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

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.

Assignments

• Homework assignment #1, due Sep 9th, in PS and in PDF.
• Homework assignment #2, due Sep 16th, in PS and in PDF.
• Homework assignment #3, due Sep 23rd, in PS and in PDF.
• Homework assignment #4, due Sep 30th, in PS and in PDF.
• Homework assignment #5, due Oct 14th, in PS and in PDF.
• Homework assignment #6, due Oct 21st, in PS and in PDF.
• Homework assignment #7, due Nov 4th, in PS and in PDF.
• Homework assignment #8, due Dec 2nd, in PS and in PDF.
• Homework assignment #9, due Dec 9th, in PS and in PDF.

Exams

Here are PS and PDF versions of a mock midterm. Here are the actual midterm in PS and in PDF and solutions in PS and in PDF.

Here is a mock final test, in PS and in PDF. Here is the actual final test, in PS and in PDF.

Demos, m-files, etc.

• The diary of the first MATLAB session and the corresponding figures graph1.eps and fig1.ps.
• Applications of wavelets using MATLAB: