Wavelets and signal processing
Lectures MWF 11:10-12:00, 9 Evans Hall
Office hours WF 14:00-15:00, 821 Evans Hall
- 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
- 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.
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.
- 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
- Homework assignment #8, due Dec 2nd, in PS and in
- Homework assignment #9, due Dec 9th, in PS and in
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.
The diary of
the first MATLAB session and the corresponding figures
Applications of wavelets using MATLAB:
Last modified: Dec 13, 2005