6.986 Wavelets and Applications (H)

L MW11-12:30, room 1-273
Professor Vivek Goyal, vgoyal@mit.edu
Prereq.: 6.341 and 18.06 or permission of instructor
3-0-9

This subject qualifies as a communication, control, and signal processing concentration subject.

Wavelet transforms and associated techniques have made the transition from esoteric research topics to essential tools in signal processing. This course aims to provide an ability to understand and apply wavelet-based techniques and recognize when they are appropriate. The course emphasizes discrete-time signal representations that can be computed efficiently, as these are most often used in practice. It also develops the foundations for research in signal processing.

Main topics: * Review of Hilbert spaces, normed spaces, orthonormal bases, frames, etc.

* Review of Fourier series, Fourier transforms, classical sampling, multirate signal processing.

* The time-frequency plane. Local Fourier analysis.

* Two-channel filter banks.

* Iterated filter banks including discrete wavelet transforms and wavelet packet transforms.

* Wavelets from filter banks: Smoothness, approximation properties.

* Approximation in bases: Linear approximation, non-linear approximation, approximation power for piecewise smooth signals.

* Statistical modeling with wavelets.

* Denoising: classical Wiener filtering, wavelet thresholding.

* Compression: Basics of quantization, transform coding, wavelet-based coding, JPEG2000, rate-distortion behavior of wavelet codes.

* Sampling revisited: Sampling not based on Fourier bandwidth.

Course structure: The course will have conventional homework assignments, problems to be completed with Matlab, a midterm exam, and a final project (no final exam). The project may be completed individually or in small groups.