Purdue University
School of Electrical and Computer Engineering

ECE648 Wavelet, Time-Frequency, and Multirate Signal Processing

Spring 2005
Prof. Ilya Pollak

Course Information.
Lecture notes for the first few lectures.

---

Problem Set 1.

---

Problem Set 2.

---

Problem Set 3.

---

Required textbook: S.G. Mallat. "A Wavelet Tour of Signal Processing." 2nd Edition. Academic Press, 1999. ISBN 0-12-466606-X

Other strongly recommended books:

G. Strang, T. Nguyen. "Wavelets and Filter Banks". Wellesley-Cambridge Press, 1997. ISBN 0-9614088-7-1

I. Daubechies. "Ten Lectures on Wavelets". Philadelphia: Society for Industrial and Applied Mathematics, 1992. ISBN 0-89871-274-2

M. Vetterli, J. Kovacevic. "Wavelets and Subband Coding." Prentice Hall, 1995. ISBN 0-13-097080-8

The main theme of this course is sparse representations of signals. Orthogonal bases are of particular interest because they can efficiently approximate certain types of signals with just a few vectors. Two examples of such applications which will be studied in this course are image compression and estimation of noisy signals.

We will consider several approximation schemes. For example, a linear approximation projects the signal over M vectors chosen a priori. Better approximations are obtained by choosing the M basis vectors adaptively, depending on the signal. A further degree of freedom is introduced by choosing the basis itself adaptively, from a family of bases. In this context, a number of very successful recent algorithms for nonlinear approximation, noise removal, and feature extraction will be considered, such as wavelet thresholding, best basis search, and matching pursuit.

Several types of orthogonal bases will be studied: wavelet bases, wavelet packet bases, and local cosine bases. We will study their properties, as well as fast algorithms for calculating the coefficients of a signal with respect to a basis and for reconstructing the signal from its coefficients. We will also address time-frequency properties of various signal decompositions--which is an important tool in deciding what decomposition is appropriate for a particular application.

The pre-requisite for this course is thorough understanding of fundamentals of signal processing--such as Fourier analysis and sampling. These topics will be reviewed at the beginning of the course.