1 Department of Mathematics, Technical University of Denmark2 Scientific Computing, Department of Informatics and Mathematical Modeling, Technical University of Denmark3 Department of Informatics and Mathematical Modeling, Technical University of Denmark4 Applied functional analysis, Department of Mathematics, Technical University of Denmark5 Department of Electrical Engineering, Technical University of Denmark6 Department of Applied Mathematics and Computer Science, Technical University of Denmark
Gabor analysis is a method for analyzing signals through the use of a set of basic building blocks. The building blocks consists of a certain function (the window) that is shifted in time and frequency. The Gabor expansion of a signal contains information on the behavior of the signal in certain frequency bands at certain times. Gabor theory can be formulated for both functions on the real line and for discrete signals of finite length. The two theories are largely the same because many aspects come from the same underlying theory of locally compact Abelian groups. The two types of Gabor systems can also be related by sampling and periodization. This thesis extends on this theory by showing new results for window construction. It also provides a discussion of the problems associated to discrete Gabor bases. The sampling and periodization connection is handy because it allows Gabor systems on the real line to be well approximated by finite and discrete Gabor frames. This method of approximation is especially attractive because efficient numerical methods exists for doing computations with finite, discrete Gabor systems. This thesis presents new algorithms for the efficient computation of finite, discrete Gabor coefficients. Reconstruction of a signal from its Gabor coefficients is done by the use of a so-called dual window. This thesis presents a number of iterative algorithms to compute dual and self-dual windows. The Linear Time Frequency Toolbox is a Matlab/Octave/C toolbox for doing basic discrete time/frequency and Gabor analysis. It is intended to be both an educational and a computational tool. The toolbox was developed as part of this Ph.D. project to provide a solid foundation for the field of computational Gabor analysis.