Syllabus for Math 5467 (Introduction to the  Mathematics of Wavelets), Spring 2002

Instructor: Willard Miller
Office: Vincent Hall 513
Phone: 612-624-7379

Classroom:  Vincent Hall 211

Class times:  10:10am-11:00am MWF

Prerequisites:   SP - [2243 or 2373 or 2573], [2283 or 2574 or 3283 or instr consent]; [[2263 or 2374], 4567] recommended; QP - instr consent

Office Hours:   M 9:05-9:55 am, Tu 2:30-3:25 pm,  W 11:15-12:05 pm


Text: Wavelets and Filter Banks, Gilbert Strang and Truong Nguyen, ISBN 0-9614088-7-1,  Wellesley-Cambridge Press, 1996

Midterm Tests:  F 15 March and F 19 April

Final Exam:  4:00-6:00 pm, Friday, May 17, Vincent Hall 211

Material covered in course:

 Background theory/experience in wavelets. Inner product spaces, operator theory, Fourier transforms, multi-scale analysis, discrete wavelets, self-similarity. Computing techniques.

We will start at the beginning and cover the basics thoroughly. All of the later topics will be treated in some form. We will make use of the Wavelets Toolbox in MATLAB for some of the homework.  Filter banks from signal processing will be used to motivate the theory, and there will be applications to image processing. This is an interdisciplinary course, with a strong math core, meant for students in mathematics, science and engineering.


I will give out assignments in class and will announce due dates in class, as well.

Homework                  20%
First Midterm             20%
Second Midterm        20%
Final Exam                  40%


Grades will be deterrmined  through an interaction of objective standards and experience with the class. I don't preassign the number of students who will receive a specific grade. On the other hand, neither will I preassign the gradelines before seeing the distribution of grades. Gradelines will be announced on the web, as soon as possible after
the quiz or exam.

Incompletes will be given only in cases where the student has completed all but a small fraction of the course with a grade of C or better and a severe unexpected event prevents completion of the course. In particular, if you get behind, you cannot ``bail out'' by taking an incomplete. The last day to cancel, without permission from your College office, is the last day of the sixth week.


Lecture Notes and Supplementary Notes for the Course  (Postscript File)(PDF File)

Homework Problem Set #1 (Postscript File)     (PDF FILE)

   Solutions (Postscript file)       (PDF FILE)

Homework Problem Set #2 (Postscript File)     (PDF FILE)

  Solutions (Postscript file)       (PDF FILE)

Homework Problem Set #3 (Postscript File)     (PDF FILE)

   Solutions (Postscript file)       (PDF FILE)

Homework Problem Set #4 (Postscript File)     (PDF FILE)

   Solutions (PDF FILE)

Homework Problem Set #5 (Postscript File)     (PDF FILE)

Solutions to Midterm 2  (Postscript File)    (PDF FILE)

Homework Problem Set #6 (Postscript File)     (PDF FILE)

 Homework Problem Set #7 (Postscript File)     (PDF FILE)


Course Syllabus:  Introduction to the Mathematics of Wavelets
O. Motivation through signal processing, analysis and synthesis

I. Vector Spaces with Inner Product

Schwarz inequality
Orthogonality, Orthonormal bases
Hilbert spaces.
L² and l². The Lebesgue integral.
Orthogonal projections
Gram-Schmidt orthogonalization
      Linear operators and matrices
Least squares, applications

II. Fourier Series

Real and complex Fourier series, Fourier series on intervals of varying length, Fourier series for odd and even functions
Convergence results
Riemann-Lebesgue Lemma
Pointwise convergence, Gibbs phenomena
Mean convergence, Bessel's inequality, Parseval's equality,
      Integration and differentiation of Fourier series

III. The Fourier Transform

The transform as a limit of Fourier series
Convergence results
Riemann-Lebesgue Lemma
Pointwise convergence,
L¹ and L² convergence, Plancherel formula,
      Properties of the Fourier transform
      Relations between Fourier series and Fourier integrals: sampling, aliasing,
      The Fourier integral and the uncertainty relation of quantum mechanics, Poisson Summation formula

IV.  Discrete Fourier Transform

Definition and relation to Fourier series
Properties of the transform
Fast Fourier Transform (FFT)
Efficiency of the FFT algorithm
Approximation to the Fourier Transform

V. Linear Filters

Definition, Discrete and continuous filters
Time invariant filters and convolution
Filters in the time domain and in the frequency domain
The Z-transform and Fourier series
       Low pass and high pass filters
Analysis and synthesis of signals, downsampling and upsampling
Filter banks, orthogonal filter banks and perfect reconstruction of signals
Spectral factorization, Maxflat filters

VI. Multiresolution Analysis

Wavelets, multiple scales
Haar wavelets as motivation
Scaling functions, The dilation equation, The wavelet equation
Wavelets from filters
Lowpass iteration and the Cascade Algorithm
Daubechies wavelets
Scaling Function by recursion, Evaluation at dyadic points
Infinite product formula for the scaling function

VII. Wavelet Theory

Accuracy of approximation, Convergence
Smoothness of scaling functions and wavelets

VIII.  Other Topics

The Windowed Fourier transform and The Wavelet Transform
Bases and Frames, Windowed frames
Biorthogonal Filters and Wavelets
Multifilters and Multiwavelets

IX.  Applications of Wavelets

Image compression
Digitized Fingerprints
Speech and Audio Compression
Differential equations

MATLAB exercises