M.Sc Thesis

M.Sc StudentSubbanna Nagesh
SubjectNon-Canonical Discrete Gabor Representations and Their
DepartmentDepartment of Electrical and Computer Engineering
Supervisors PROF. Yonina Eldar


In his original scheme of signal representation, Denis Gabor introduced a technique that utilized a set of time-shifted and frequency modulated window functions to represent signals in the combined time-frequency domain.  In this form of signal representation, popularly known as the Gabor expansion of signals, it is the oversampling case that has been a prominent technique for time-frequency representation of signals.  However, there are some problems with the traditional Gabor expansions in the oversampling case.  Utilizing a combination of non-canonical frames and matrix properties, we propose a new technique for expansion of discrete signals in the time-frequency domain, alleviating some of the problems of the traditional methods. 

In this thesis, we focus on discrete Gabor expansions.  We examine the advantages/disadvantages of canonical duals of Gabor frames and extend the concept of non-canonical frames to discrete Gabor expansions.  We illustrate several advantages of non-canonical Gabor frames and show that the canonical dual is merely a special case of the non-canonical dual.  In particular, our more general form of the Gabor frame retains almost all the properties of the canonical frame, besides the frame structure.  Utilizing properties of Gabor frames, we show that the non-canonical dual has several advantages over the canonical dual in the form of a better computational efficiency, greater flexibility and higher stability of the expansion.  We derive necessary and sufficient conditions for existence of the non-canonical dual for both integer  and rational oversampling cases in the signal and the Zak transform domains.  We then extend the concept of non-canonical dual to discrete multiwindow Gabor frames and derive equivalent conditions.


The latter part of the thesis is devoted to studying the utility of  non-canonical multiwindow Gabor functions to index DNA and proteins sequences and manage macromolecule databases.  Here, we solve the problem where the object is to look for a good set of coefficients, given an expansion basis (or frame).  We use the non-canonical multiwindow Gabor functions to extract local periodicities, which are then used as fingerprints for indexing macromolecules.  We also use the multiwindow Gabor coefficients for lossy, but effective compression of sequences.   Based on correlation, we develop a search technique to retrieve similar subsequences from the macromolecular sequences.