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6.003Homework1DueatthebeginningofrecitationonWednesday,February10,2010.Problems1. IndependentandDependentVariablesAssume that theheightofawaterwave isgivenby g(xvt) where x isdistance, v isvelocity, and t is time. Assume that theheightof thewave isa sinusoidal functionofdistanceateach instantof time. Alsoassume that thepositivepeakshaveaheightof1meter(relativetotheaveragewater level)andthattheyoccurat integermultiplesof2meterswhenthetimet= 3 seconds.a. Determineanexpression for theheightofthewave h(x, t) asa functionofdistance

xand time t if thewave is traveling in thepositive xdirectionat5meters/second.Whatisthefunctiong( ) forthiscase?

b. Determineanexpression for theheightofthewave h(x, t) asa functionofdistancexandtime t ifthewave istraveling inthenegative xdirectionat4meters/second.Whatisthefunctiong( ) forthiscase?

c. Determine the speedof thewave if successivepositivepeaksat x = 1.3metersareseparatedby0.75 seconds.

2. EvenandOddTheevenandoddpartsofasignalx[n] aredefinedbythefollowing:

xe[n] = xe[n] (i.e.,xe isanevenfunctionofn)xo[n] = xo[n] (i.e.,xo isanoddfunctionofn)x[n] = xe[n] + [n] xo

Letxrepresentthesignalwhosesamplesaregivenbyx[n] =

2

1n

n0 .0 otherwise

a. Determinetheevenandoddpartsofthesignalx.b. Showthatyouranswerinpartaisunique.c. Plottheresultsofparta.

3. Geometricsums1

a. Expand inapowerseries.Forwhatrangeofadoesyouranswerconverge?1a

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1

10

a(t)

t

26.003Homework1 / Spring2010N1

nb. Find a closed-form expression for a . Forwhat range of a does your answern=0

converge?1

c. Expand(1a)2 inapowerseries.Forwhatrangeofadoesyouranswerconverge?

ratedby

Tseconds

in

xc.

a. Determine an expression for xc(t) in terms of the samples xd[n] and the functionsLet yc(t) representacontinuous-time signalderived from thediscrete-time signal yd[n]

c. Determineanexpression for dyc(t) in termsof the samples yd[n] and the functionsdt

a(t),b(t),andc(t).5. MissingParametersConsiderthefollowingsystem.

4. ReconstructingCTSignals fromSamplesLeta(t),b(t),andc(t) representthefollowingfunctionsoftime.

1

10

b(t)

t

1

10

c(t)

t

Let xc(t) representacontinuous-timesignalderived fromthediscrete-time signal xd[n]usingazero-orderhold,as illustratedbelow,whereconsecutivesamplesofxd aresepa

n

xd[n]

0 2 4 6 8 10t

xc(t)

0 2T 4T 6T 8T 10T

xd[0]

xd[1]

a(t),b(t),andc(t).using apiecewise linear interpolator, so that sucessive samplesof yd are connectedbystraightlinesegments.

n

yd[n]

0 2 4 6 8 10t

yc(t)

0 2T 4T 6T 8T 10T

yd[0]

yd[1]

b. Determineanexpressionforyc(t) intermsofthesamplesyd[n] andthefunctionsa(t),b(t),andc(t).

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6.003Homework1 / Spring2010 3+

R

1 32R

R

X Y

AssumethatX istheunit-samplesignal,x[n] = [n]. Determinethevaluesofandforwhichy[n] isthefollowingsequence(i.e.,y[0],y[1],y[2],...):

3 7 15 310 , 1 , , , , . . .

2 4 8 16 ,

EngineeringDesignProblems6. ChoosingabankConsidertwobanks.Bank#1offersa3%annualinterestrate,butchargesa$1servicechargeeachyear,includingtheyearwhentheaccountwasopened.Bank#2offersa2%annualinterestrate,andhasnoannualservicecharge.Letyi[n] representthebalanceinbankiatthebeginningofyearnandxi[n] representtheamountofmoneyyoudepositinbankiduringyearn.Assumethatdepositsduringyearnarecreditedtothebalanceattheendofthatyearbutearnnointerestuntilthefollowingyear.a. Use difference equations to express the relation between deposits and balances foreachbank.

b. Assume that you deposit $100 in each bank andmake no further deposits. Solveyourdifference equations in part anumerically (usingMatlab,Octave, orPython)todetermineyourbalance ineachbank forthenext25years. Makeaplotofthesebalances. Whichaccounthas the largerbalance5yearsafter the initial investment(oneyearwithout interestand4yearswith interest). Whichaccounthasthe largerbalanceafter25years(i.e.,atthebeginningofthe26th year)[Hint:SeetheAppendixforhelpwithprogramming.]

7. DrugdosingWhendrugsareusedtotreatamedicalcondition,doctorsoftenrecommendstartingwithahigherdose on thefirstday than on subsequentdays. In thisproblem,we considera simplemodel tounderstandwhy. Assume that thehumanbody is a tank ofbloodandthatdrugsinstantlydissolveinthebloodwheningested.Furtherassumethatdrugvanishesfromtheblood(eitherbecauseitisbrokendownorbecauseitisflushedbythekidneys)ataratethatisproportionaltodrugconcentration.Let x[n] represent theamountofdrug takenonday n,and let y[n] represent the totalamountofdruginthebloodondayn,justafterthedosex[n] hasdissolvedintheblood,sothat

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

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46.003Homework1 / Spring2010a. Determinetheamountofdruginthebloodthatwouldresultaftertakingoneunitofdrugeachdayformanyconsecutivedays,i.e.,determinelimny[n] whenx[n] = 1.

b. Assumethatthereisinitiallynodrugintheblood.Then,startingonday0,oneunitofdrug is takeneachday. Determine thefirstdaywhen theamountofdrug inthebloodwillequalorexceedhalfofitsfinalvalue.

c. Considerthefollowingtableofdosesandresultingamountofdrugintheblood:n x[n] y[n]1 0 00 1 1.001 1 1.732 1 2.273 1 2.664 1 2.955 1 3.16

6 1 3.327 1 3.43

Notice that the blood concentration ramps up over the first few days. Suggest adifferent initialdose x[0] thatwill result inamore constantamountofdrug in theblood(withx[n] remainingat1 foralln1).Makeatabletoshowyourresult.

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Appendix: FibonaccicodeYoumayusePythonand/orMatlab/Octavetosolveproblemsinthishomeworkassignment.Octaveisafree-softwarelinear-algebrasolver,withasyntaxthatissimilartothatofMatlab.Octaveisavailableformostplatforms. Seewww.octave.org.The following code calculates, prints, and plots the first 20Fibonacci numbers (i.e., f[0]throughf[19]).

ExampleMatlab/Octavecodey(1) = 1; % initial conditionsy(2) = 1; % indices start at 1 (not 0)for i = 3:20

y(i) = y(i-1)+y(i-2)endy % print ystem(0:19,y)

ExamplePythoncodefrom pylab import *y = [1,1] # initial conditionsfor i in range(2,20):

y.append(y[i-1]+y[i-2]) print ystem(range(20),y)show()

http://www.octave.org/http://www.octave.org/

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6.003Signals and SystemsSpring 2010

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