Time-frequency localized functions and operators in Gabor analysis

  • Analyzing a signal in regard to its time-varying frequency content is a classical method in science. Recently this method has grown into a rich mathematical theory within applied harmonic analysis, with applications ranging from radio and mobile communications to medical image processing and geophysics. This thesis addresses the following three fundamental, but not yet fully understood questions. The first one is the construction of Gabor frames with smooth and compactlysupported window functions defined on the Euclidean plane for separable lattices. In addition, we review the crucial applications of the theory of operator algebra representations for proving the general statement of the density theorem for Gabor frames. Second, we study identification of incompletely known linear operators based on the observation of restricted input and output signals. We develop a general setup for identification of general classes of time-frequency localizing operators based on a discretization method. The third question is the uncertainty principle for the joint time-frequency representation of functions on finite Abelian groups. Algebraic properties of such groups lead to results relating the support sizes of functions and their short-time Fourier transforms, with applications in the construction of a class of equal norm tight Gabor frames that are maximally robust to erasures, and consequences in the theory of recovering and storing signals with sparse time-frequency representations.

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Meta data
Publishing Institution:IRC-Library, Information Resource Center der Jacobs University Bremen
Granting Institution:Jacobs Univ.
Author:Peter Rashkov
Referee:Götz Pfander, Peter Oswald, Stephan Dahlke
Advisor:Götz Pfander
Persistent Identifier (URN):urn:nbn:de:101:1-2013052411339
Document Type:PhD Thesis
Language:English
Date of Successful Oral Defense:2010/05/11
Date of First Publication:2010/06/03
PhD Degree:Mathematics
School:SES School of Engineering and Science
Library of Congress Classification:T Technology / TK Electrical engineering. Electronics. Nuclear engineering / TK5101-6720 Telecommunication. Including telegraphy, telephone, radio, radar, television [and positioning technology] / TK5101-5105.8887 Telecommunication / TK5102.9 Signal processing
Call No:Thesis 2010/11

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