6 003 Homework 1 Due at the beginning of recitation on Wednesday September 16 2009 Problems 1 Waves Assume that the height of a water wave is given by g x vt where x is distance v is velocity and t is time The function g is defined in the following plot and is zero outside the indicated range g r r 3 2 1 0 1 2 3 a Assume x 0 and v 1 Sketch f1 t g t which illustrates the height of the wave 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 differences between f1 and 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 Odd Let x n represent the signal shown in the following plot where all values outside the indicated range are zero x n 1 n 3 2 1 0 1 2 3 1 Assume that x n can be written as the sum of an even part xe n xe n and an odd part xo n xo n a Sketch xe n and xo n b Prove that xo n is unique 6 003 Homework 1 Fall 2009 2 3 Periodicity Consider two DT signals 3 n x1 n cos and 4 5 n x2 n cos 4 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 sums a Expand 1 in a power series For what range of a does your answer converge 1 a b Find a closed form expression for N 1 X an For what range of a does your answer n 0 converge c Expand 1 in a power series For what range of a does your answer converge 1 a 2 Engineering Design Problems 1 Choosing a bank Consider two banks Bank 1 offers a 3 annual interest rate but charges a 1 service charge 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 in bank i at the beginning of year n and xi n represent the amount of money you deposit in bank i during year n Assume that deposits during year n are credited to the balance at 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 for each bank b Assume that you deposit 100 in each bank and make no further deposits Solve your 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 these balances 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 larger balance after 25 years i e at the beginning of the 26th year Hint See the Appendix for help with programming 6 003 Homework 1 Fall 2009 3 2 More Rabbits Let 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 difference equation should have the form g n x n a0 g n 1 a1 g n 2 where the coefficients ai are constants and x is given by n 1 if n 0 x n 0 otherwise b The sequence g can be described by a population problem that is very similar to the one that describes the Fibonacci sequence Write a description of the population problem described by g c The sequence g n grows approximately geometrically i e g n abn for large values of n Determine a and b d Determine an exact expression for g n of the form g n abn cdn Hours Please estimate the number of hours you spent working on this homework assignment Also feel free to comment on these problems Appendix Fibonacci code You 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 code y 1 1 y 2 1 for i 3 20 y i y i 1 y i 2 end y stem 0 19 y initial conditions indices start at 1 not 0 print y Example Python code from pylab import y 1 1 for i in range 2 20 y append y i 1 y i 2 print y stem range 20 y show initial conditions
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