysuchanalternativeT2.

Isthereanotherpossiblechoiceofp(t)? IfyouanswerYes,thenspecifysuchanalternativep(t).

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1(c) (3 points) With T1 picked as in 1(a), how would you modify your choice of T2 andp(t)from1(b)toensurethat

r(t) =xc(2.7t).

ModifiedT2 =

Modifiedp(t) =5


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1(d) (4points)Assumethatp(t) isnowchosensothat itsCTFT,P(j), isasshownbelow.DetermineavalueofT2 toensurethatq[n] =y[n].

P(j)6

103

-2103 2103

AnappropriatechoiceisT2 =6


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Problem2 (18points)Foreachofthefollowingparts,writedownwhetherthestatementisTrueorFalse(circle

whicheverisappropriate),givingaclearexplanationorcounterexample. (Takecarewiththis!)

2(a) (4points) Supposex[n] is a zero-meandiscrete-time (DT) wide-sense stationary (WSS)random process. If its autocorrelation function Rxx[m] is 0 for |m| 2 but nonzero form=1,0,1,thenthelinearminimummean-square-error(LMMSE)estimatorofx[n+ 1]frommeasurementsofx[n]andx[n1],namely

x[n+ 1]=a0x[n] +a1x[n1],willnecessarilyhavea1 =0.TRUE FALSEExplanation/counterexample:

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2(b) (4points)IfthepowerspectraldensitySyy(j )ofacontinuous-time(CT)WSSrandomprocessy(t)isgivenby

17+2Syy(j ) =

23+2thenthemeanvalueoftheprocessiszero,i.e.,y =E[y(t)]=0.TRUE FALSEExplanation/counterexample:

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2(c) (4 points) If the autocovariance function Cvv [m] of a DT WSS random process v[n] isgivenby

m

Cvv [m] =1| | ,3

thentheLMMSEestimatorofv[n+1]fromallpastmeasurements,whichwewriteasv[n+ 1]= hkv[nk]+d ,

k=0willhavehk =0forallk1, i.e.,onlythecoefficientsh0 anddcanbenonzero.TRUE FALSEExplanation/counterexample:

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2(d) (3points)Theprocessv[n]in2(c) isergodicinmeanvalue.TRUE FALSEExplanation/counterexample:

2(e) (3points)Ifz[n] =v[n] +W,wherev[n]istheprocessin2(c),andwhereW isarandomvariablewithmean0andvariance2 >0,thentheprocessz[n]isergodicinmeanvalue.WTRUE FALSEExplanation/counterexample:

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Problem3 (25points)

q[n+ 1] = Aq[n] +bx[n] +hw[n],y[n] = cTq[n] +v[n].

where q1[n] 3412 , b= , h= 0 , cT = 0 1 ,

1 112q[n] = A=

q2[n] , 0 2

3(a) Determinethetwonaturalfrequenciesofthesystem(i.e.,theeigenvaluesofA),andforeach of them specify whether the associated mode satisfies the properties listed on thenextpage.

(Writeyouranswerson thenextpage.)11


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3(a) (continued)(8points)

Thetwoeigenvaluesare: 1 = 2 =List below whichever of the eigenvalues, if either, has an associated mode that satisfiestheindicatedcondition:

(i) decaysasymptoticallyto0 inthezero-inputresponse:(ii) isreachablefromthe inputx[n](withw[n]keptatzero):

(iii) isreachablefromthe inputw[n](withx[n]keptatzero):(iv) isobservablefromtheoutputy[n]:

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3(b) (2points)Yourspecificationoftheobserver,toobtainanestimateq[n]ofthestateq[n](explainyourchoice):

3(c) (2points)Withq[n] =q[n]q[n],explaincarefullywhythecomponentsq1[n]andq2[n]ofq[n] at time n are uncorrelated with the noise terms w[n] and v[n] at time n (or equivalently,ofcourse! explainwhythecomponentsofq[n+1]areuncorrelatedwithw[n+1]andv[n+1]):

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3(d) (4 points) The state estimation error in 3(c) is governed by a state-space model of theform

q[n+ 1]=Bq[n] +fw[n] +gv[n].DetermineB,f andg intermsofpreviouslyspecifiedquantities.

