Gabor frames with trigonometric spline dual windows

Kim, Inmi

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https://hdl.handle.net/2142/26039 Copy

Description

Title

Gabor frames with trigonometric spline dual windows

Author(s)

Kim, Inmi

Issue Date

2011-08-25T22:10:11Z

Director of Research (if dissertation) or Advisor (if thesis)

Laugesen, Richard S.

Doctoral Committee Chair(s)

Rosenblatt, Joseph

Committee Member(s)

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

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

• Gabor frames
• dual frames
• windows

Abstract

A Gabor system is a collection of modulated and translated copies of a window function. If we have a signal in $L^2(\mathbb{R})$, 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.

Graduation Semester

2011-08

Permalink

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

Copyright and License Information

Copyright 2011 Inmi Kim

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