Math 118

Wavelets and signal processing

Lectures MWF 11:10-12:00, 9 Evans Hall

Office hours WF 14:00-15:00, 821 Evans Hall

Course Info (.pdf version here)
Resources (books, notes, more)
Academic honesty
Assignments
Solutions
Exams
Demos, mfiles, etc

Topics covered

08/29: Introduction to representation of functions.
08/31: Inner product spaces. L^2[a,b] and l^2.
09/02: Pointwise, uniform and L^2-convergence.
09/07: Orthogonality. Orthonormal bases. Orthogonal projectors. Linear maps and their adjoints.
09/09: Fourier series: complex and real forms.
09/12: Fourier series: examples, Bessel's inequality and Parseval's identity.
09/14: Fourier series: decomposing even and odd functions; convergence.
09/16: Fourier series: uniform and pointwise convergence.
09/19: Fourier transform: basic properties, inversion, Plancherel's formula.
09/21: Linear, time-invariant, causal filters.
09/23: Shannon-Whittaker sampling theorem. Heisenberg uncertainty principle.
09/26: Discrete Fourier transform (DFT).
09/28: Eigenvalues of the Fourier transform and of the DFT.
09/30: Interplay between the Fourier transform on the circle and the DFT. Aliasing.
10/03: Back to Heisenberg. Differences between the line, the circle, and the DFT.
10/05: More about DFT. Fast DFT (aka FFT).
10/07: Discrete signals and filters. Z-transform.
10/10: Haar scaling function, wavelet and multiresolution analysis (MRA).
10/12: Haar decomposition.
10/14: Haar reconstruction. Data compression.
10/17: MRA: general setup.
10/19: Scaling functions and generation of wavelets.
10/21: Midterm.
10/24: Discussion of the midterm.
10/26: Scaling functions: infinite product formula; orthogonality of shifts.
10/28: Scaling functions: cascade algorithm.
10/31: Daubechies' scaling functions.
11/02: Daubechies' scaling functions and wavelets.
11/04: Vanishing moments of wavelets.
11/07: Computations of scaling functions and wavelets.
11/09: Programming in MATLAB (review).
11/14: Computational complexity of wavelet decomposition/reconstruction; signal extenstions; other computational issues.
11/16: MATLAB wavelet toolbox, part 1.
11/18: Wavelets in higher dimensions. Signal denoising. Feature detection.
11/21: Continuous wavelet transform.
11/23: MATLAB wavelet toolbox, part 2.
11/28: The unitary extension principle.
11/30: The transfer operator and refinability.
12/02: The transfer operator and smoothness. Connections to convolution with cardinal B-splines.
12/05: Introduction to splines: recurrence relation for B-splines, underlying knot sequences, partition of unity.
12/07: Evaluation, differentiation, integration of splines. Multiple knots and smoothness. Interpolation and approximation using splines. Control polygons.
12/09: MATLAB spline toolbox.

Resources

Books.
- Wavelets and filter banks, by Gilbert Strang and Truong Nguyen, 1997.
- A first course on wavelets, by Eugenio Hernandez and Guido Weiss, 1996.
- A first course in wavelets with Fourier analysis, by Albert Boggess and Francis J. Narcowich, 2001.
The third book serves as the main textbook for the course.
Lecture notes on wavelets by Amos Ron.
Lecture notes on splines by Carl de Boor.
CS 515, a course on wavelets and splines taught at UW-Madison.
MATLAB primer.

The instructor welcomes cooperation among students and the use of books . However, handing in homework that makes use of other people's work (be it from a fellow student, a book or paper, or whatever) without explicit acknowledgement is considered academic misconduct.