6.003 Homework 1. Problems. Due at the beginning of recitation on Wednesday, February 10, 2010.

Transcription

1 6.003 Homework Due a he beginning of reciaion on Wednesday, February 0, 200. Problems. Independen and Dependen Variables Assume ha he heigh of a waer wave is given by g(x v) where x is disance, v is velociy, and is ime. Assume ha he heigh of he wave is a sinusoidal funcion of disance a each insan of ime. Also assume ha he posiive peaks have a heigh of meer (relaive o he average waer level) and ha hey occur a ineger muliples of 2 meers when he ime = 3 seconds. a. Deermine an expression for he heigh of he wave h(x, ) as a funcion of disance x and ime if he wave is raveling in he posiive x direcion a 5 meers/second. Wha is he funcion g( ) for his case? b. Deermine an expression for he heigh of he wave h(x, ) as a funcion of disance x and ime if he wave is raveling in he negaive x direcion a 4 meers/second. Wha is he funcion g( ) for his case? c. Deermine he speed of he wave if successive posiive peaks a x =.3 meers are separaed by 0.75 seconds. 2. Even and Odd The even and odd pars of a signal x[n] are defined by he following: x e [ n] = x e [n] (i.e., x e is an even funcion of n) x o [ n] = x o [n] (i.e., x o is an odd funcion of n) x[n] = x e [n] + x o [n] Le x represen he signal whose samples are given by { ( x[n] = ) n 2 n 0. 0 oherwise a. Deermine he even and odd pars of he signal x. b. Show ha your answer in par a is unique. c. Plo he resuls of par a. 3. Geomeric sums a. Expand in a power series. For wha range of a does your answer converge? a

2 6.003 Homework / Spring N n b. Find a closed-form expression for a. For wha range of a does your answer converge? c. Expand ( a) 2 n=0 in a power series. For wha range of a does your answer converge? 4. Reconsrucing CT Signals from Samples Le a(), b(), and c() represen he following funcions of ime. a() b() c() Le x c () represen a coninuous-ime signal derived from he discree-ime signal x d [n] using a zero-order hold, as illusraed below, where consecuive samples of x d are sepa raed by T seconds in x c. x d [n] x c () x d [0] x d [] n 0 2T 4T 6T 8T 0T a. Deermine an expression for x c () in erms of he samples x d [n] and he funcions a(), b(), and c(). Le y c () represen a coninuous-ime signal derived from he discree-ime signal y d [n] using a piecewise linear inerpolaor, so ha sucessive samples of y d are conneced by sraigh line segmens. y d [n] y c () y d [0] y d [] n 0 2T 4T 6T 8T 0T b. Deermine an expression for y c () in erms of he samples y d [n] and he funcions a(), b(), and c(). c. Deermine an expression for dy c() in erms of he samples yd [n] and he funcions d a(), b(), and c(). 5. Missing Parameers Consider he following sysem.

3 6.003 Homework / Spring X R + α β 2 3R R Y Assume ha X is he uni-sample signal, x[n] = δ[n]. Deermine he values of α and β for which y[n] is he following sequence (i.e., y[0], y[], y[2],...): ,,,,, , Engineering Design Problems 6. Choosing a bank Consider wo banks. Bank # offers a 3% annual ineres rae, bu charges a $ service charge each year, including he year when he accoun was opened. Bank #2 offers a 2% annual ineres rae, and has no annual service charge. Le y i [n] represen he balance in bank i a he beginning of year n and x i [n] represen he amoun of money you deposi in bank i during year n. Assume ha deposis during year n are credied o he balance a he end of ha year bu earn no ineres unil he following year. a. Use difference equaions o express he relaion beween deposis and balances for each bank. b. Assume ha you deposi $00 in each bank and make no furher deposis. Solve your difference equaions in par a numerically (using Malab, Ocave, or Pyhon) o deermine your balance in each bank for he nex 25 years. Make a plo of hese balances. Which accoun has he larger balance 5 years afer he iniial invesmen (one year wihou ineres and 4 years wih ineres). Which accoun has he larger balance afer 25 years (i.e., a he beginning of he 26 h year) [Hin: See he Appendix for help wih programming.] 7. Drug dosing When drugs are used o rea a medical condiion, docors ofen recommend saring wih a higher dose on he firs day han on subsequen days. In his problem, we consider a simple model o undersand why. Assume ha he human body is a ank of blood and ha drugs insanly dissolve in he blood when ingesed. Furher assume ha drug vanishes from he blood (eiher because i is broken down or because i is flushed by he kidneys) a a rae ha is proporional o drug concenraion. Le x[n] represen he amoun of drug aken on day n, and le y[n] represen he oal amoun of drug in he blood on day n, jus afer he dose x[n] has dissolved in he blood, so ha y[n] = x[n] + αy[n ].

4 6.003 Homework / Spring 200 a. Deermine he amoun of drug in he blood ha would resul afer aking one uni of drug each day for many consecuive days, i.e., deermine lim n y[n] when x[n] =. b. Assume ha here is iniially no drug in he blood. Then, saring on day 0, one uni of drug is aken each day. Deermine he firs day when he amoun of drug in he blood will equal or exceed half of is final value. c. Consider he following able of doses and resuling amoun of drug in he blood: n x[n] y[n] Noice ha he blood concenraion ramps up over he firs few days. Sugges a differen iniial dose x[0] ha will resul in a more consan amoun of drug in he blood (wih x[n] remaining a for all n ). Make a able o show your resul. 4

5 Appendix: Fibonacci code You may use Pyhon and/or Malab/Ocave o solve problems in his homework assignmen. Ocave is a free-sofware linear-algebra solver, wih a synax ha is similar o ha of Malab. Ocave is available for mos plaforms. See The following code calculaes, prins, and plos he firs 20 Fibonacci numbers (i.e., f[0] hrough f[9]). Example Malab/Ocave code y() = ; % iniial condiions y(2) = ; % indices sar a (no 0) for i = 3:20 y(i) = y(i-)+y(i-2) end y % prin y sem(0:9,y) Example Pyhon code from pylab impor * y = [,] for i in range(2,20): y.append(y[i-]+y[i-2]) prin y sem(range(20),y) show() # iniial condiions

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