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Course by: Richard Baraniuk. E-mail the author

Homework #1

Module by: Richard Baraniuk, Justin Romberg. E-mail the authors

Summary: Homework 1 problem set for Elec301 at Rice University.

due date:

Noon, Thursday, September 5, 2002

Assignment 1

Homework, tests, and solutions from previous offerings of this course are off limits, under the honor code.

Problem 1

Form a study group of 3-4 members. With your group, discuss and synthesize the major themes of this week of lectures. Turn in a one page summary of your discussion. You need turn in only one summary per group, but include the names of all group members. Please do not write up just a "table of contents."

Problem 2

Construct a WWW page (with your picture) and email Mike Wakin (wakin@rice.edu) your name (as you want it to appear on the class web page) and the URL. If you need assistance setting up your page or taking/scanning a picture (both are easy!), ask your classmates.

Problem 3: Learning Styles

Follow this learning styles link (also found on the Elec 301 web page) and learn about the basics of learning styles. Write a short summary of what you learned. Also, complete the "Index of learning styles" self-scoring test on the web and bring your results to class.

Problem 4

Make sure you know the material in Lathi, Chapter B, Sections 1-4, 6.1, 6.2, 7. Specifically, be sure to review topics such as:

• complex arithmetic (adding, multiplying, powers)
• finding (complex) roots of polynomials
• complex plane and plotting roots
• vectors (adding, inner products)

Problem 5: Complex Number Applet

Reacquaint yourself with complex numbers by going to the course applets web page and clicking on the Complex Numbers applet (may take a few seconds to load).

(a) Change the default add function to exponential (exp). Click on the complex plane to get a blue arrow, which is your complex number zz. Click again anywhere on the complex plane to get a yellow arrow, which is equal to ez z . Now drag the tip of the blue arrow along the unit circle on with |z|=1 z 1 (smaller circle). For which values of zz on the unit circle does ez z also lie on the unit circle? Why?

(b) Experiment with the functions absolute (abs), real part (re), and imaginary part (im) and report your findings.

Problem 6: Complex Arithmetic

Reduce the following to the Cartesian form, a+jb a b . Do not use your calculator!

(a) -1j220 -1 2 20

(b) 1+2j3+4j 12 34

(c) 1+3j3j 1 3 3

(d) j

(e) jj

Problem 7: Roots of Polynomials

Find the roots of each of the following polynomials (show your work). Use MATLAB to check your answer with the roots command and to plot the roots in the complex plane. Mark the root locations with an 'o'. Put all of the roots on the same plot and identify the corresponding polynomial (aa, bb, etc...).

(a) z24z z 2 4 z

(b) z24z+4 z 2 4 z 4

(c) z24z+8 z 2 4 z 8

(d) z2+8 z 2 8

(e) z2+4z+8 z 2 4 z 8

(f) 2z2+4z+8 2 z 2 4 z 8

Problem 8: Nth Roots of Unity

ej2πN 2 N is called an Nth Root of Unity.

(a) Why?

(b) Let z=ej2π7 z 2 7 . Draw zz2z7 z z 2 z 7 in the complex plane.

(c) Let z=ej4π7 z 4 7 . Draw zz2z7 z z 2 z 7 in the complex plane.

Problem 9: Writing Vectors in Terms of Other Vectors

A pair of vectors uC2 u 2 and vC2 v 2 are called linearly independent if αu+βv=0   if and only if   α=β=0 α u β v 0   if and only if   α β 0 It is a fact that we can write any vector in C2 2 as a weighted sum (or linear combination) of any two linearly independent vectors, where the weights αα and ββ are complex-valued.

(a) Write 3+4j6+2j 34 62 as a linear combination of 12 1 2 and -53 -5 3 . That is, find αα and β β such that 3+4j6+2j=α12+β-53 34 62 α 1 2 β -5 3

(b) More generally, write x=( x1 x2 ) x x 1 x 2 as a linear combination of 12 1 2 and -53 -5 3 . We will denote the answer for a given xx as αx α x and βx β x .

(c) Write the answer to (a) in matrix form, i.e. find a 2×2 matrix AA such that A( x1 x2 )=αxβx A x 1 x 2 α x β x

(d) Repeat (b) and (c) for a general set of linearly independent vectors u u and v v.

Problem 10: Fun with Fractals

A Julia set JJ is obtained by characterizing points in the complex plane. Specifically, let fx=x2+μ f x x 2 μ with μμ complex, and define g 0 x=x g 0 x x g 1 x=f g 0 x=fx g 1 x f g 0 x f x g 2 x=f g 1 x=ffx g 2 x f g 1 x f f x g n x=f g n 1 x g n x f g n 1 x Then for each xx in the complex plane, we say xJ x J if the sequence | g 0 x|| g 1 x|| g 2 x| g 0 x g 1 x g 2 x does not tend to infinity. Notice that if xJ x J , then each element of the sequence g 0 x g 1 x g 2 x g 0 x g 1 x g 2 x also belongs to JJ.

For most values of μμ, the boundary of a Julia set is a fractal curve - it contains "jagged" detail no matter how far you zoom in on it. The well-known Mandelbrot set contains all values of μμ for which the corresponding Julia set is connected.

(a) Let μ=-1 μ -1 . Is x=1 x 1 in JJ?

(b) Let μ=0 μ 0 . What conditions on xx ensure that xx belongs to JJ?

(c) Create an approximate picture of a Julia set in MATLAB. The easiest way is to create a matrix of complex numbers, decide for each number whether it belongs to JJ, and plot the results using the imagesc command. To determine whether a number belongs to JJ, it is helpful to define a limit NN on the number of iterations of gg. For a given xx, if the magnitude | g n x| g n x remains below some threshold MM for all 0nN 0 n N , we say that xx belongs to JJ. The code below will help you get started:



N = 100;      % Max # of iterations
M = 2;        % Magnitude threshold
mu = -0.75;   % Julia parameter
realVals = [-1.6:0.01:1.6];
imagVals = [-1.2:0.01:1.2];

xVals = ones(length(imagVals),1) * realVals + ...
j*imagVals'*ones(1,length(realVals));

Jmap = ones(size(xVals));
g = xVals;    % Start with g0

% Insert code here to fill in elements of Jmap.  Leave a '1'
% in locations where x belongs to J, insert '0' in the
% locations otherwise.  It is not necessary to store all 100
% iterations of g!

imagesc(realVals, imagVals, Jmap);
colormap gray;
xlabel('Re(x)');
ylabel('Imag(x)');



This creates the following picture for μ=-0.75 μ -0.75 , N=100 N 100 , and M=2 M 2 .

Using the same values for NN, MM, and xx, create a picture of the Julia set for μ=-0.3910.587j μ -0.391 0.587 . Print out this picture and hand it in with your MATLAB code.

Just for Fun:

Try assigning different color values to Jmap. For example, let Jmap indicate the first iteration when the magnitude exceeds MM. Tip: try imagesc(log(Jmap)) and colormap jet for a neat picture.

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