B= , f = , g=

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3(e) (5points)Isitpossibletoarbitrarilyvarythenaturalfrequenciesofthestateestimationerror evolution equation in 3(d) by controlling the observer gains 1 and 2? Explicitlynotehowyouranswerhereisconsistentwithyouranswerto3(a)(iv).

What constraints, if any, on 1 and 2 must be satisfied to make the error evolutionequationasymptoticallystable?

Wouldthechoice2 =0allowyoutoobtainagoodstateestimate? explain.

Ifyouhavedonethingscorrectly,youshouldfindthatchoosing1 =43 makesthematrixBinpart3(d)adiagonalmatrix.Keep 1 fixedat3 fortherestof thisproblem,and4alsoassume2 ischosensothattheerrorevolutionequation isasymptoticallystable.

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3(f) (4points)Underthegivenassumptions,themean-squaredestimationerrorsattainconstant steady-state values, E(sions for 1 2,

212q

2q

and E(1 2.expressing them as functions of 2. [Hint: At steady state,

2q222q[n]) [n]) Find explicit expres = =q qand

qE(21[n+1])=E(q21[n])andE(q22[n+1])=E(q22[n]).]

2q 1 2q= 2 =

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Problem4 (20points)6f(x|H0) f(x H1)6 | 1

214- -

2x 1 1 x2

4(a) (4points)Sketch(x) = fX|H(x|H1) asafunctionofxfor2< x


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4(b) (6points)(i) Forthresholdatsomevaluestrictlyabove2,determinePD andPFA:

PD = PFA =

(ii) For atsomevaluestrictlybetween0and2,determinePD andPFA:

PD = PFA =

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4(b) (continued)(iii) For atsomevaluestrictlybelow0,determinePD andPFA:

PD = PFA =

4(c) (2 points) If the specified limit on PFA is = 0.3, which of the choices in 4(b) can wepick,andwhatistheassociatedPD?

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4(d) (8points)What is the probability thatweget(X)=0 if H0 holds? And what is theprobabilityweget(X) = 0ifH1 holds?

P((X) = 0 H0) = P((X) = 0 H1) =| |

Announce H0when(x)=0. When(x)>0,announce H1withprobability,andotherwiseannounce H0. WhatarePD andPFA withthisrandomizeddecisionrule?

PD = PFA =

TomaximizePD whilekeepingPFA0.3withthisdecisionrule,choose=

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Problem5 (20points)r[n]x[n]

- + -6

- K(z) = 1z1

y[n]

H1r[n] g[n] n=n0 - >- -


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5(a)(i) (continued)

Specifyf[n]orF(z):

Threshold=

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5(a)(ii) WritedownanexpressionforP(H1H0)andfortheminimumprobabilityoferror|inthecasewherethetwohypothesesareequallylikely,p0 =p1. Youcanwritethese

intermsofthestandardfunction21 t

Q() = 2 dt2 e

P(H1H0) =|

Theminimumprobabilityoferror is:

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5(a)(iii) Ifthevalueofischangedtoanewvalue=/2,wecangetthesameprobabilityoferror as priortothe change ifthe noise variancechangestosomenewvalue2.Express intermsof:

=

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5(b) (10 points) Suppose now that x[n] is a zero-mean, i.i.d., Gaussian noise process, withvariance2 ateach instantoftime,andthaty[n] isthesignalweare interested in. We

r[n]x[n]- K(z) = 1z1 - + -

6y[n]

H1r[n] g[n] n=n0 - >- -


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5(b) (continued)

Specifyf[n]orF(z):Threshold=

Alsowritedown(intermsofand)therelevantsignalenergytonoisepowerratiothatgovernstheperformanceofthissystem:

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6.011 Introduction to Communication, Control, and Signal Processing

Spring 2010

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MIT6 011S10 Finadfgdfl Ans

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