## Files in this item

FilesDescriptionFormat

application/pdf

Kim_Inmi.pdf (12Mb)
(no description provided)PDF

## Description

 Title: Gabor frames with trigonometric spline dual windows Author(s): Kim, Inmi Director of Research: Laugesen, Richard S. Doctoral Committee Chair(s): Rosenblatt, Joseph Doctoral Committee Member(s): Laugesen, Richard S.; Erdogan, M. Burak; Li, Xiaochun; Kirr, Eduard-Wilhelm Department / Program: Mathematics Discipline: Mathematics Degree Granting Institution: University of Illinois at Urbana-Champaign Degree: Ph.D. Genre: Dissertation Subject(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. Issue Date: 2011-08-25 URI: http://hdl.handle.net/2142/26039 Rights Information: Copyright 2011 Inmi Kim Date Available in IDEALS: 2011-08-25 Date Deposited: 2011-08
﻿