Math 522 Homework Assignments

MATH 522
Wavelets

Homework Assignments and Supplementary Materials

To view an Acrobat PDF version of each Solution Set, click on the symbol .
(Free Acrobat reader can be found here.)

Homework Assignments
(Each Problem is worth 10 pts.)

No. Assignment Due Page and No.'s

# 1
Aug 28 (Thursday)
[but to be turned in Tuesday Sept. 2]

Verify that Cⁿ is a vector space and in fact an inner product space.
Show that the set of all 3x2 matrices form a vector space over the corresponding scalar field.
Also work Problem #2, page 33 (worth 20 pts).
Using the inner product for C[0,2 pi], show that cos(x) and sin(2x) are orthogonal.
Graduate Students & Extra Credit: Show that any orthogonal collection of vectors in an inner product space is linearly independent.

# 2 Sept 9 (Tuesday)

Page 33: 9, 10, 14 (use any way you wish to plot), 15
Graduate Students & Extra Credit: #18

# 3 Sept 16 (Tuesday)

Page 33: 12,13

# 4 Oct. 7 (Tuesday)

Using Matlab or your preferred language, plot the Dirichlet kernel for values N=4, 8, 16, 32.

page 83-84: 15,16,17,18

E.C.: page 85-86: #28 [20 pts]

# 5 Oct. 7 (Tuesday)

The e-mail describing the homework reading assignment is located here.

# 6 Oct. 28 (Tuesday)
Solution
This is a little different than what was asked for for last Thursday. Let
x = [9 5 0 1 -6 -5 1 4 0 -5 7 9 0 -1 5 3]
and perform the Haar decomposition (dividing by 2; do not use square roots!):

First build the tree of wavelet coefficients along with the one scale coefficient. Do not retain the tree of averages as you proceed.

From the wavelet tree alone (and the one scale coefficient), decode and reconstruct the multiresolution tree of scale coefficients (i.e. the averages) that you just discarded.

Plot the reconstructed coarser level approximations on the same graph as the original vector x.

From the tree of Wavelet coefficients, pick those that are largest and on a new plot of the vector x, mark the locations where the wavelet coefficients are large.

# 7 Nov. 25 (Tuesday)

As outlined during class:
Page 127: #6.    You should just use the elementary properties of the Fourier transform that we established earlier. (Also read p. 123 on Gaussians and dispersion.)
[typo from earlier corrected: "Hilbert" replaced by "Fourier"]

Page 223: #8 (a),(b). Make sure that you distinguish between sinc in class and that used in the text, which we refer to instead as Sinc.    (Hint)

Details on the Math 522 Term Paper

Course Grading (from Math 522 Homepage)
"Graduate Students will be required to write a term paper (counting 25% of the Final Exam) on wavelet applications in the student's subject area, as well as additional test and homework problems."
Topics
What I was looking for was a paper describing how the Fourier transform, multiresolution analysis, or wavelets are used or planned to be used in your research or general area. Any engineering or scientific topic is fine. If you have trouble picking a topic let me know on Tuesday.
Length
The paper should be 4-5 typeset pages of prose, but you are encouraged to add additional images or diagrams. Any of the formats (Latex, TeX, Word, PDF, rtf) for the paper are fine. The amount of effort that I am anticipating that you spend on this is approximately that of 2 homework assignments.
Due Date
At the beginning of the Final Exam (9 am on December 11).

   Worksheets, Matlab Tutorials and Documentation

"demo"

Although there is on-line help in Matlab, you can find helpful tutorials and quick-start documents at the following links:

Manual of basics to get started (University of Florida): MatLab Primer
"Getting Started with Matlab" from MathWorks in hyperlinked, cross-referenced PDF document format [Getting Started] and on-line at the URL [On-Line]
Official Matlab source html and PDF documents for technical information from MathWorks.

Matlab Scripts

No. Description Link

# 1
Fourier Series convergence demos
HW_a and HW_b

This page maintained by Robert Sharpley (sharpley@math.sc.edu) and last updated Aug. 28, 2003.
This page ©2002-2003, The Board of Trustees of the University of South Carolina.
URL: http://www.math.sc.edu/~sharpley/math522

Return to Math 522 Home Page

No.	Assignment Due	Page and No.'s
# 1	Aug 28 (Thursday) [but to be turned in Tuesday Sept. 2]	Verify that Cⁿ is a vector space and in fact an inner product space. Show that the set of all 3x2 matrices form a vector space over the corresponding scalar field. Also work Problem #2, page 33 (worth 20 pts). Using the inner product for C[0,2 pi], show that cos(x) and sin(2x) are orthogonal. Graduate Students & Extra Credit: Show that any orthogonal collection of vectors in an inner product space is linearly independent.
# 2	Sept 9 (Tuesday)	Page 33: 9, 10, 14 (use any way you wish to plot), 15 Graduate Students & Extra Credit: #18
# 3	Sept 16 (Tuesday)	Page 33: 12,13
# 4	Oct. 7 (Tuesday)	Using Matlab or your preferred language, plot the Dirichlet kernel for values N=4, 8, 16, 32. page 83-84: 15,16,17,18 E.C.: page 85-86: #28 [20 pts]
# 5	Oct. 7 (Tuesday)	The e-mail describing the homework reading assignment is located here.
# 6	Oct. 28 (Tuesday) Solution	This is a little different than what was asked for for last Thursday. Let x = [9 5 0 1 -6 -5 1 4 0 -5 7 9 0 -1 5 3] and perform the Haar decomposition (dividing by 2; do not use square roots!): First build the tree of wavelet coefficients along with the one scale coefficient. Do not retain the tree of averages as you proceed. From the wavelet tree alone (and the one scale coefficient), decode and reconstruct the multiresolution tree of scale coefficients (i.e. the averages) that you just discarded. Plot the reconstructed coarser level approximations on the same graph as the original vector x. From the tree of Wavelet coefficients, pick those that are largest and on a new plot of the vector x, mark the locations where the wavelet coefficients are large.
# 7	Nov. 25 (Tuesday)	As outlined during class: Page 127: #6. You should just use the elementary properties of the Fourier transform that we established earlier. (Also read p. 123 on Gaussians and dispersion.) [typo from earlier corrected: "Hilbert" replaced by "Fourier"] Page 223: #8 (a),(b). Make sure that you distinguish between sinc in class and that used in the text, which we refer to instead as Sinc. (Hint)

No.	Description	Link
# 1	Fourier Series convergence demos	HW_a and HW_b



This page maintained by Robert Sharpley (sharpley@math.sc.edu) and last updated Aug. 28, 2003. This page ©2002-2003, The Board of Trustees of the University of South Carolina. URL: http://www.math.sc.edu/~sharpley/math522