Constructing dual windows for Gabor frames

This thesis is concerned with computational and theoretical aspects of Gabor analysis and frame theory, in particular, with the study of so-called dual frames of Gabor frames in L2(R). A frame is a system of “simple” functions, or building blocks, that provides ways of analyzing signals. From a frame one obtains convenient series expansions of all L2-signals as “infinite” linear combinations of the building blocks with coefficients found by an associated frame, called a dual frame. A Gabor frame is a frame where the building blocks are regular time and frequency shifts of a single function. The functions generating a Gabor frame and a dual Gabor frame are called a (Gabor) window and a dual (Gabor) window, respectively. The objective of this thesis is the construction of dual windows for a class of Gabor windows. To obtain good time-localization, we will insist that both the Gabor window and the dual Gabor window have compact support. Furthermore, we usually require the construction of the dual window to preserve desirable properties from the the window, e.g., smoothness, symmetry, and boundedness. We are particularly concerned with smoothness of the windows since this controls the decay of the generators in the frequency domain. The classical method of constructing dual windows is by painless non-orthonormal expansions by Daubechies, Grossmann and Meyer. While this construction really is “painless” (easy, that is), it often causes the Gabor system to be more redundant than what one ideally wants. In this thesis we step away from the standard “painless” construction as a way to provide dual Gabor frames with desirable properties, while still maintaining a low redundancy. The two new scenarios we consider are called Gabor frames with 2-overlap and N-overlap, respectively. In the case of 2-overlap, we derive an explicit formula for all dual windows with sufficiently small support. This, in turn, yields a parametrization of all such dual windows by a measurable function defined on the unit interval. We derive easily verifiable conditions on the “parameter” function such that the dual inherits boundedness, symmetry, and smoothness. Furthermore, we establish the general optimally obtain able smoothness of the dual window, and investigate specific cases where smoothness of the dual window can be increased. In the case of N-overlap, we develop a new methodology for the construction of compactly supported dual windows which inherit smoothness. The construction is based on assumptions that are much stronger than in the 2-overlap setting. Moreover, the construction is (naturally) less explicit than in the 2-overlap case. On the other hand, we are still able to parametrize all dual windows with sufficiently small support by measurable functions; this time we use N−1 functions defined on the unit interval. We also derive easily verifiable conditions on the “parameter” functions such that the dual window inherits smoothness. The dual windows are computed using a matrix multiplication formula, and we give an example of the construction of “nice” dual windows for B-splines. To give other researchers better opportunity to investigate and continue to work with our methods we provide the Maple worksheets that were used to create the figures in this thesis. We give a short description of the functionality of the worksheets and provide the full code in appendices.