**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

**Homepage: **www.ima.umn.edu/~miller

**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.

**Assignments:**

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

**Grading:**

Homework 20%

First Midterm 20%

Second Midterm 20%

Final Exam 40%

**Miscellaneous:**

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)**

**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

**I. Vector Spaces with Inner Product**

**Definitions**

**Bases**

**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**

**Definitions**

**Real and complex Fourier series, Fourier series on intervals
of varying length, Fourier series for odd and even functions**

**Examples**

**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**

** Examples**

** 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**

**Causality**

**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**

**Definitions**

**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**