6 081 Spring Semester 2007 Work for Week 2 1 MASSACHVSETTS INSTITVTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science 6 081 Introduction to EECS I Spring Semester 2007 Work for Week 2 Issued Tuesday Feb 13 This handout contains Software Lab for Tuesday February 13 Pre Lab exercises due Thursday February 15 at 2PM you should come and do them in lab on Wednesday February 14 Robot Lab for Thursday February 15 read this before coming to Lab Post lab exercises due Thursday Feb 22 There is no lecture on Tuesday Feb 20 MIT virtual Monday Higher order procedures Behaviors and utility functions This week s work gives you practice with higher order procedures i e procedures that manipulate other procedures Tuesday s lecture and post class lab cover Python s support for functional programming including list comprehension Wednesday s homework and Thursday s lab apply higher order procedures to the tasks of specifying robot behaviors with the aid of utility functions Tuesday Software Lab Higher order procedures This session gives you practice with Python s basic tools for functional programming higher order procedures and list comprehension A higher order procedure is a procedure that takes takes procedures as inputs and or returns procedures as results The basic example we ll work with now is a procedure called plot which takes a function f as input and graphs it Start by loading the file feb 13 class py It begins with the following code import the graphing package from SoaR import SoaR from SoaR Util GraphingWindow import GraphingWindow import sine and cosine import Numeric from Numeric import sin from Numeric import cos import the version of addition for use with reduce import operator from operator import add as add set up a variable for the plotting window plotWindow None 6 081 Spring Semester 2007 Work for Week 2 2 plot a given function f def plot f global plotWindow if plotWindow plotWindow close xmax 10 xvals i 100 for i in range 0 int xmax 100 yvals f x for x in xvals ymax reduce max abs y for y in yvals graphmax 1 1 ymax plotWindow GraphingWindow 500 500 0 xmax graphmax graphmax Plot plotWindow graphContinuous f plotWindow helpIdle The plot procedure opens a graphing window and plots f in the interval from 0 to 10 The system then waits hangs until you close the plotting window with the mouse 1 Note that the input f to plot is a procedure Question 1 What do you expect plot lambda x x to produce Try it and see Question 2 Make a plot of the sine function Do you have to write plot lambda x sin x or can you simply write plot sin Try it and see Make sure to ask if the result isn t what you expect Observe how plot uses list comprehension and reduce with max to compute the ranges for the axes If you ve programmed before you probably expected that this would be done using some kind of loop One of the hallmarks of functional style is to use functions operating on sets of elements rather than writing explicit loops with for or while Now let s plot something more interesting superpositions of sine waves of increasing frequency sin x 1 1 1 sin 2x sin 3x sin 4x 2 3 4 out to more and more terms Let s call saw n the function whose value at x is the sum of the first n 1 terms of this sequence Then we can compute saw as def saw n return lambda x reduce add 1 0 j sin j x for j in range 1 n This definition is included in the file you loaded Here saw is a procedure that takes n as input and returns a function of x which we can pass to plot Observe again how we ve used list comprehension and reduce with add rather than write an explicit loop to sum the terms Question 3 Plot saw n for various values of n starting with n 2 which is just a sine wave do you see why up to large values of n like 100 It should become obvious why we ve named the procedure saw 1 We re embarrassed about setting things up this way It s a kludge to get around a bad interaction between the SoaR robot system and Python s Idle editor 6 081 Spring Semester 2007 Work for Week 2 3 As you ll learn later in 18 03 and or 6 003 any periodic function can be expressed as a Fourier series i e a superposition of sines and cosines of the form a1 sin x b1 cos x a2 sin 2x b2 cos 2x ak sin kx bk cos kx Question 4 Define and plot the function squareW ave n that produces partial sums of the Fourier expansion of a square wave sin x 1 1 1 sin 3x sin 5x sin 7x 3 5 7 Do this with list comprehension using a construct of the form expression in j for j in range if condition Question 5 Do the same thing for a triangle wave whose Fourier expansion is cos x 1 1 1 cos 3x cos 5x cos 7x 9 25 49 Now let s rewrite these programs using an even more functional style similar to what you ll do in lab this week The following procedure addf included in the file you loaded takes as inputs two functions f and g and returns a function whose value at any x is f x g x def addf f g return lambda x f x g x The important thing to observe is that the inputs to addf as well as the result are functions i e addf transforms functions to functions Similarly we have scalef which takes a function f and a number s and return the function whose value at x is s f x and expandf which produces the function whose value at x is f s x Convince yourself that an equivalent way to write the saw procedure is def saw2 n return reduce addf scalef expandf sin j 1 0 j for j in range 1 n and make some plots The definitions of addf scalef expandf and saw2 are included in the file you loaded Question 6 Write similar definitions for square waves and triangle waves and check the results by plotting them For the rest of this period play around with these procedures as you have time doing things that interest you Here are some ideas to try You needn t do all of them 6 081 Spring Semester 2007 Work for Week 2 4 Question 7 Change the square triangle and sawtooth wave definitions to use cosines in place of sines and vice versa and see what kind curves this produces Or try some other Fourier series Question 8 Generalize the plot procedure so that you can specify the range for x Question 9 Change the plot procedure to use the procedure plotWindow graphDiscrete rather than plotWindow graphContinuous This is the sort of plotting we ll be doing …
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