6.003 Homework 1Due at the beginning of recitation on Wednesday, February 10, 2010.Problems1. Independent and Dependent VariablesAssume that the height of a water wave is given by g(x − vt) where x is distance, v isvelocity, and t is time. Assume that the height of the wave is a sinusoidal function ofdistance at each instant of time. Also assume that the positive peaks have a height of1 meter (relative to the average water level) and that they occur at integer multiples of2 meters when the time t = 3 seconds.a. Determine an expression for the height of the wave h(x, t) as a function of distancex and time t if the wave is traveling in the positive x direction at 5 meters/second.What is the function g(·) for this case?b. Determine an expression for the height of the wave h(x, t) as a function of distancex and time t if the wave is traveling in the negative x direction at 4 meters/second.What is the function g(·) for this case?c. Determine the speed of the wave if successive positive peaks at x = 1.3 meters areseparated by 0.75 seconds.2. Even and OddThe even and odd parts of a signal x[n] are defined by the following:• xe[−n] = xe[n] (i.e., xeis an even function of n)• xo[−n] = −xo[n] (i.e., xois an odd function of n)• x[n] = xe[n] + xo[n]Let x represent the signal whose samples are given byx[n] =12nn ≥ 00 otherwise.a. Determine the even and odd parts of the signal x.b. Show that your answer in part a is unique.c. Plot the results of part a.3. Geometric sumsa. Expand11 − ain a power series. For what range of a does your answer converge?6.003 Homework 1 / Spring 2010 2b. 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?4. Reconstructing CT Signals from SamplesLet a(t), b(t), and c(t) represent the following functions of time.110a(t)t110b(t)t110c(t)tLet xc(t) represent a continuous-time signal derived from the discrete-time signal xd[n]using a zero-order hold, as illustrated below, where consecutive samples of xdare sepa-rated by T seconds in xc.nxd[n]0 2 4 6 8 10txc(t)02T 4T 6T 8T 10Txd[0]xd[1]a. Determine an expression for xc(t) in terms of the samples xd[n] and the functionsa(t), b(t), and c(t). Your expression should match xc(t) at all points in time exceptpossibly at integer multiples of T , where xc(t) is discontinuous. [Extra credit: Brieflydiscuss whether it is possible to develop an expression that matches xc(t) at all points,including those where xc(t) is discontinuous.]Let yc(t) represent a continuous-time signal derived from the discrete-time signal yd[n]using a piecewise linear interpolator, so that sucessive samples of ydare connected bystraight line segments.nyd[n]0 2 4 6 8 10tyc(t)02T 4T 6T 8T 10Tyd[0]yd[1]b. Determine an expression for yc(t) in terms of the samples yd[n] and the functions a(t),b(t), and c(t). [Your expression need not match x(t) at integer multiples of T .]c. Determine an expression fordyc(t)dtin terms of the samples yd[n] and the functionsa(t), b(t), and c(t). [Your expression need not match x(t) at integer multiples of T .]6.003 Homework 1 / Spring 2010 35. Missing ParametersConsider the following system.+αβR1 −32RRX Y−Assume that X is the unit-sample signal, x[n] = δ[ n]. Determine the values of α and βfor which y[n] is the following sequence (i.e., y[0], y[1], y[2], . . .):0 , 1 ,32,74,158,3116, . . .Engineering Design Problems6. 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.]7. Drug dosingWhen drugs are used to treat a medical condition, doctors often recommend starting witha higher dose on the first day than on subsequent days. In this problem, we considera simple model to understand why. Assume that the human body is a tank of bloodand that drugs instantly dissolve in the blood when ingested. Further assume that drugvanishes from the blood (either because it is broken down or because it is flushed by thekidneys) at a rate that is proportional to drug concentration.Let x[n] represent the amount of drug taken on day n, and let y[n] represent the totalamount of drug in the blood on day n, just after the dose x[n] has dissolved in the blood,so that6.003 Homework 1 / Spring 2010 4y[n] = x[n] + αy[n − 1] .a. Determine the amount of drug in the blood that would result after taking one unit ofdrug each day for many consecutive days, i.e., determine limn→∞y[n] when x[n] = 1.b. Assume that there is initially no drug in the blood. Then, starting on day 0, one unitof drug is taken each day. Determine the first day when the amount of drug in theblood will equal or exceed half of its final value.c. Consider the following table of doses and resulting amount of drug in the blood:n x[n] y[n]−1 0 00 1 1.001 1 1.732 1 2.273 1 2.664 1 2.955 1 3.166 1 3.327 1 3.43Notice that the blood concentration ramps up over the first few days. Suggest adifferent initial dose x[0] that will result in a more constant amount of drug in theblood (with x[n] remaining at 1 for all n ≥ 1). Make a table to show your result.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
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