Gabor frames with trigonometric spline dual windows

A Gabor system is a collection of modulated and translated copies of a window function. If we have a signal in L²(\mathbb{R}), it can be analyzed with a Gabor system generated by a certain window $g$ and then synthesized with a Gabor system generated by another window $h$. If this leads us to a perfect reconstruction, we say that $g$ and $h$ are dual Gabor windows. Few explicit examples of dual window pairs are known. This thesis constructs explicit examples of Gabor dual windows with trigonometric form. The windows have fixed support and have an arbitrary smoothness. Also, in the discrete time domain, the trigonometric form allows us to evaluate the Gabor coefficients efficiently using the Discrete Fourier Transform. For the higher dimensional cases, we find window examples for a large class of modulation parameter lattices, including shear lattices. Also, a sufficient condition on the norm of the modulation lattice to have explicit dual Gabor windows is presented, for every dimension.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2011

Author dc:creator

• Kim, Inmi

Contributors dc:contributor

• Laugesen, Richard S.
• Rosenblatt, Joseph
• Erdogan, M. Burak
• Li, Xiaochun
• Kirr, Eduard-Wilhelm

• Gabor frames
• dual frames
• windows

Statement dc:rights

• Copyright 2011 Inmi Kim

Language dc:language

en

Handle dc:identifier

http://hdl.handle.net/2142/26039

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/26039