6 01 Fall Semester 2007 Assignment Week 5 Issued Tuesday Oct 2 1 MASSACHVSETTS INSTITVTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science 6 01 Introduction to EECS I Fall Semester 2007 Assignment Week 5 Issued Tuesday Oct 2 To do this week NOTE TWO LECTURES TWO SOFTWARE LABS in Tuesday software lab 10 2 1 Start writing code and test cases for the numbered questions in the software lab Paste all your code including your test cases into the box provided in the Software Lab Part 5 1 problem on the on line Tutor This will not be graded before lecture on Thursday 10 4 1 Nothing in Thursday software lab 10 4 1 Start writing code and test cases for the numbered questions in the software lab Paste all your code including your test cases into the box provided in the Software Lab Part 5 2 problem on the on line Tutor This will not be graded 2 Get the handout for next Thursday s lab No Lecture Tuesday 10 9 before the lab Thursday 10 11 1 Do the on line Tutor problems for week 5 that are due on Thursday Part 5 3 2 Do the writeup for the 10 2 and 10 4 software labs in this handout providing written answers including code and test cases for every numbered question in this handout before lecture Tuesday 10 16 1 Do the writeup for the 10 11 design lab NOT in this handout 2 Do the nanoquiz it will be based on the material in the lecture notes and the on line Tutor problems due on Thursday 6 01 Fall Semester 2007 Assignment Week 5 Issued Tuesday Oct 2 2 On Athena or the lab laptops make sure you execute athrun 6 01 update so that you can get the Desktop 6 01 lab5 directory which has the files mentioned in this handout You need the file differenceEquationWithInput py You need the file soar graph py During this lab please use the lab laptop for Tuesday s software lab If you are using your own laptop for Thursday s software lab download the files from the course Web site on the Calendar page Tuesday Software Lab Speed of difference equation solving In this lab we will be using python s features for timing software Since our experience is that the timing functions are not very portable please work in pairs and use the lab laptops In earlier lectures you learned that the solution to a homogeneous difference equation can be represented in closed form using the equation s natural frequencies In today s lecture you learned that when a difference equation has an input that is the equation is not homogeneous the solution is more complicated to compute and that computation is probably best accomplished using a program One general form for a difference equation with input we will see another form just below is y n 1 y n 1 2 y n 2 K y n K o x n 1 x n 1 L x n L where x n is the input and is a given sequence To determine y n from the above difference equation it is necessary to specify the values of the coefficients 1 K and o L the initial values y 0 y K 1 and value of x n for all n One way to specify the input x is use a procedure For example consider defining x n using the python procedure def x n return 1 In this case x n 1 for all n In today s lab we will be examining and modifying an implementation of a difference equation class and will then use that class to develop a feel for orders of growth Examine the difference equation class defined in differenceEquationWithInput py You will notice that the difference class uses a different format for specifying a difference equation one that turns out to be much more easily manipulated when using transform techniques to be described in a later lecture Specifically the difference equation format used in the difference equation class is K X k 0 ak y n k L X bl x n l l 0 A note should be made about the order of the coefficients In the polynomial class the coefficients for a polynomial z2 2z 3 would be given as a list 1 2 3 In following that convention the 6 01 Fall Semester 2007 Assignment Week 5 Issued Tuesday Oct 2 3 coefficients of a difference equation go from higher to lower order so that a list of the y coefficients would be ak ak 1 ao Note that this same seemingly backward convention is true for initial conditions as well the intial values are y K 1 y 0 This may seem confusing as the kth coefficient of a difference equation is not the kth element of the list We agree but the common convention is against us Consider the Fibonacci example In order to create an instance of the difference equation class that can be used to compute the Fibonacci numbers it is necessary to create an instance associated with the difference equation y n y n 1 y n 2 Here there is no input and therefore x n 0 for all n Loading differenceEquationWithInput py and then typing the following command in the interpreter fib DifferenceEquationWithInput 1 0 1 1 1 1 lambda n 0 will create an difference equation instance which will generate the Fibonacci numbers Question 1 How are the k s and l s related to the ak s and the bl s in the above two forms of difference equations If you examine the DifferenceEquationWithInput class you will notice that two methods are implemented for computing the nth value of the solution to the difference equation One method uses iteration and the other uses recursion To test your understanding of the class implementation please answer the following questions Question 2 What does the line vals nextVal vals 1 in the valIter function accomplish Question 3 What does the line dotProd self outCoefficients 1 self valRecur n i 1 for i in range self order in the valRecur function accomplish A clever plot It can often be revealing to plot the series of values a difference equation generates Please refer to the week 4 assignment for instructions on how to plot using SoaR the assignment is on the course calendar page if you do not have it handy Question 4 Demonstrate that you understand how the difference equation class works by using the class to solve and plot the solution to the difference equation y n 0 5y n 1 x n 1 where y 0 0 and x n 1 for all n Question 5 Now use the difference equation class to solve and plot the solution to the difference equation y n 0 5y n 1 x n 1 where y 0 0 and x n 1 for all n Question 6 Explain why the two plots look the way they do and explain the differences between the two plots 6 01 Fall Semester 2007 Assignment Week …
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