6.003 Homework 1Due at the beginning of recitation on Wednesday, September 16, 2009.Problems1. WavesAssume that the height of a water wave is given by g(x − vt) where x is distance, v isvelocity, and t is time. The function g is defined in the following plot and is zero outsidethe indicated range.g(r)r−3 −2 −1 0 1 2 3a. Assume x = 0 and v = 1. Sketch f1(t) = g(−t), which illustrates the height of thewave as a function of time, as it passes the point x = 0.b. Assume x = 0 and v = 2 and sketch f2(t) = g(−2t). Explain the important differencesbetween f1and f2.c. Sketch f3(x) = g(x − 1). Suggest a physical interpretation for f3(x).d. Sketch f4(t) = g(1 − t). Suggest a physical interpretation for f4(t).e. Is the wave traveling in the positive or negative x direction? Explain.2. Even and OddLet x[n] represent the signal shown in the following plot, where all values outside theindicated range are zero.x[n]n−3 −2 −1 0 1 2 3−11Assume that x[n] can be written as the sum of an even part (xe[n] = xe[−n]) and an oddpart (xo[n] = −xo[−n]).a. Sketch xe[n] and xo[n].b. Prove that xo[n] is unique.6.003 Homework 1 / Fall 2009 23. PeriodicityConsider two DT signals:x1[n] = cos3πn4andx2[n] = cos5πn4.How many of the following statements are true?− x1[n] is periodic with a period of 8.− x2[n] is periodic with a period of 8.− x1[n] = x2[n].Briefly explain.4. Geometric sumsa. Expand11 − ain a power series. For what range of a does your answer converge?b. Find a closed-form expression forN −1Xn=0an. For what range of a does your answerconverge?c. Expand1(1 − a)2in a power series. For what range of a does your answer converge?Engineering Design Problems1. Choosing a bankConsider two banks. Bank #1 offers a 3% annual interest rate, but charges a $1 servicecharge each year, including the year when the account was opened. Bank #2 offers a 2%annual interest rate, and has no annual service charge. Let yi[n] represent the balance inbank i at the beginning of year n and xi[n] represent the amount of money you depositin bank i during year n. Assume that deposits during year n are credited to the balanceat the end of that year but earn no interest until the following year.a. Use difference equations to express the relation between deposits and balances foreach bank.b. Assume that you deposit $100 in each bank and make no further deposits. Solveyour difference equations in part a numerically (using Matlab, Octave, or Python)to determine your balance in each bank for the next 25 years. Make a plot of thesebalances. Which account has the larger balance 5 years after the initial investment(one year without interest and 4 years with interest). Which account has the largerbalance after 25 years (i.e., at the beginning of the 26thyear) [Hint: See the Appendixfor help with programming.]6.003 Homework 1 / Fall 2009 32. More RabbitsLet g represent the following sequence:1, 1, 3, 5, 11, 21, 43, 85, 171, 341, 683, 1365, · · ·which is closely related to the Fibonacci sequence.a. Determine the difference equation that generates the sequence g. The differenceequation should have the formg[n] = x[n] + a0g[n − 1] + a1g[n − 2] + · · ·where the coefficients aiare constants and x is given byx[n] =n1 if n = 00 otherwise.b. The sequence g can be described by a population problem that is very similar tothe one that describes the Fibonacci sequence. Write a description of the populationproblem described by g.c. The sequence g[n] grows approximately geometrically, i.e.,g[n] ≈ abnfor large values of n. Determine a and b.d. Determine an exact expression for g[n] of the formg[n] = abn+ cdn.HoursPlease estimate the number of hours you spent working on this homework assignment.Also, feel free to comment on these problems.Appendix: Fibonacci codeYou may use Python and/or Matlab/Octave to solve problems in this homework assignment.Octave is a free-software linear-algebra solver, with a syntax that is similar to that of Matlab.Octave is available for most platforms. See www.octave.org.The following code calculates, prints, and plots the first 20 Fibonacci numbers (i.e., f [0]through f[19]).Example Matlab/Octave codey(1) = 1; % initial conditionsy(2) = 1; % indices start at 1 (not 0)for i = 3:20y(i) = y(i-1)+y(i-2)endy % print ystem(0:19,y)Example Python codefrom pylab import *y = [1,1] # initial conditionsfor i in range(2,20):y.append(y[i-1]+y[i-2])print
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